From cbb47a8053684ab2975ad848270a34bbfb183399 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Sun, 20 Aug 2023 21:16:10 +0200 Subject: [PATCH] update --- doc/pub/week34/html/._week34-bs000.html | 100 +- doc/pub/week34/html/._week34-bs001.html | 102 +- doc/pub/week34/html/._week34-bs002.html | 100 +- doc/pub/week34/html/._week34-bs003.html | 100 +- doc/pub/week34/html/._week34-bs004.html | 100 +- doc/pub/week34/html/._week34-bs005.html | 100 +- doc/pub/week34/html/._week34-bs006.html | 100 +- doc/pub/week34/html/._week34-bs007.html | 100 +- doc/pub/week34/html/._week34-bs008.html | 100 +- doc/pub/week34/html/._week34-bs009.html | 100 +- doc/pub/week34/html/._week34-bs010.html | 100 +- doc/pub/week34/html/._week34-bs011.html | 100 +- doc/pub/week34/html/._week34-bs012.html | 100 +- doc/pub/week34/html/._week34-bs013.html | 100 +- doc/pub/week34/html/._week34-bs014.html | 100 +- doc/pub/week34/html/._week34-bs015.html | 100 +- doc/pub/week34/html/._week34-bs016.html | 104 +- doc/pub/week34/html/._week34-bs017.html | 100 +- doc/pub/week34/html/._week34-bs018.html | 100 +- doc/pub/week34/html/._week34-bs019.html | 100 +- doc/pub/week34/html/._week34-bs020.html | 100 +- doc/pub/week34/html/._week34-bs021.html | 100 +- doc/pub/week34/html/._week34-bs022.html | 100 +- doc/pub/week34/html/._week34-bs023.html | 100 +- doc/pub/week34/html/._week34-bs024.html | 100 +- doc/pub/week34/html/._week34-bs025.html | 100 +- doc/pub/week34/html/._week34-bs026.html | 100 +- doc/pub/week34/html/._week34-bs027.html | 104 +- doc/pub/week34/html/._week34-bs028.html | 105 +- doc/pub/week34/html/._week34-bs029.html | 100 +- doc/pub/week34/html/._week34-bs030.html | 361 ++- doc/pub/week34/html/._week34-bs031.html | 382 ++- doc/pub/week34/html/._week34-bs032.html | 355 ++- doc/pub/week34/html/._week34-bs033.html | 799 +++++- doc/pub/week34/html/._week34-bs034.html | 383 +-- doc/pub/week34/html/._week34-bs035.html | 354 +-- doc/pub/week34/html/._week34-bs036.html | 128 +- doc/pub/week34/html/._week34-bs037.html | 1010 +------- doc/pub/week34/html/._week34-bs038.html | 130 +- doc/pub/week34/html/._week34-bs039.html | 124 +- doc/pub/week34/html/._week34-bs040.html | 149 +- doc/pub/week34/html/._week34-bs041.html | 131 +- doc/pub/week34/html/._week34-bs042.html | 122 +- doc/pub/week34/html/._week34-bs043.html | 119 +- doc/pub/week34/html/._week34-bs044.html | 226 +- doc/pub/week34/html/._week34-bs045.html | 138 +- doc/pub/week34/html/._week34-bs046.html | 151 +- doc/pub/week34/html/._week34-bs047.html | 145 +- doc/pub/week34/html/._week34-bs048.html | 203 +- doc/pub/week34/html/._week34-bs049.html | 128 +- doc/pub/week34/html/._week34-bs050.html | 229 +- doc/pub/week34/html/._week34-bs051.html | 230 +- doc/pub/week34/html/._week34-bs052.html | 134 +- doc/pub/week34/html/._week34-bs053.html | 120 +- doc/pub/week34/html/._week34-bs054.html | 205 +- doc/pub/week34/html/._week34-bs055.html | 227 +- doc/pub/week34/html/._week34-bs056.html | 130 +- doc/pub/week34/html/._week34-bs057.html | 143 +- doc/pub/week34/html/._week34-bs058.html | 139 +- doc/pub/week34/html/._week34-bs059.html | 232 +- doc/pub/week34/html/._week34-bs060.html | 213 +- doc/pub/week34/html/._week34-bs061.html | 343 ++- doc/pub/week34/html/week34-bs.html | 100 +- doc/pub/week34/html/week34-reveal.html | 440 +--- doc/pub/week34/html/week34-solarized.html | 425 +--- doc/pub/week34/html/week34.html | 425 +--- doc/pub/week34/ipynb/ipynb-week34-src.tar.gz | Bin 103516 -> 103516 bytes doc/pub/week34/ipynb/week34.ipynb | 2353 ++++++++++-------- doc/src/week34/week34.do.txt | 9 +- 69 files changed, 6436 insertions(+), 8214 deletions(-) diff --git a/doc/pub/week34/html/._week34-bs000.html b/doc/pub/week34/html/._week34-bs000.html index 9aaab6cf1..19b4a2d84 100644 --- a/doc/pub/week34/html/._week34-bs000.html +++ b/doc/pub/week34/html/._week34-bs000.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
  • Installing R, C++, cython or Julia
  • Installing R, C++, cython, Numba etc
  • Numpy examples and Important Matrix and vector handling packages
    • -
  • Basic Matrix Features
    • -
  •    Some famous Matrices
    • -
  •    More Basic Matrix Features
    • -
  • Numpy and arrays
    • -
  • Matrices in Python
    • -
  • Meet the Pandas
    • -
  • Friday August 27
    • -
  •    Simple linear regression model using scikit-learn
    • -
  •    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
    • -
  •    Organizing our data
    • -
  •    Seeing the wood for the trees
    • -
  •    And what about using neural networks?
    • -
  • A first summary
    • -
  • Why Linear Regression (aka Ordinary Least Squares and family)
    • -
  • Regression analysis, overarching aims
    • -
  • Regression analysis, overarching aims II
    • -
  • Examples
    • -
  • General linear models
    • -
  • Rewriting the fitting procedure as a linear algebra problem
    • -
  • Rewriting the fitting procedure as a linear algebra problem, more details
    • -
  • Generalizing the fitting procedure as a linear algebra problem
    • -
  • Generalizing the fitting procedure as a linear algebra problem
    • -
  • Optimizing our parameters
    • -
  • Our model for the nuclear binding energies
    • -
  • Optimizing our parameters, more details
    • -
  • Interpretations and optimizing our parameters
    • -
  • Interpretations and optimizing our parameters
    • -
  • Some useful matrix and vector expressions
    • -
  • Interpretations and optimizing our parameters
    • -
  • Own code for Ordinary Least Squares
    • -
  • Adding error analysis and training set up
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • Fitting an Equation of State for Dense Nuclear Matter
    • -
  • The code
    • -
  • Splitting our Data in Training and Test data
    • -
  • Exercises
    • -
  • Exercise 1: Setting up various Python environments
    • -
  • Exercise 2: making your own data and exploring scikit-learn
    • -
  • Exercise 3: Normalizing our data
    • +
  • Numpy and arrays
    • +
  • Matrices in Python
    • +
  • Meet the Pandas
    • +
  •    Simple linear regression model using scikit-learn
    • +
  •    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
    • +
  •    Organizing our data
    • +
  •    And what about using neural networks?
    • +
  • A first summary
    • +
  • Why Linear Regression (aka Ordinary Least Squares and family)
    • +
  • Regression analysis, overarching aims
    • +
  • Regression analysis, overarching aims II
    • +
  • Examples
    • +
  • General linear models
    • +
  • Rewriting the fitting procedure as a linear algebra problem
    • +
  • Rewriting the fitting procedure as a linear algebra problem, more details
    • +
  • Generalizing the fitting procedure as a linear algebra problem
    • +
  • Generalizing the fitting procedure as a linear algebra problem
    • +
  • Optimizing our parameters
    • +
  • Our model for the nuclear binding energies
    • +
  • Optimizing our parameters, more details
    • +
  • Interpretations and optimizing our parameters
    • +
  • Interpretations and optimizing our parameters
    • +
  • Some useful matrix and vector expressions
    • +
  • Interpretations and optimizing our parameters
    • +
  • Own code for Ordinary Least Squares
    • +
  • Adding error analysis and training set up
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • Fitting an Equation of State for Dense Nuclear Matter
    • +
  • The code
    • +
  • Splitting our Data in Training and Test data
    • +
  • Exercises
    • +
  • Exercise 1: Setting up various Python environments
    • +
  • Exercise 2: making your own data and exploring scikit-learn
    • +
  • Exercise 3: Split data in test and training data
    • @@ -394,7 +378,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 66
    • +
  • 62
  • »
    • diff --git a/doc/pub/week34/html/._week34-bs001.html b/doc/pub/week34/html/._week34-bs001.html index 6f10f533d..4059ee806 100644 --- a/doc/pub/week34/html/._week34-bs001.html +++ b/doc/pub/week34/html/._week34-bs001.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
  • Installing R, C++, cython or Julia
  • Installing R, C++, cython, Numba etc
  • Numpy examples and Important Matrix and vector handling packages
    • -
  • Basic Matrix Features
    • -
  •    Some famous Matrices
    • -
  •    More Basic Matrix Features
    • -
  • Numpy and arrays
    • -
  • Matrices in Python
    • -
  • Meet the Pandas
    • -
  • Friday August 27
    • -
  •    Simple linear regression model using scikit-learn
    • -
  •    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
    • -
  •    Organizing our data
    • -
  •    Seeing the wood for the trees
    • -
  •    And what about using neural networks?
    • -
  • A first summary
    • -
  • Why Linear Regression (aka Ordinary Least Squares and family)
    • -
  • Regression analysis, overarching aims
    • -
  • Regression analysis, overarching aims II
    • -
  • Examples
    • -
  • General linear models
    • -
  • Rewriting the fitting procedure as a linear algebra problem
    • -
  • Rewriting the fitting procedure as a linear algebra problem, more details
    • -
  • Generalizing the fitting procedure as a linear algebra problem
    • -
  • Generalizing the fitting procedure as a linear algebra problem
    • -
  • Optimizing our parameters
    • -
  • Our model for the nuclear binding energies
    • -
  • Optimizing our parameters, more details
    • -
  • Interpretations and optimizing our parameters
    • -
  • Interpretations and optimizing our parameters
    • -
  • Some useful matrix and vector expressions
    • -
  • Interpretations and optimizing our parameters
    • -
  • Own code for Ordinary Least Squares
    • -
  • Adding error analysis and training set up
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • The \( \chi^2 \) function
    • -
  • Fitting an Equation of State for Dense Nuclear Matter
    • -
  • The code
    • -
  • Splitting our Data in Training and Test data
    • -
  • Exercises
    • -
  • Exercise 1: Setting up various Python environments
    • -
  • Exercise 2: making your own data and exploring scikit-learn
    • -
  • Exercise 3: Normalizing our data
    • +
  • Numpy and arrays
    • +
  • Matrices in Python
    • +
  • Meet the Pandas
    • +
  •    Simple linear regression model using scikit-learn
    • +
  •    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
    • +
  •    Organizing our data
    • +
  •    And what about using neural networks?
    • +
  • A first summary
    • +
  • Why Linear Regression (aka Ordinary Least Squares and family)
    • +
  • Regression analysis, overarching aims
    • +
  • Regression analysis, overarching aims II
    • +
  • Examples
    • +
  • General linear models
    • +
  • Rewriting the fitting procedure as a linear algebra problem
    • +
  • Rewriting the fitting procedure as a linear algebra problem, more details
    • +
  • Generalizing the fitting procedure as a linear algebra problem
    • +
  • Generalizing the fitting procedure as a linear algebra problem
    • +
  • Optimizing our parameters
    • +
  • Our model for the nuclear binding energies
    • +
  • Optimizing our parameters, more details
    • +
  • Interpretations and optimizing our parameters
    • +
  • Interpretations and optimizing our parameters
    • +
  • Some useful matrix and vector expressions
    • +
  • Interpretations and optimizing our parameters
    • +
  • Own code for Ordinary Least Squares
    • +
  • Adding error analysis and training set up
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • The \( \chi^2 \) function
    • +
  • Fitting an Equation of State for Dense Nuclear Matter
    • +
  • The code
    • +
  • Splitting our Data in Training and Test data
    • +
  • Exercises
    • +
  • Exercise 1: Setting up various Python environments
    • +
  • Exercise 2: making your own data and exploring scikit-learn
    • +
  • Exercise 3: Split data in test and training data
    • @@ -355,7 +339,7 @@ MathJax.Hub.Config({
    1. The sessions on Tuesdays and Wednesdays last four hours for each group (four groups in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects.
      2. -
    2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week.
      3. +
    3. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week. Videos and teaching material will be announced in due time.
    4. There are four groups:
      5. diff --git a/doc/pub/week34/html/._week34-bs002.html b/doc/pub/week34/html/._week34-bs002.html index b3b507791..b3f6f153d 100644 --- a/doc/pub/week34/html/._week34-bs002.html +++ b/doc/pub/week34/html/._week34-bs002.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5. Installing R, C++, cython or Julia
    6. Installing R, C++, cython, Numba etc
    7. Numpy examples and Important Matrix and vector handling packages
      8. -
    8. Basic Matrix Features
      9. -
    9.    Some famous Matrices
      10. -
    10.    More Basic Matrix Features
      11. -
    11. Numpy and arrays
      12. -
    12. Matrices in Python
      13. -
    13. Meet the Pandas
      14. -
    14. Friday August 27
      15. -
    15.    Simple linear regression model using scikit-learn
      16. -
    16.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      17. -
    17.    Organizing our data
      18. -
    18.    Seeing the wood for the trees
      19. -
    19.    And what about using neural networks?
      20. -
    20. A first summary
      21. -
    21. Why Linear Regression (aka Ordinary Least Squares and family)
      22. -
    22. Regression analysis, overarching aims
      23. -
    23. Regression analysis, overarching aims II
      24. -
    24. Examples
      25. -
    25. General linear models
      26. -
    26. Rewriting the fitting procedure as a linear algebra problem
      27. -
    27. Rewriting the fitting procedure as a linear algebra problem, more details
      28. -
    28. Generalizing the fitting procedure as a linear algebra problem
      29. -
    29. Generalizing the fitting procedure as a linear algebra problem
      30. -
    30. Optimizing our parameters
      31. -
    31. Our model for the nuclear binding energies
      32. -
    32. Optimizing our parameters, more details
      33. -
    33. Interpretations and optimizing our parameters
      34. -
    34. Interpretations and optimizing our parameters
      35. -
    35. Some useful matrix and vector expressions
      36. -
    36. Interpretations and optimizing our parameters
      37. -
    37. Own code for Ordinary Least Squares
      38. -
    38. Adding error analysis and training set up
      39. -
    39. The \( \chi^2 \) function
      40. -
    40. The \( \chi^2 \) function
      41. -
    41. The \( \chi^2 \) function
      42. -
    42. The \( \chi^2 \) function
      43. -
    43. The \( \chi^2 \) function
      44. -
    44. The \( \chi^2 \) function
      45. -
    45. Fitting an Equation of State for Dense Nuclear Matter
      46. -
    46. The code
      47. -
    47. Splitting our Data in Training and Test data
      48. -
    48. Exercises
      49. -
    49. Exercise 1: Setting up various Python environments
      50. -
    50. Exercise 2: making your own data and exploring scikit-learn
      51. -
    51. Exercise 3: Normalizing our data
      52. +
    52. Numpy and arrays
      53. +
    53. Matrices in Python
      54. +
    54. Meet the Pandas
      55. +
    55.    Simple linear regression model using scikit-learn
      56. +
    56.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      57. +
    57.    Organizing our data
      58. +
    58.    And what about using neural networks?
      59. +
    59. A first summary
      60. +
    60. Why Linear Regression (aka Ordinary Least Squares and family)
      61. +
    61. Regression analysis, overarching aims
      62. +
    62. Regression analysis, overarching aims II
      63. +
    63. Examples
      64. +
    64. General linear models
      65. +
    65. Rewriting the fitting procedure as a linear algebra problem
      66. +
    66. Rewriting the fitting procedure as a linear algebra problem, more details
      67. +
    67. Generalizing the fitting procedure as a linear algebra problem
      68. +
    68. Generalizing the fitting procedure as a linear algebra problem
      69. +
    69. Optimizing our parameters
      70. +
    70. Our model for the nuclear binding energies
      71. +
    71. Optimizing our parameters, more details
      72. +
    72. Interpretations and optimizing our parameters
      73. +
    73. Interpretations and optimizing our parameters
      74. +
    74. Some useful matrix and vector expressions
      75. +
    75. Interpretations and optimizing our parameters
      76. +
    76. Own code for Ordinary Least Squares
      77. +
    77. Adding error analysis and training set up
      78. +
    78. The \( \chi^2 \) function
      79. +
    79. The \( \chi^2 \) function
      80. +
    80. The \( \chi^2 \) function
      81. +
    81. The \( \chi^2 \) function
      82. +
    82. The \( \chi^2 \) function
      83. +
    83. The \( \chi^2 \) function
      84. +
    84. Fitting an Equation of State for Dense Nuclear Matter
      85. +
    85. The code
      86. +
    86. Splitting our Data in Training and Test data
      87. +
    87. Exercises
      88. +
    88. Exercise 1: Setting up various Python environments
      89. +
    89. Exercise 2: making your own data and exploring scikit-learn
      90. +
    90. Exercise 3: Split data in test and training data
      91. @@ -382,7 +366,7 @@ MathJax.Hub.Config({
    91. 11
    92. 12
    93. ...
      94. -
    94. 66
      95. +
    95. 62
    96. »
      97. diff --git a/doc/pub/week34/html/._week34-bs003.html b/doc/pub/week34/html/._week34-bs003.html index a5ea46b15..2c7bc9acd 100644 --- a/doc/pub/week34/html/._week34-bs003.html +++ b/doc/pub/week34/html/._week34-bs003.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    97. Installing R, C++, cython or Julia
    98. Installing R, C++, cython, Numba etc
    99. Numpy examples and Important Matrix and vector handling packages
      100. -
    100. Basic Matrix Features
      101. -
    101.    Some famous Matrices
      102. -
    102.    More Basic Matrix Features
      103. -
    103. Numpy and arrays
      104. -
    104. Matrices in Python
      105. -
    105. Meet the Pandas
      106. -
    106. Friday August 27
      107. -
    107.    Simple linear regression model using scikit-learn
      108. -
    108.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      109. -
    109.    Organizing our data
      110. -
    110.    Seeing the wood for the trees
      111. -
    111.    And what about using neural networks?
      112. -
    112. A first summary
      113. -
    113. Why Linear Regression (aka Ordinary Least Squares and family)
      114. -
    114. Regression analysis, overarching aims
      115. -
    115. Regression analysis, overarching aims II
      116. -
    116. Examples
      117. -
    117. General linear models
      118. -
    118. Rewriting the fitting procedure as a linear algebra problem
      119. -
    119. Rewriting the fitting procedure as a linear algebra problem, more details
      120. -
    120. Generalizing the fitting procedure as a linear algebra problem
      121. -
    121. Generalizing the fitting procedure as a linear algebra problem
      122. -
    122. Optimizing our parameters
      123. -
    123. Our model for the nuclear binding energies
      124. -
    124. Optimizing our parameters, more details
      125. -
    125. Interpretations and optimizing our parameters
      126. -
    126. Interpretations and optimizing our parameters
      127. -
    127. Some useful matrix and vector expressions
      128. -
    128. Interpretations and optimizing our parameters
      129. -
    129. Own code for Ordinary Least Squares
      130. -
    130. Adding error analysis and training set up
      131. -
    131. The \( \chi^2 \) function
      132. -
    132. The \( \chi^2 \) function
      133. -
    133. The \( \chi^2 \) function
      134. -
    134. The \( \chi^2 \) function
      135. -
    135. The \( \chi^2 \) function
      136. -
    136. The \( \chi^2 \) function
      137. -
    137. Fitting an Equation of State for Dense Nuclear Matter
      138. -
    138. The code
      139. -
    139. Splitting our Data in Training and Test data
      140. -
    140. Exercises
      141. -
    141. Exercise 1: Setting up various Python environments
      142. -
    142. Exercise 2: making your own data and exploring scikit-learn
      143. -
    143. Exercise 3: Normalizing our data
      144. +
    144. Numpy and arrays
      145. +
    145. Matrices in Python
      146. +
    146. Meet the Pandas
      147. +
    147.    Simple linear regression model using scikit-learn
      148. +
    148.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      149. +
    149.    Organizing our data
      150. +
    150.    And what about using neural networks?
      151. +
    151. A first summary
      152. +
    152. Why Linear Regression (aka Ordinary Least Squares and family)
      153. +
    153. Regression analysis, overarching aims
      154. +
    154. Regression analysis, overarching aims II
      155. +
    155. Examples
      156. +
    156. General linear models
      157. +
    157. Rewriting the fitting procedure as a linear algebra problem
      158. +
    158. Rewriting the fitting procedure as a linear algebra problem, more details
      159. +
    159. Generalizing the fitting procedure as a linear algebra problem
      160. +
    160. Generalizing the fitting procedure as a linear algebra problem
      161. +
    161. Optimizing our parameters
      162. +
    162. Our model for the nuclear binding energies
      163. +
    163. Optimizing our parameters, more details
      164. +
    164. Interpretations and optimizing our parameters
      165. +
    165. Interpretations and optimizing our parameters
      166. +
    166. Some useful matrix and vector expressions
      167. +
    167. Interpretations and optimizing our parameters
      168. +
    168. Own code for Ordinary Least Squares
      169. +
    169. Adding error analysis and training set up
      170. +
    170. The \( \chi^2 \) function
      171. +
    171. The \( \chi^2 \) function
      172. +
    172. The \( \chi^2 \) function
      173. +
    173. The \( \chi^2 \) function
      174. +
    174. The \( \chi^2 \) function
      175. +
    175. The \( \chi^2 \) function
      176. +
    176. Fitting an Equation of State for Dense Nuclear Matter
      177. +
    177. The code
      178. +
    178. Splitting our Data in Training and Test data
      179. +
    179. Exercises
      180. +
    180. Exercise 1: Setting up various Python environments
      181. +
    181. Exercise 2: making your own data and exploring scikit-learn
      182. +
    182. Exercise 3: Split data in test and training data
      183. @@ -387,7 +371,7 @@ MathJax.Hub.Config({
    183. 12
    184. 13
    185. ...
      186. -
    186. 66
      187. +
    187. 62
    188. »
      189. diff --git a/doc/pub/week34/html/._week34-bs004.html b/doc/pub/week34/html/._week34-bs004.html index 8e35c8c24..3ce4433bd 100644 --- a/doc/pub/week34/html/._week34-bs004.html +++ b/doc/pub/week34/html/._week34-bs004.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    189. Installing R, C++, cython or Julia
    190. Installing R, C++, cython, Numba etc
    191. Numpy examples and Important Matrix and vector handling packages
      192. -
    192. Basic Matrix Features
      193. -
    193.    Some famous Matrices
      194. -
    194.    More Basic Matrix Features
      195. -
    195. Numpy and arrays
      196. -
    196. Matrices in Python
      197. -
    197. Meet the Pandas
      198. -
    198. Friday August 27
      199. -
    199.    Simple linear regression model using scikit-learn
      200. -
    200.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      201. -
    201.    Organizing our data
      202. -
    202.    Seeing the wood for the trees
      203. -
    203.    And what about using neural networks?
      204. -
    204. A first summary
      205. -
    205. Why Linear Regression (aka Ordinary Least Squares and family)
      206. -
    206. Regression analysis, overarching aims
      207. -
    207. Regression analysis, overarching aims II
      208. -
    208. Examples
      209. -
    209. General linear models
      210. -
    210. Rewriting the fitting procedure as a linear algebra problem
      211. -
    211. Rewriting the fitting procedure as a linear algebra problem, more details
      212. -
    212. Generalizing the fitting procedure as a linear algebra problem
      213. -
    213. Generalizing the fitting procedure as a linear algebra problem
      214. -
    214. Optimizing our parameters
      215. -
    215. Our model for the nuclear binding energies
      216. -
    216. Optimizing our parameters, more details
      217. -
    217. Interpretations and optimizing our parameters
      218. -
    218. Interpretations and optimizing our parameters
      219. -
    219. Some useful matrix and vector expressions
      220. -
    220. Interpretations and optimizing our parameters
      221. -
    221. Own code for Ordinary Least Squares
      222. -
    222. Adding error analysis and training set up
      223. -
    223. The \( \chi^2 \) function
      224. -
    224. The \( \chi^2 \) function
      225. -
    225. The \( \chi^2 \) function
      226. -
    226. The \( \chi^2 \) function
      227. -
    227. The \( \chi^2 \) function
      228. -
    228. The \( \chi^2 \) function
      229. -
    229. Fitting an Equation of State for Dense Nuclear Matter
      230. -
    230. The code
      231. -
    231. Splitting our Data in Training and Test data
      232. -
    232. Exercises
      233. -
    233. Exercise 1: Setting up various Python environments
      234. -
    234. Exercise 2: making your own data and exploring scikit-learn
      235. -
    235. Exercise 3: Normalizing our data
      236. +
    236. Numpy and arrays
      237. +
    237. Matrices in Python
      238. +
    238. Meet the Pandas
      239. +
    239.    Simple linear regression model using scikit-learn
      240. +
    240.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      241. +
    241.    Organizing our data
      242. +
    242.    And what about using neural networks?
      243. +
    243. A first summary
      244. +
    244. Why Linear Regression (aka Ordinary Least Squares and family)
      245. +
    245. Regression analysis, overarching aims
      246. +
    246. Regression analysis, overarching aims II
      247. +
    247. Examples
      248. +
    248. General linear models
      249. +
    249. Rewriting the fitting procedure as a linear algebra problem
      250. +
    250. Rewriting the fitting procedure as a linear algebra problem, more details
      251. +
    251. Generalizing the fitting procedure as a linear algebra problem
      252. +
    252. Generalizing the fitting procedure as a linear algebra problem
      253. +
    253. Optimizing our parameters
      254. +
    254. Our model for the nuclear binding energies
      255. +
    255. Optimizing our parameters, more details
      256. +
    256. Interpretations and optimizing our parameters
      257. +
    257. Interpretations and optimizing our parameters
      258. +
    258. Some useful matrix and vector expressions
      259. +
    259. Interpretations and optimizing our parameters
      260. +
    260. Own code for Ordinary Least Squares
      261. +
    261. Adding error analysis and training set up
      262. +
    262. The \( \chi^2 \) function
      263. +
    263. The \( \chi^2 \) function
      264. +
    264. The \( \chi^2 \) function
      265. +
    265. The \( \chi^2 \) function
      266. +
    266. The \( \chi^2 \) function
      267. +
    267. The \( \chi^2 \) function
      268. +
    268. Fitting an Equation of State for Dense Nuclear Matter
      269. +
    269. The code
      270. +
    270. Splitting our Data in Training and Test data
      271. +
    271. Exercises
      272. +
    272. Exercise 1: Setting up various Python environments
      273. +
    273. Exercise 2: making your own data and exploring scikit-learn
      274. +
    274. Exercise 3: Split data in test and training data
      275. @@ -388,7 +372,7 @@ MathJax.Hub.Config({
    275. 13
    276. 14
    277. ...
      278. -
    278. 66
      279. +
    279. 62
    280. »
      281. diff --git a/doc/pub/week34/html/._week34-bs005.html b/doc/pub/week34/html/._week34-bs005.html index e368bfb65..dead6245d 100644 --- a/doc/pub/week34/html/._week34-bs005.html +++ b/doc/pub/week34/html/._week34-bs005.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    281. Installing R, C++, cython or Julia
    282. Installing R, C++, cython, Numba etc
    283. Numpy examples and Important Matrix and vector handling packages
      284. -
    284. Basic Matrix Features
      285. -
    285.    Some famous Matrices
      286. -
    286.    More Basic Matrix Features
      287. -
    287. Numpy and arrays
      288. -
    288. Matrices in Python
      289. -
    289. Meet the Pandas
      290. -
    290. Friday August 27
      291. -
    291.    Simple linear regression model using scikit-learn
      292. -
    292.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      293. -
    293.    Organizing our data
      294. -
    294.    Seeing the wood for the trees
      295. -
    295.    And what about using neural networks?
      296. -
    296. A first summary
      297. -
    297. Why Linear Regression (aka Ordinary Least Squares and family)
      298. -
    298. Regression analysis, overarching aims
      299. -
    299. Regression analysis, overarching aims II
      300. -
    300. Examples
      301. -
    301. General linear models
      302. -
    302. Rewriting the fitting procedure as a linear algebra problem
      303. -
    303. Rewriting the fitting procedure as a linear algebra problem, more details
      304. -
    304. Generalizing the fitting procedure as a linear algebra problem
      305. -
    305. Generalizing the fitting procedure as a linear algebra problem
      306. -
    306. Optimizing our parameters
      307. -
    307. Our model for the nuclear binding energies
      308. -
    308. Optimizing our parameters, more details
      309. -
    309. Interpretations and optimizing our parameters
      310. -
    310. Interpretations and optimizing our parameters
      311. -
    311. Some useful matrix and vector expressions
      312. -
    312. Interpretations and optimizing our parameters
      313. -
    313. Own code for Ordinary Least Squares
      314. -
    314. Adding error analysis and training set up
      315. -
    315. The \( \chi^2 \) function
      316. -
    316. The \( \chi^2 \) function
      317. -
    317. The \( \chi^2 \) function
      318. -
    318. The \( \chi^2 \) function
      319. -
    319. The \( \chi^2 \) function
      320. -
    320. The \( \chi^2 \) function
      321. -
    321. Fitting an Equation of State for Dense Nuclear Matter
      322. -
    322. The code
      323. -
    323. Splitting our Data in Training and Test data
      324. -
    324. Exercises
      325. -
    325. Exercise 1: Setting up various Python environments
      326. -
    326. Exercise 2: making your own data and exploring scikit-learn
      327. -
    327. Exercise 3: Normalizing our data
      328. +
    328. Numpy and arrays
      329. +
    329. Matrices in Python
      330. +
    330. Meet the Pandas
      331. +
    331.    Simple linear regression model using scikit-learn
      332. +
    332.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      333. +
    333.    Organizing our data
      334. +
    334.    And what about using neural networks?
      335. +
    335. A first summary
      336. +
    336. Why Linear Regression (aka Ordinary Least Squares and family)
      337. +
    337. Regression analysis, overarching aims
      338. +
    338. Regression analysis, overarching aims II
      339. +
    339. Examples
      340. +
    340. General linear models
      341. +
    341. Rewriting the fitting procedure as a linear algebra problem
      342. +
    342. Rewriting the fitting procedure as a linear algebra problem, more details
      343. +
    343. Generalizing the fitting procedure as a linear algebra problem
      344. +
    344. Generalizing the fitting procedure as a linear algebra problem
      345. +
    345. Optimizing our parameters
      346. +
    346. Our model for the nuclear binding energies
      347. +
    347. Optimizing our parameters, more details
      348. +
    348. Interpretations and optimizing our parameters
      349. +
    349. Interpretations and optimizing our parameters
      350. +
    350. Some useful matrix and vector expressions
      351. +
    351. Interpretations and optimizing our parameters
      352. +
    352. Own code for Ordinary Least Squares
      353. +
    353. Adding error analysis and training set up
      354. +
    354. The \( \chi^2 \) function
      355. +
    355. The \( \chi^2 \) function
      356. +
    356. The \( \chi^2 \) function
      357. +
    357. The \( \chi^2 \) function
      358. +
    358. The \( \chi^2 \) function
      359. +
    359. The \( \chi^2 \) function
      360. +
    360. Fitting an Equation of State for Dense Nuclear Matter
      361. +
    361. The code
      362. +
    362. Splitting our Data in Training and Test data
      363. +
    363. Exercises
      364. +
    364. Exercise 1: Setting up various Python environments
      365. +
    365. Exercise 2: making your own data and exploring scikit-learn
      366. +
    366. Exercise 3: Split data in test and training data
      367. @@ -377,7 +361,7 @@ MathJax.Hub.Config({
    367. 14
    368. 15
    369. ...
      370. -
    370. 66
      371. +
    371. 62
    372. »
      373. diff --git a/doc/pub/week34/html/._week34-bs006.html b/doc/pub/week34/html/._week34-bs006.html index 6424e0c56..e017e4ba6 100644 --- a/doc/pub/week34/html/._week34-bs006.html +++ b/doc/pub/week34/html/._week34-bs006.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    373. Installing R, C++, cython or Julia
    374. Installing R, C++, cython, Numba etc
    375. Numpy examples and Important Matrix and vector handling packages
      376. -
    376. Basic Matrix Features
      377. -
    377.    Some famous Matrices
      378. -
    378.    More Basic Matrix Features
      379. -
    379. Numpy and arrays
      380. -
    380. Matrices in Python
      381. -
    381. Meet the Pandas
      382. -
    382. Friday August 27
      383. -
    383.    Simple linear regression model using scikit-learn
      384. -
    384.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      385. -
    385.    Organizing our data
      386. -
    386.    Seeing the wood for the trees
      387. -
    387.    And what about using neural networks?
      388. -
    388. A first summary
      389. -
    389. Why Linear Regression (aka Ordinary Least Squares and family)
      390. -
    390. Regression analysis, overarching aims
      391. -
    391. Regression analysis, overarching aims II
      392. -
    392. Examples
      393. -
    393. General linear models
      394. -
    394. Rewriting the fitting procedure as a linear algebra problem
      395. -
    395. Rewriting the fitting procedure as a linear algebra problem, more details
      396. -
    396. Generalizing the fitting procedure as a linear algebra problem
      397. -
    397. Generalizing the fitting procedure as a linear algebra problem
      398. -
    398. Optimizing our parameters
      399. -
    399. Our model for the nuclear binding energies
      400. -
    400. Optimizing our parameters, more details
      401. -
    401. Interpretations and optimizing our parameters
      402. -
    402. Interpretations and optimizing our parameters
      403. -
    403. Some useful matrix and vector expressions
      404. -
    404. Interpretations and optimizing our parameters
      405. -
    405. Own code for Ordinary Least Squares
      406. -
    406. Adding error analysis and training set up
      407. -
    407. The \( \chi^2 \) function
      408. -
    408. The \( \chi^2 \) function
      409. -
    409. The \( \chi^2 \) function
      410. -
    410. The \( \chi^2 \) function
      411. -
    411. The \( \chi^2 \) function
      412. -
    412. The \( \chi^2 \) function
      413. -
    413. Fitting an Equation of State for Dense Nuclear Matter
      414. -
    414. The code
      415. -
    415. Splitting our Data in Training and Test data
      416. -
    416. Exercises
      417. -
    417. Exercise 1: Setting up various Python environments
      418. -
    418. Exercise 2: making your own data and exploring scikit-learn
      419. -
    419. Exercise 3: Normalizing our data
      420. +
    420. Numpy and arrays
      421. +
    421. Matrices in Python
      422. +
    422. Meet the Pandas
      423. +
    423.    Simple linear regression model using scikit-learn
      424. +
    424.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      425. +
    425.    Organizing our data
      426. +
    426.    And what about using neural networks?
      427. +
    427. A first summary
      428. +
    428. Why Linear Regression (aka Ordinary Least Squares and family)
      429. +
    429. Regression analysis, overarching aims
      430. +
    430. Regression analysis, overarching aims II
      431. +
    431. Examples
      432. +
    432. General linear models
      433. +
    433. Rewriting the fitting procedure as a linear algebra problem
      434. +
    434. Rewriting the fitting procedure as a linear algebra problem, more details
      435. +
    435. Generalizing the fitting procedure as a linear algebra problem
      436. +
    436. Generalizing the fitting procedure as a linear algebra problem
      437. +
    437. Optimizing our parameters
      438. +
    438. Our model for the nuclear binding energies
      439. +
    439. Optimizing our parameters, more details
      440. +
    440. Interpretations and optimizing our parameters
      441. +
    441. Interpretations and optimizing our parameters
      442. +
    442. Some useful matrix and vector expressions
      443. +
    443. Interpretations and optimizing our parameters
      444. +
    444. Own code for Ordinary Least Squares
      445. +
    445. Adding error analysis and training set up
      446. +
    446. The \( \chi^2 \) function
      447. +
    447. The \( \chi^2 \) function
      448. +
    448. The \( \chi^2 \) function
      449. +
    449. The \( \chi^2 \) function
      450. +
    450. The \( \chi^2 \) function
      451. +
    451. The \( \chi^2 \) function
      452. +
    452. Fitting an Equation of State for Dense Nuclear Matter
      453. +
    453. The code
      454. +
    454. Splitting our Data in Training and Test data
      455. +
    455. Exercises
      456. +
    456. Exercise 1: Setting up various Python environments
      457. +
    457. Exercise 2: making your own data and exploring scikit-learn
      458. +
    458. Exercise 3: Split data in test and training data
      459. @@ -392,7 +376,7 @@ MathJax.Hub.Config({
    459. 15
    460. 16
    461. ...
      462. -
    462. 66
      463. +
    463. 62
    464. »
      465. diff --git a/doc/pub/week34/html/._week34-bs007.html b/doc/pub/week34/html/._week34-bs007.html index f7b15e20b..480c13442 100644 --- a/doc/pub/week34/html/._week34-bs007.html +++ b/doc/pub/week34/html/._week34-bs007.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    465. Installing R, C++, cython or Julia
    466. Installing R, C++, cython, Numba etc
    467. Numpy examples and Important Matrix and vector handling packages
      468. -
    468. Basic Matrix Features
      469. -
    469.    Some famous Matrices
      470. -
    470.    More Basic Matrix Features
      471. -
    471. Numpy and arrays
      472. -
    472. Matrices in Python
      473. -
    473. Meet the Pandas
      474. -
    474. Friday August 27
      475. -
    475.    Simple linear regression model using scikit-learn
      476. -
    476.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      477. -
    477.    Organizing our data
      478. -
    478.    Seeing the wood for the trees
      479. -
    479.    And what about using neural networks?
      480. -
    480. A first summary
      481. -
    481. Why Linear Regression (aka Ordinary Least Squares and family)
      482. -
    482. Regression analysis, overarching aims
      483. -
    483. Regression analysis, overarching aims II
      484. -
    484. Examples
      485. -
    485. General linear models
      486. -
    486. Rewriting the fitting procedure as a linear algebra problem
      487. -
    487. Rewriting the fitting procedure as a linear algebra problem, more details
      488. -
    488. Generalizing the fitting procedure as a linear algebra problem
      489. -
    489. Generalizing the fitting procedure as a linear algebra problem
      490. -
    490. Optimizing our parameters
      491. -
    491. Our model for the nuclear binding energies
      492. -
    492. Optimizing our parameters, more details
      493. -
    493. Interpretations and optimizing our parameters
      494. -
    494. Interpretations and optimizing our parameters
      495. -
    495. Some useful matrix and vector expressions
      496. -
    496. Interpretations and optimizing our parameters
      497. -
    497. Own code for Ordinary Least Squares
      498. -
    498. Adding error analysis and training set up
      499. -
    499. The \( \chi^2 \) function
      500. -
    500. The \( \chi^2 \) function
      501. -
    501. The \( \chi^2 \) function
      502. -
    502. The \( \chi^2 \) function
      503. -
    503. The \( \chi^2 \) function
      504. -
    504. The \( \chi^2 \) function
      505. -
    505. Fitting an Equation of State for Dense Nuclear Matter
      506. -
    506. The code
      507. -
    507. Splitting our Data in Training and Test data
      508. -
    508. Exercises
      509. -
    509. Exercise 1: Setting up various Python environments
      510. -
    510. Exercise 2: making your own data and exploring scikit-learn
      511. -
    511. Exercise 3: Normalizing our data
      512. +
    512. Numpy and arrays
      513. +
    513. Matrices in Python
      514. +
    514. Meet the Pandas
      515. +
    515.    Simple linear regression model using scikit-learn
      516. +
    516.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      517. +
    517.    Organizing our data
      518. +
    518.    And what about using neural networks?
      519. +
    519. A first summary
      520. +
    520. Why Linear Regression (aka Ordinary Least Squares and family)
      521. +
    521. Regression analysis, overarching aims
      522. +
    522. Regression analysis, overarching aims II
      523. +
    523. Examples
      524. +
    524. General linear models
      525. +
    525. Rewriting the fitting procedure as a linear algebra problem
      526. +
    526. Rewriting the fitting procedure as a linear algebra problem, more details
      527. +
    527. Generalizing the fitting procedure as a linear algebra problem
      528. +
    528. Generalizing the fitting procedure as a linear algebra problem
      529. +
    529. Optimizing our parameters
      530. +
    530. Our model for the nuclear binding energies
      531. +
    531. Optimizing our parameters, more details
      532. +
    532. Interpretations and optimizing our parameters
      533. +
    533. Interpretations and optimizing our parameters
      534. +
    534. Some useful matrix and vector expressions
      535. +
    535. Interpretations and optimizing our parameters
      536. +
    536. Own code for Ordinary Least Squares
      537. +
    537. Adding error analysis and training set up
      538. +
    538. The \( \chi^2 \) function
      539. +
    539. The \( \chi^2 \) function
      540. +
    540. The \( \chi^2 \) function
      541. +
    541. The \( \chi^2 \) function
      542. +
    542. The \( \chi^2 \) function
      543. +
    543. The \( \chi^2 \) function
      544. +
    544. Fitting an Equation of State for Dense Nuclear Matter
      545. +
    545. The code
      546. +
    546. Splitting our Data in Training and Test data
      547. +
    547. Exercises
      548. +
    548. Exercise 1: Setting up various Python environments
      549. +
    549. Exercise 2: making your own data and exploring scikit-learn
      550. +
    550. Exercise 3: Split data in test and training data
      551. @@ -398,7 +382,7 @@ MathJax.Hub.Config({
    551. 16
    552. 17
    553. ...
      554. -
    554. 66
      555. +
    555. 62
    556. »
      557. diff --git a/doc/pub/week34/html/._week34-bs008.html b/doc/pub/week34/html/._week34-bs008.html index 226078733..837446f48 100644 --- a/doc/pub/week34/html/._week34-bs008.html +++ b/doc/pub/week34/html/._week34-bs008.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    557. Installing R, C++, cython or Julia
    558. Installing R, C++, cython, Numba etc
    559. Numpy examples and Important Matrix and vector handling packages
      560. -
    560. Basic Matrix Features
      561. -
    561.    Some famous Matrices
      562. -
    562.    More Basic Matrix Features
      563. -
    563. Numpy and arrays
      564. -
    564. Matrices in Python
      565. -
    565. Meet the Pandas
      566. -
    566. Friday August 27
      567. -
    567.    Simple linear regression model using scikit-learn
      568. -
    568.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      569. -
    569.    Organizing our data
      570. -
    570.    Seeing the wood for the trees
      571. -
    571.    And what about using neural networks?
      572. -
    572. A first summary
      573. -
    573. Why Linear Regression (aka Ordinary Least Squares and family)
      574. -
    574. Regression analysis, overarching aims
      575. -
    575. Regression analysis, overarching aims II
      576. -
    576. Examples
      577. -
    577. General linear models
      578. -
    578. Rewriting the fitting procedure as a linear algebra problem
      579. -
    579. Rewriting the fitting procedure as a linear algebra problem, more details
      580. -
    580. Generalizing the fitting procedure as a linear algebra problem
      581. -
    581. Generalizing the fitting procedure as a linear algebra problem
      582. -
    582. Optimizing our parameters
      583. -
    583. Our model for the nuclear binding energies
      584. -
    584. Optimizing our parameters, more details
      585. -
    585. Interpretations and optimizing our parameters
      586. -
    586. Interpretations and optimizing our parameters
      587. -
    587. Some useful matrix and vector expressions
      588. -
    588. Interpretations and optimizing our parameters
      589. -
    589. Own code for Ordinary Least Squares
      590. -
    590. Adding error analysis and training set up
      591. -
    591. The \( \chi^2 \) function
      592. -
    592. The \( \chi^2 \) function
      593. -
    593. The \( \chi^2 \) function
      594. -
    594. The \( \chi^2 \) function
      595. -
    595. The \( \chi^2 \) function
      596. -
    596. The \( \chi^2 \) function
      597. -
    597. Fitting an Equation of State for Dense Nuclear Matter
      598. -
    598. The code
      599. -
    599. Splitting our Data in Training and Test data
      600. -
    600. Exercises
      601. -
    601. Exercise 1: Setting up various Python environments
      602. -
    602. Exercise 2: making your own data and exploring scikit-learn
      603. -
    603. Exercise 3: Normalizing our data
      604. +
    604. Numpy and arrays
      605. +
    605. Matrices in Python
      606. +
    606. Meet the Pandas
      607. +
    607.    Simple linear regression model using scikit-learn
      608. +
    608.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      609. +
    609.    Organizing our data
      610. +
    610.    And what about using neural networks?
      611. +
    611. A first summary
      612. +
    612. Why Linear Regression (aka Ordinary Least Squares and family)
      613. +
    613. Regression analysis, overarching aims
      614. +
    614. Regression analysis, overarching aims II
      615. +
    615. Examples
      616. +
    616. General linear models
      617. +
    617. Rewriting the fitting procedure as a linear algebra problem
      618. +
    618. Rewriting the fitting procedure as a linear algebra problem, more details
      619. +
    619. Generalizing the fitting procedure as a linear algebra problem
      620. +
    620. Generalizing the fitting procedure as a linear algebra problem
      621. +
    621. Optimizing our parameters
      622. +
    622. Our model for the nuclear binding energies
      623. +
    623. Optimizing our parameters, more details
      624. +
    624. Interpretations and optimizing our parameters
      625. +
    625. Interpretations and optimizing our parameters
      626. +
    626. Some useful matrix and vector expressions
      627. +
    627. Interpretations and optimizing our parameters
      628. +
    628. Own code for Ordinary Least Squares
      629. +
    629. Adding error analysis and training set up
      630. +
    630. The \( \chi^2 \) function
      631. +
    631. The \( \chi^2 \) function
      632. +
    632. The \( \chi^2 \) function
      633. +
    633. The \( \chi^2 \) function
      634. +
    634. The \( \chi^2 \) function
      635. +
    635. The \( \chi^2 \) function
      636. +
    636. Fitting an Equation of State for Dense Nuclear Matter
      637. +
    637. The code
      638. +
    638. Splitting our Data in Training and Test data
      639. +
    639. Exercises
      640. +
    640. Exercise 1: Setting up various Python environments
      641. +
    641. Exercise 2: making your own data and exploring scikit-learn
      642. +
    642. Exercise 3: Split data in test and training data
      643. @@ -390,7 +374,7 @@ MathJax.Hub.Config({
    643. 17
    644. 18
    645. ...
      646. -
    646. 66
      647. +
    647. 62
    648. »
      649. diff --git a/doc/pub/week34/html/._week34-bs009.html b/doc/pub/week34/html/._week34-bs009.html index 43f68979b..44a93e1f9 100644 --- a/doc/pub/week34/html/._week34-bs009.html +++ b/doc/pub/week34/html/._week34-bs009.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    649. Installing R, C++, cython or Julia
    650. Installing R, C++, cython, Numba etc
    651. Numpy examples and Important Matrix and vector handling packages
      652. -
    652. Basic Matrix Features
      653. -
    653.    Some famous Matrices
      654. -
    654.    More Basic Matrix Features
      655. -
    655. Numpy and arrays
      656. -
    656. Matrices in Python
      657. -
    657. Meet the Pandas
      658. -
    658. Friday August 27
      659. -
    659.    Simple linear regression model using scikit-learn
      660. -
    660.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      661. -
    661.    Organizing our data
      662. -
    662.    Seeing the wood for the trees
      663. -
    663.    And what about using neural networks?
      664. -
    664. A first summary
      665. -
    665. Why Linear Regression (aka Ordinary Least Squares and family)
      666. -
    666. Regression analysis, overarching aims
      667. -
    667. Regression analysis, overarching aims II
      668. -
    668. Examples
      669. -
    669. General linear models
      670. -
    670. Rewriting the fitting procedure as a linear algebra problem
      671. -
    671. Rewriting the fitting procedure as a linear algebra problem, more details
      672. -
    672. Generalizing the fitting procedure as a linear algebra problem
      673. -
    673. Generalizing the fitting procedure as a linear algebra problem
      674. -
    674. Optimizing our parameters
      675. -
    675. Our model for the nuclear binding energies
      676. -
    676. Optimizing our parameters, more details
      677. -
    677. Interpretations and optimizing our parameters
      678. -
    678. Interpretations and optimizing our parameters
      679. -
    679. Some useful matrix and vector expressions
      680. -
    680. Interpretations and optimizing our parameters
      681. -
    681. Own code for Ordinary Least Squares
      682. -
    682. Adding error analysis and training set up
      683. -
    683. The \( \chi^2 \) function
      684. -
    684. The \( \chi^2 \) function
      685. -
    685. The \( \chi^2 \) function
      686. -
    686. The \( \chi^2 \) function
      687. -
    687. The \( \chi^2 \) function
      688. -
    688. The \( \chi^2 \) function
      689. -
    689. Fitting an Equation of State for Dense Nuclear Matter
      690. -
    690. The code
      691. -
    691. Splitting our Data in Training and Test data
      692. -
    692. Exercises
      693. -
    693. Exercise 1: Setting up various Python environments
      694. -
    694. Exercise 2: making your own data and exploring scikit-learn
      695. -
    695. Exercise 3: Normalizing our data
      696. +
    696. Numpy and arrays
      697. +
    697. Matrices in Python
      698. +
    698. Meet the Pandas
      699. +
    699.    Simple linear regression model using scikit-learn
      700. +
    700.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      701. +
    701.    Organizing our data
      702. +
    702.    And what about using neural networks?
      703. +
    703. A first summary
      704. +
    704. Why Linear Regression (aka Ordinary Least Squares and family)
      705. +
    705. Regression analysis, overarching aims
      706. +
    706. Regression analysis, overarching aims II
      707. +
    707. Examples
      708. +
    708. General linear models
      709. +
    709. Rewriting the fitting procedure as a linear algebra problem
      710. +
    710. Rewriting the fitting procedure as a linear algebra problem, more details
      711. +
    711. Generalizing the fitting procedure as a linear algebra problem
      712. +
    712. Generalizing the fitting procedure as a linear algebra problem
      713. +
    713. Optimizing our parameters
      714. +
    714. Our model for the nuclear binding energies
      715. +
    715. Optimizing our parameters, more details
      716. +
    716. Interpretations and optimizing our parameters
      717. +
    717. Interpretations and optimizing our parameters
      718. +
    718. Some useful matrix and vector expressions
      719. +
    719. Interpretations and optimizing our parameters
      720. +
    720. Own code for Ordinary Least Squares
      721. +
    721. Adding error analysis and training set up
      722. +
    722. The \( \chi^2 \) function
      723. +
    723. The \( \chi^2 \) function
      724. +
    724. The \( \chi^2 \) function
      725. +
    725. The \( \chi^2 \) function
      726. +
    726. The \( \chi^2 \) function
      727. +
    727. The \( \chi^2 \) function
      728. +
    728. Fitting an Equation of State for Dense Nuclear Matter
      729. +
    729. The code
      730. +
    730. Splitting our Data in Training and Test data
      731. +
    731. Exercises
      732. +
    732. Exercise 1: Setting up various Python environments
      733. +
    733. Exercise 2: making your own data and exploring scikit-learn
      734. +
    734. Exercise 3: Split data in test and training data
      735. @@ -392,7 +376,7 @@ MathJax.Hub.Config({
    735. 18
    736. 19
    737. ...
      738. -
    738. 66
      739. +
    739. 62
    740. »
      741. diff --git a/doc/pub/week34/html/._week34-bs010.html b/doc/pub/week34/html/._week34-bs010.html index dd8d98177..f4eb43050 100644 --- a/doc/pub/week34/html/._week34-bs010.html +++ b/doc/pub/week34/html/._week34-bs010.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    741. Installing R, C++, cython or Julia
    742. Installing R, C++, cython, Numba etc
    743. Numpy examples and Important Matrix and vector handling packages
      744. -
    744. Basic Matrix Features
      745. -
    745.    Some famous Matrices
      746. -
    746.    More Basic Matrix Features
      747. -
    747. Numpy and arrays
      748. -
    748. Matrices in Python
      749. -
    749. Meet the Pandas
      750. -
    750. Friday August 27
      751. -
    751.    Simple linear regression model using scikit-learn
      752. -
    752.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      753. -
    753.    Organizing our data
      754. -
    754.    Seeing the wood for the trees
      755. -
    755.    And what about using neural networks?
      756. -
    756. A first summary
      757. -
    757. Why Linear Regression (aka Ordinary Least Squares and family)
      758. -
    758. Regression analysis, overarching aims
      759. -
    759. Regression analysis, overarching aims II
      760. -
    760. Examples
      761. -
    761. General linear models
      762. -
    762. Rewriting the fitting procedure as a linear algebra problem
      763. -
    763. Rewriting the fitting procedure as a linear algebra problem, more details
      764. -
    764. Generalizing the fitting procedure as a linear algebra problem
      765. -
    765. Generalizing the fitting procedure as a linear algebra problem
      766. -
    766. Optimizing our parameters
      767. -
    767. Our model for the nuclear binding energies
      768. -
    768. Optimizing our parameters, more details
      769. -
    769. Interpretations and optimizing our parameters
      770. -
    770. Interpretations and optimizing our parameters
      771. -
    771. Some useful matrix and vector expressions
      772. -
    772. Interpretations and optimizing our parameters
      773. -
    773. Own code for Ordinary Least Squares
      774. -
    774. Adding error analysis and training set up
      775. -
    775. The \( \chi^2 \) function
      776. -
    776. The \( \chi^2 \) function
      777. -
    777. The \( \chi^2 \) function
      778. -
    778. The \( \chi^2 \) function
      779. -
    779. The \( \chi^2 \) function
      780. -
    780. The \( \chi^2 \) function
      781. -
    781. Fitting an Equation of State for Dense Nuclear Matter
      782. -
    782. The code
      783. -
    783. Splitting our Data in Training and Test data
      784. -
    784. Exercises
      785. -
    785. Exercise 1: Setting up various Python environments
      786. -
    786. Exercise 2: making your own data and exploring scikit-learn
      787. -
    787. Exercise 3: Normalizing our data
      788. +
    788. Numpy and arrays
      789. +
    789. Matrices in Python
      790. +
    790. Meet the Pandas
      791. +
    791.    Simple linear regression model using scikit-learn
      792. +
    792.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      793. +
    793.    Organizing our data
      794. +
    794.    And what about using neural networks?
      795. +
    795. A first summary
      796. +
    796. Why Linear Regression (aka Ordinary Least Squares and family)
      797. +
    797. Regression analysis, overarching aims
      798. +
    798. Regression analysis, overarching aims II
      799. +
    799. Examples
      800. +
    800. General linear models
      801. +
    801. Rewriting the fitting procedure as a linear algebra problem
      802. +
    802. Rewriting the fitting procedure as a linear algebra problem, more details
      803. +
    803. Generalizing the fitting procedure as a linear algebra problem
      804. +
    804. Generalizing the fitting procedure as a linear algebra problem
      805. +
    805. Optimizing our parameters
      806. +
    806. Our model for the nuclear binding energies
      807. +
    807. Optimizing our parameters, more details
      808. +
    808. Interpretations and optimizing our parameters
      809. +
    809. Interpretations and optimizing our parameters
      810. +
    810. Some useful matrix and vector expressions
      811. +
    811. Interpretations and optimizing our parameters
      812. +
    812. Own code for Ordinary Least Squares
      813. +
    813. Adding error analysis and training set up
      814. +
    814. The \( \chi^2 \) function
      815. +
    815. The \( \chi^2 \) function
      816. +
    816. The \( \chi^2 \) function
      817. +
    817. The \( \chi^2 \) function
      818. +
    818. The \( \chi^2 \) function
      819. +
    819. The \( \chi^2 \) function
      820. +
    820. Fitting an Equation of State for Dense Nuclear Matter
      821. +
    821. The code
      822. +
    822. Splitting our Data in Training and Test data
      823. +
    823. Exercises
      824. +
    824. Exercise 1: Setting up various Python environments
      825. +
    825. Exercise 2: making your own data and exploring scikit-learn
      826. +
    826. Exercise 3: Split data in test and training data
      827. @@ -389,7 +373,7 @@ Python is the recurring programming language.
    827. 19
    828. 20
    829. ...
      830. -
    830. 66
      831. +
    831. 62
    832. »
      833. diff --git a/doc/pub/week34/html/._week34-bs011.html b/doc/pub/week34/html/._week34-bs011.html index 7a0f68b07..bb0e8db2e 100644 --- a/doc/pub/week34/html/._week34-bs011.html +++ b/doc/pub/week34/html/._week34-bs011.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    833. Installing R, C++, cython or Julia
    834. Installing R, C++, cython, Numba etc
    835. Numpy examples and Important Matrix and vector handling packages
      836. -
    836. Basic Matrix Features
      837. -
    837.    Some famous Matrices
      838. -
    838.    More Basic Matrix Features
      839. -
    839. Numpy and arrays
      840. -
    840. Matrices in Python
      841. -
    841. Meet the Pandas
      842. -
    842. Friday August 27
      843. -
    843.    Simple linear regression model using scikit-learn
      844. -
    844.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      845. -
    845.    Organizing our data
      846. -
    846.    Seeing the wood for the trees
      847. -
    847.    And what about using neural networks?
      848. -
    848. A first summary
      849. -
    849. Why Linear Regression (aka Ordinary Least Squares and family)
      850. -
    850. Regression analysis, overarching aims
      851. -
    851. Regression analysis, overarching aims II
      852. -
    852. Examples
      853. -
    853. General linear models
      854. -
    854. Rewriting the fitting procedure as a linear algebra problem
      855. -
    855. Rewriting the fitting procedure as a linear algebra problem, more details
      856. -
    856. Generalizing the fitting procedure as a linear algebra problem
      857. -
    857. Generalizing the fitting procedure as a linear algebra problem
      858. -
    858. Optimizing our parameters
      859. -
    859. Our model for the nuclear binding energies
      860. -
    860. Optimizing our parameters, more details
      861. -
    861. Interpretations and optimizing our parameters
      862. -
    862. Interpretations and optimizing our parameters
      863. -
    863. Some useful matrix and vector expressions
      864. -
    864. Interpretations and optimizing our parameters
      865. -
    865. Own code for Ordinary Least Squares
      866. -
    866. Adding error analysis and training set up
      867. -
    867. The \( \chi^2 \) function
      868. -
    868. The \( \chi^2 \) function
      869. -
    869. The \( \chi^2 \) function
      870. -
    870. The \( \chi^2 \) function
      871. -
    871. The \( \chi^2 \) function
      872. -
    872. The \( \chi^2 \) function
      873. -
    873. Fitting an Equation of State for Dense Nuclear Matter
      874. -
    874. The code
      875. -
    875. Splitting our Data in Training and Test data
      876. -
    876. Exercises
      877. -
    877. Exercise 1: Setting up various Python environments
      878. -
    878. Exercise 2: making your own data and exploring scikit-learn
      879. -
    879. Exercise 3: Normalizing our data
      880. +
    880. Numpy and arrays
      881. +
    881. Matrices in Python
      882. +
    882. Meet the Pandas
      883. +
    883.    Simple linear regression model using scikit-learn
      884. +
    884.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      885. +
    885.    Organizing our data
      886. +
    886.    And what about using neural networks?
      887. +
    887. A first summary
      888. +
    888. Why Linear Regression (aka Ordinary Least Squares and family)
      889. +
    889. Regression analysis, overarching aims
      890. +
    890. Regression analysis, overarching aims II
      891. +
    891. Examples
      892. +
    892. General linear models
      893. +
    893. Rewriting the fitting procedure as a linear algebra problem
      894. +
    894. Rewriting the fitting procedure as a linear algebra problem, more details
      895. +
    895. Generalizing the fitting procedure as a linear algebra problem
      896. +
    896. Generalizing the fitting procedure as a linear algebra problem
      897. +
    897. Optimizing our parameters
      898. +
    898. Our model for the nuclear binding energies
      899. +
    899. Optimizing our parameters, more details
      900. +
    900. Interpretations and optimizing our parameters
      901. +
    901. Interpretations and optimizing our parameters
      902. +
    902. Some useful matrix and vector expressions
      903. +
    903. Interpretations and optimizing our parameters
      904. +
    904. Own code for Ordinary Least Squares
      905. +
    905. Adding error analysis and training set up
      906. +
    906. The \( \chi^2 \) function
      907. +
    907. The \( \chi^2 \) function
      908. +
    908. The \( \chi^2 \) function
      909. +
    909. The \( \chi^2 \) function
      910. +
    910. The \( \chi^2 \) function
      911. +
    911. The \( \chi^2 \) function
      912. +
    912. Fitting an Equation of State for Dense Nuclear Matter
      913. +
    913. The code
      914. +
    914. Splitting our Data in Training and Test data
      915. +
    915. Exercises
      916. +
    916. Exercise 1: Setting up various Python environments
      917. +
    917. Exercise 2: making your own data and exploring scikit-learn
      918. +
    918. Exercise 3: Split data in test and training data
      919. @@ -413,7 +397,7 @@ specifically, after this course you will
    919. 20
    920. 21
    921. ...
      922. -
    922. 66
      923. +
    923. 62
    924. »
      925. diff --git a/doc/pub/week34/html/._week34-bs012.html b/doc/pub/week34/html/._week34-bs012.html index 63566c2fc..e1297ade0 100644 --- a/doc/pub/week34/html/._week34-bs012.html +++ b/doc/pub/week34/html/._week34-bs012.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    925. Installing R, C++, cython or Julia
    926. Installing R, C++, cython, Numba etc
    927. Numpy examples and Important Matrix and vector handling packages
      928. -
    928. Basic Matrix Features
      929. -
    929.    Some famous Matrices
      930. -
    930.    More Basic Matrix Features
      931. -
    931. Numpy and arrays
      932. -
    932. Matrices in Python
      933. -
    933. Meet the Pandas
      934. -
    934. Friday August 27
      935. -
    935.    Simple linear regression model using scikit-learn
      936. -
    936.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      937. -
    937.    Organizing our data
      938. -
    938.    Seeing the wood for the trees
      939. -
    939.    And what about using neural networks?
      940. -
    940. A first summary
      941. -
    941. Why Linear Regression (aka Ordinary Least Squares and family)
      942. -
    942. Regression analysis, overarching aims
      943. -
    943. Regression analysis, overarching aims II
      944. -
    944. Examples
      945. -
    945. General linear models
      946. -
    946. Rewriting the fitting procedure as a linear algebra problem
      947. -
    947. Rewriting the fitting procedure as a linear algebra problem, more details
      948. -
    948. Generalizing the fitting procedure as a linear algebra problem
      949. -
    949. Generalizing the fitting procedure as a linear algebra problem
      950. -
    950. Optimizing our parameters
      951. -
    951. Our model for the nuclear binding energies
      952. -
    952. Optimizing our parameters, more details
      953. -
    953. Interpretations and optimizing our parameters
      954. -
    954. Interpretations and optimizing our parameters
      955. -
    955. Some useful matrix and vector expressions
      956. -
    956. Interpretations and optimizing our parameters
      957. -
    957. Own code for Ordinary Least Squares
      958. -
    958. Adding error analysis and training set up
      959. -
    959. The \( \chi^2 \) function
      960. -
    960. The \( \chi^2 \) function
      961. -
    961. The \( \chi^2 \) function
      962. -
    962. The \( \chi^2 \) function
      963. -
    963. The \( \chi^2 \) function
      964. -
    964. The \( \chi^2 \) function
      965. -
    965. Fitting an Equation of State for Dense Nuclear Matter
      966. -
    966. The code
      967. -
    967. Splitting our Data in Training and Test data
      968. -
    968. Exercises
      969. -
    969. Exercise 1: Setting up various Python environments
      970. -
    970. Exercise 2: making your own data and exploring scikit-learn
      971. -
    971. Exercise 3: Normalizing our data
      972. +
    972. Numpy and arrays
      973. +
    973. Matrices in Python
      974. +
    974. Meet the Pandas
      975. +
    975.    Simple linear regression model using scikit-learn
      976. +
    976.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      977. +
    977.    Organizing our data
      978. +
    978.    And what about using neural networks?
      979. +
    979. A first summary
      980. +
    980. Why Linear Regression (aka Ordinary Least Squares and family)
      981. +
    981. Regression analysis, overarching aims
      982. +
    982. Regression analysis, overarching aims II
      983. +
    983. Examples
      984. +
    984. General linear models
      985. +
    985. Rewriting the fitting procedure as a linear algebra problem
      986. +
    986. Rewriting the fitting procedure as a linear algebra problem, more details
      987. +
    987. Generalizing the fitting procedure as a linear algebra problem
      988. +
    988. Generalizing the fitting procedure as a linear algebra problem
      989. +
    989. Optimizing our parameters
      990. +
    990. Our model for the nuclear binding energies
      991. +
    991. Optimizing our parameters, more details
      992. +
    992. Interpretations and optimizing our parameters
      993. +
    993. Interpretations and optimizing our parameters
      994. +
    994. Some useful matrix and vector expressions
      995. +
    995. Interpretations and optimizing our parameters
      996. +
    996. Own code for Ordinary Least Squares
      997. +
    997. Adding error analysis and training set up
      998. +
    998. The \( \chi^2 \) function
      999. +
    999. The \( \chi^2 \) function
      1000. +
    1000. The \( \chi^2 \) function
      1001. +
    1001. The \( \chi^2 \) function
      1002. +
    1002. The \( \chi^2 \) function
      1003. +
    1003. The \( \chi^2 \) function
      1004. +
    1004. Fitting an Equation of State for Dense Nuclear Matter
      1005. +
    1005. The code
      1006. +
    1006. Splitting our Data in Training and Test data
      1007. +
    1007. Exercises
      1008. +
    1008. Exercise 1: Setting up various Python environments
      1009. +
    1009. Exercise 2: making your own data and exploring scikit-learn
      1010. +
    1010. Exercise 3: Split data in test and training data
      1011. @@ -404,7 +388,7 @@ MathJax.Hub.Config({
    1011. 21
    1012. 22
    1013. ...
      1014. -
    1014. 66
      1015. +
    1015. 62
    1016. »
      1017. diff --git a/doc/pub/week34/html/._week34-bs013.html b/doc/pub/week34/html/._week34-bs013.html index 8049abdfa..139c98ceb 100644 --- a/doc/pub/week34/html/._week34-bs013.html +++ b/doc/pub/week34/html/._week34-bs013.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1017. Installing R, C++, cython or Julia
    1018. Installing R, C++, cython, Numba etc
    1019. Numpy examples and Important Matrix and vector handling packages
      1020. -
    1020. Basic Matrix Features
      1021. -
    1021.    Some famous Matrices
      1022. -
    1022.    More Basic Matrix Features
      1023. -
    1023. Numpy and arrays
      1024. -
    1024. Matrices in Python
      1025. -
    1025. Meet the Pandas
      1026. -
    1026. Friday August 27
      1027. -
    1027.    Simple linear regression model using scikit-learn
      1028. -
    1028.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1029. -
    1029.    Organizing our data
      1030. -
    1030.    Seeing the wood for the trees
      1031. -
    1031.    And what about using neural networks?
      1032. -
    1032. A first summary
      1033. -
    1033. Why Linear Regression (aka Ordinary Least Squares and family)
      1034. -
    1034. Regression analysis, overarching aims
      1035. -
    1035. Regression analysis, overarching aims II
      1036. -
    1036. Examples
      1037. -
    1037. General linear models
      1038. -
    1038. Rewriting the fitting procedure as a linear algebra problem
      1039. -
    1039. Rewriting the fitting procedure as a linear algebra problem, more details
      1040. -
    1040. Generalizing the fitting procedure as a linear algebra problem
      1041. -
    1041. Generalizing the fitting procedure as a linear algebra problem
      1042. -
    1042. Optimizing our parameters
      1043. -
    1043. Our model for the nuclear binding energies
      1044. -
    1044. Optimizing our parameters, more details
      1045. -
    1045. Interpretations and optimizing our parameters
      1046. -
    1046. Interpretations and optimizing our parameters
      1047. -
    1047. Some useful matrix and vector expressions
      1048. -
    1048. Interpretations and optimizing our parameters
      1049. -
    1049. Own code for Ordinary Least Squares
      1050. -
    1050. Adding error analysis and training set up
      1051. -
    1051. The \( \chi^2 \) function
      1052. -
    1052. The \( \chi^2 \) function
      1053. -
    1053. The \( \chi^2 \) function
      1054. -
    1054. The \( \chi^2 \) function
      1055. -
    1055. The \( \chi^2 \) function
      1056. -
    1056. The \( \chi^2 \) function
      1057. -
    1057. Fitting an Equation of State for Dense Nuclear Matter
      1058. -
    1058. The code
      1059. -
    1059. Splitting our Data in Training and Test data
      1060. -
    1060. Exercises
      1061. -
    1061. Exercise 1: Setting up various Python environments
      1062. -
    1062. Exercise 2: making your own data and exploring scikit-learn
      1063. -
    1063. Exercise 3: Normalizing our data
      1064. +
    1064. Numpy and arrays
      1065. +
    1065. Matrices in Python
      1066. +
    1066. Meet the Pandas
      1067. +
    1067.    Simple linear regression model using scikit-learn
      1068. +
    1068.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1069. +
    1069.    Organizing our data
      1070. +
    1070.    And what about using neural networks?
      1071. +
    1071. A first summary
      1072. +
    1072. Why Linear Regression (aka Ordinary Least Squares and family)
      1073. +
    1073. Regression analysis, overarching aims
      1074. +
    1074. Regression analysis, overarching aims II
      1075. +
    1075. Examples
      1076. +
    1076. General linear models
      1077. +
    1077. Rewriting the fitting procedure as a linear algebra problem
      1078. +
    1078. Rewriting the fitting procedure as a linear algebra problem, more details
      1079. +
    1079. Generalizing the fitting procedure as a linear algebra problem
      1080. +
    1080. Generalizing the fitting procedure as a linear algebra problem
      1081. +
    1081. Optimizing our parameters
      1082. +
    1082. Our model for the nuclear binding energies
      1083. +
    1083. Optimizing our parameters, more details
      1084. +
    1084. Interpretations and optimizing our parameters
      1085. +
    1085. Interpretations and optimizing our parameters
      1086. +
    1086. Some useful matrix and vector expressions
      1087. +
    1087. Interpretations and optimizing our parameters
      1088. +
    1088. Own code for Ordinary Least Squares
      1089. +
    1089. Adding error analysis and training set up
      1090. +
    1090. The \( \chi^2 \) function
      1091. +
    1091. The \( \chi^2 \) function
      1092. +
    1092. The \( \chi^2 \) function
      1093. +
    1093. The \( \chi^2 \) function
      1094. +
    1094. The \( \chi^2 \) function
      1095. +
    1095. The \( \chi^2 \) function
      1096. +
    1096. Fitting an Equation of State for Dense Nuclear Matter
      1097. +
    1097. The code
      1098. +
    1098. Splitting our Data in Training and Test data
      1099. +
    1099. Exercises
      1100. +
    1100. Exercise 1: Setting up various Python environments
      1101. +
    1101. Exercise 2: making your own data and exploring scikit-learn
      1102. +
    1102. Exercise 3: Split data in test and training data
      1103. @@ -406,7 +390,7 @@ MathJax.Hub.Config({
    1103. 22
    1104. 23
    1105. ...
      1106. -
    1106. 66
      1107. +
    1107. 62
    1108. »
      1109. diff --git a/doc/pub/week34/html/._week34-bs014.html b/doc/pub/week34/html/._week34-bs014.html index 1de18d9e5..8fe5ee44c 100644 --- a/doc/pub/week34/html/._week34-bs014.html +++ b/doc/pub/week34/html/._week34-bs014.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1109. Installing R, C++, cython or Julia
    1110. Installing R, C++, cython, Numba etc
    1111. Numpy examples and Important Matrix and vector handling packages
      1112. -
    1112. Basic Matrix Features
      1113. -
    1113.    Some famous Matrices
      1114. -
    1114.    More Basic Matrix Features
      1115. -
    1115. Numpy and arrays
      1116. -
    1116. Matrices in Python
      1117. -
    1117. Meet the Pandas
      1118. -
    1118. Friday August 27
      1119. -
    1119.    Simple linear regression model using scikit-learn
      1120. -
    1120.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1121. -
    1121.    Organizing our data
      1122. -
    1122.    Seeing the wood for the trees
      1123. -
    1123.    And what about using neural networks?
      1124. -
    1124. A first summary
      1125. -
    1125. Why Linear Regression (aka Ordinary Least Squares and family)
      1126. -
    1126. Regression analysis, overarching aims
      1127. -
    1127. Regression analysis, overarching aims II
      1128. -
    1128. Examples
      1129. -
    1129. General linear models
      1130. -
    1130. Rewriting the fitting procedure as a linear algebra problem
      1131. -
    1131. Rewriting the fitting procedure as a linear algebra problem, more details
      1132. -
    1132. Generalizing the fitting procedure as a linear algebra problem
      1133. -
    1133. Generalizing the fitting procedure as a linear algebra problem
      1134. -
    1134. Optimizing our parameters
      1135. -
    1135. Our model for the nuclear binding energies
      1136. -
    1136. Optimizing our parameters, more details
      1137. -
    1137. Interpretations and optimizing our parameters
      1138. -
    1138. Interpretations and optimizing our parameters
      1139. -
    1139. Some useful matrix and vector expressions
      1140. -
    1140. Interpretations and optimizing our parameters
      1141. -
    1141. Own code for Ordinary Least Squares
      1142. -
    1142. Adding error analysis and training set up
      1143. -
    1143. The \( \chi^2 \) function
      1144. -
    1144. The \( \chi^2 \) function
      1145. -
    1145. The \( \chi^2 \) function
      1146. -
    1146. The \( \chi^2 \) function
      1147. -
    1147. The \( \chi^2 \) function
      1148. -
    1148. The \( \chi^2 \) function
      1149. -
    1149. Fitting an Equation of State for Dense Nuclear Matter
      1150. -
    1150. The code
      1151. -
    1151. Splitting our Data in Training and Test data
      1152. -
    1152. Exercises
      1153. -
    1153. Exercise 1: Setting up various Python environments
      1154. -
    1154. Exercise 2: making your own data and exploring scikit-learn
      1155. -
    1155. Exercise 3: Normalizing our data
      1156. +
    1156. Numpy and arrays
      1157. +
    1157. Matrices in Python
      1158. +
    1158. Meet the Pandas
      1159. +
    1159.    Simple linear regression model using scikit-learn
      1160. +
    1160.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1161. +
    1161.    Organizing our data
      1162. +
    1162.    And what about using neural networks?
      1163. +
    1163. A first summary
      1164. +
    1164. Why Linear Regression (aka Ordinary Least Squares and family)
      1165. +
    1165. Regression analysis, overarching aims
      1166. +
    1166. Regression analysis, overarching aims II
      1167. +
    1167. Examples
      1168. +
    1168. General linear models
      1169. +
    1169. Rewriting the fitting procedure as a linear algebra problem
      1170. +
    1170. Rewriting the fitting procedure as a linear algebra problem, more details
      1171. +
    1171. Generalizing the fitting procedure as a linear algebra problem
      1172. +
    1172. Generalizing the fitting procedure as a linear algebra problem
      1173. +
    1173. Optimizing our parameters
      1174. +
    1174. Our model for the nuclear binding energies
      1175. +
    1175. Optimizing our parameters, more details
      1176. +
    1176. Interpretations and optimizing our parameters
      1177. +
    1177. Interpretations and optimizing our parameters
      1178. +
    1178. Some useful matrix and vector expressions
      1179. +
    1179. Interpretations and optimizing our parameters
      1180. +
    1180. Own code for Ordinary Least Squares
      1181. +
    1181. Adding error analysis and training set up
      1182. +
    1182. The \( \chi^2 \) function
      1183. +
    1183. The \( \chi^2 \) function
      1184. +
    1184. The \( \chi^2 \) function
      1185. +
    1185. The \( \chi^2 \) function
      1186. +
    1186. The \( \chi^2 \) function
      1187. +
    1187. The \( \chi^2 \) function
      1188. +
    1188. Fitting an Equation of State for Dense Nuclear Matter
      1189. +
    1189. The code
      1190. +
    1190. Splitting our Data in Training and Test data
      1191. +
    1191. Exercises
      1192. +
    1192. Exercise 1: Setting up various Python environments
      1193. +
    1193. Exercise 2: making your own data and exploring scikit-learn
      1194. +
    1194. Exercise 3: Split data in test and training data
      1195. @@ -389,7 +373,7 @@ MathJax.Hub.Config({
    1195. 23
    1196. 24
    1197. ...
      1198. -
    1198. 66
      1199. +
    1199. 62
    1200. »
      1201. diff --git a/doc/pub/week34/html/._week34-bs015.html b/doc/pub/week34/html/._week34-bs015.html index b67e2f56e..9c9a00303 100644 --- a/doc/pub/week34/html/._week34-bs015.html +++ b/doc/pub/week34/html/._week34-bs015.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1201. Installing R, C++, cython or Julia
    1202. Installing R, C++, cython, Numba etc
    1203. Numpy examples and Important Matrix and vector handling packages
      1204. -
    1204. Basic Matrix Features
      1205. -
    1205.    Some famous Matrices
      1206. -
    1206.    More Basic Matrix Features
      1207. -
    1207. Numpy and arrays
      1208. -
    1208. Matrices in Python
      1209. -
    1209. Meet the Pandas
      1210. -
    1210. Friday August 27
      1211. -
    1211.    Simple linear regression model using scikit-learn
      1212. -
    1212.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1213. -
    1213.    Organizing our data
      1214. -
    1214.    Seeing the wood for the trees
      1215. -
    1215.    And what about using neural networks?
      1216. -
    1216. A first summary
      1217. -
    1217. Why Linear Regression (aka Ordinary Least Squares and family)
      1218. -
    1218. Regression analysis, overarching aims
      1219. -
    1219. Regression analysis, overarching aims II
      1220. -
    1220. Examples
      1221. -
    1221. General linear models
      1222. -
    1222. Rewriting the fitting procedure as a linear algebra problem
      1223. -
    1223. Rewriting the fitting procedure as a linear algebra problem, more details
      1224. -
    1224. Generalizing the fitting procedure as a linear algebra problem
      1225. -
    1225. Generalizing the fitting procedure as a linear algebra problem
      1226. -
    1226. Optimizing our parameters
      1227. -
    1227. Our model for the nuclear binding energies
      1228. -
    1228. Optimizing our parameters, more details
      1229. -
    1229. Interpretations and optimizing our parameters
      1230. -
    1230. Interpretations and optimizing our parameters
      1231. -
    1231. Some useful matrix and vector expressions
      1232. -
    1232. Interpretations and optimizing our parameters
      1233. -
    1233. Own code for Ordinary Least Squares
      1234. -
    1234. Adding error analysis and training set up
      1235. -
    1235. The \( \chi^2 \) function
      1236. -
    1236. The \( \chi^2 \) function
      1237. -
    1237. The \( \chi^2 \) function
      1238. -
    1238. The \( \chi^2 \) function
      1239. -
    1239. The \( \chi^2 \) function
      1240. -
    1240. The \( \chi^2 \) function
      1241. -
    1241. Fitting an Equation of State for Dense Nuclear Matter
      1242. -
    1242. The code
      1243. -
    1243. Splitting our Data in Training and Test data
      1244. -
    1244. Exercises
      1245. -
    1245. Exercise 1: Setting up various Python environments
      1246. -
    1246. Exercise 2: making your own data and exploring scikit-learn
      1247. -
    1247. Exercise 3: Normalizing our data
      1248. +
    1248. Numpy and arrays
      1249. +
    1249. Matrices in Python
      1250. +
    1250. Meet the Pandas
      1251. +
    1251.    Simple linear regression model using scikit-learn
      1252. +
    1252.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1253. +
    1253.    Organizing our data
      1254. +
    1254.    And what about using neural networks?
      1255. +
    1255. A first summary
      1256. +
    1256. Why Linear Regression (aka Ordinary Least Squares and family)
      1257. +
    1257. Regression analysis, overarching aims
      1258. +
    1258. Regression analysis, overarching aims II
      1259. +
    1259. Examples
      1260. +
    1260. General linear models
      1261. +
    1261. Rewriting the fitting procedure as a linear algebra problem
      1262. +
    1262. Rewriting the fitting procedure as a linear algebra problem, more details
      1263. +
    1263. Generalizing the fitting procedure as a linear algebra problem
      1264. +
    1264. Generalizing the fitting procedure as a linear algebra problem
      1265. +
    1265. Optimizing our parameters
      1266. +
    1266. Our model for the nuclear binding energies
      1267. +
    1267. Optimizing our parameters, more details
      1268. +
    1268. Interpretations and optimizing our parameters
      1269. +
    1269. Interpretations and optimizing our parameters
      1270. +
    1270. Some useful matrix and vector expressions
      1271. +
    1271. Interpretations and optimizing our parameters
      1272. +
    1272. Own code for Ordinary Least Squares
      1273. +
    1273. Adding error analysis and training set up
      1274. +
    1274. The \( \chi^2 \) function
      1275. +
    1275. The \( \chi^2 \) function
      1276. +
    1276. The \( \chi^2 \) function
      1277. +
    1277. The \( \chi^2 \) function
      1278. +
    1278. The \( \chi^2 \) function
      1279. +
    1279. The \( \chi^2 \) function
      1280. +
    1280. Fitting an Equation of State for Dense Nuclear Matter
      1281. +
    1281. The code
      1282. +
    1282. Splitting our Data in Training and Test data
      1283. +
    1283. Exercises
      1284. +
    1284. Exercise 1: Setting up various Python environments
      1285. +
    1285. Exercise 2: making your own data and exploring scikit-learn
      1286. +
    1286. Exercise 3: Split data in test and training data
      1287. @@ -391,7 +375,7 @@ MathJax.Hub.Config({
    1287. 24
    1288. 25
    1289. ...
      1290. -
    1290. 66
      1291. +
    1291. 62
    1292. »
      1293. diff --git a/doc/pub/week34/html/._week34-bs016.html b/doc/pub/week34/html/._week34-bs016.html index fb8b304b8..99ee3dbc2 100644 --- a/doc/pub/week34/html/._week34-bs016.html +++ b/doc/pub/week34/html/._week34-bs016.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1293. Installing R, C++, cython or Julia
    1294. Installing R, C++, cython, Numba etc
    1295. Numpy examples and Important Matrix and vector handling packages
      1296. -
    1296. Basic Matrix Features
      1297. -
    1297.    Some famous Matrices
      1298. -
    1298.    More Basic Matrix Features
      1299. -
    1299. Numpy and arrays
      1300. -
    1300. Matrices in Python
      1301. -
    1301. Meet the Pandas
      1302. -
    1302. Friday August 27
      1303. -
    1303.    Simple linear regression model using scikit-learn
      1304. -
    1304.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1305. -
    1305.    Organizing our data
      1306. -
    1306.    Seeing the wood for the trees
      1307. -
    1307.    And what about using neural networks?
      1308. -
    1308. A first summary
      1309. -
    1309. Why Linear Regression (aka Ordinary Least Squares and family)
      1310. -
    1310. Regression analysis, overarching aims
      1311. -
    1311. Regression analysis, overarching aims II
      1312. -
    1312. Examples
      1313. -
    1313. General linear models
      1314. -
    1314. Rewriting the fitting procedure as a linear algebra problem
      1315. -
    1315. Rewriting the fitting procedure as a linear algebra problem, more details
      1316. -
    1316. Generalizing the fitting procedure as a linear algebra problem
      1317. -
    1317. Generalizing the fitting procedure as a linear algebra problem
      1318. -
    1318. Optimizing our parameters
      1319. -
    1319. Our model for the nuclear binding energies
      1320. -
    1320. Optimizing our parameters, more details
      1321. -
    1321. Interpretations and optimizing our parameters
      1322. -
    1322. Interpretations and optimizing our parameters
      1323. -
    1323. Some useful matrix and vector expressions
      1324. -
    1324. Interpretations and optimizing our parameters
      1325. -
    1325. Own code for Ordinary Least Squares
      1326. -
    1326. Adding error analysis and training set up
      1327. -
    1327. The \( \chi^2 \) function
      1328. -
    1328. The \( \chi^2 \) function
      1329. -
    1329. The \( \chi^2 \) function
      1330. -
    1330. The \( \chi^2 \) function
      1331. -
    1331. The \( \chi^2 \) function
      1332. -
    1332. The \( \chi^2 \) function
      1333. -
    1333. Fitting an Equation of State for Dense Nuclear Matter
      1334. -
    1334. The code
      1335. -
    1335. Splitting our Data in Training and Test data
      1336. -
    1336. Exercises
      1337. -
    1337. Exercise 1: Setting up various Python environments
      1338. -
    1338. Exercise 2: making your own data and exploring scikit-learn
      1339. -
    1339. Exercise 3: Normalizing our data
      1340. +
    1340. Numpy and arrays
      1341. +
    1341. Matrices in Python
      1342. +
    1342. Meet the Pandas
      1343. +
    1343.    Simple linear regression model using scikit-learn
      1344. +
    1344.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1345. +
    1345.    Organizing our data
      1346. +
    1346.    And what about using neural networks?
      1347. +
    1347. A first summary
      1348. +
    1348. Why Linear Regression (aka Ordinary Least Squares and family)
      1349. +
    1349. Regression analysis, overarching aims
      1350. +
    1350. Regression analysis, overarching aims II
      1351. +
    1351. Examples
      1352. +
    1352. General linear models
      1353. +
    1353. Rewriting the fitting procedure as a linear algebra problem
      1354. +
    1354. Rewriting the fitting procedure as a linear algebra problem, more details
      1355. +
    1355. Generalizing the fitting procedure as a linear algebra problem
      1356. +
    1356. Generalizing the fitting procedure as a linear algebra problem
      1357. +
    1357. Optimizing our parameters
      1358. +
    1358. Our model for the nuclear binding energies
      1359. +
    1359. Optimizing our parameters, more details
      1360. +
    1360. Interpretations and optimizing our parameters
      1361. +
    1361. Interpretations and optimizing our parameters
      1362. +
    1362. Some useful matrix and vector expressions
      1363. +
    1363. Interpretations and optimizing our parameters
      1364. +
    1364. Own code for Ordinary Least Squares
      1365. +
    1365. Adding error analysis and training set up
      1366. +
    1366. The \( \chi^2 \) function
      1367. +
    1367. The \( \chi^2 \) function
      1368. +
    1368. The \( \chi^2 \) function
      1369. +
    1369. The \( \chi^2 \) function
      1370. +
    1370. The \( \chi^2 \) function
      1371. +
    1371. The \( \chi^2 \) function
      1372. +
    1372. Fitting an Equation of State for Dense Nuclear Matter
      1373. +
    1373. The code
      1374. +
    1374. Splitting our Data in Training and Test data
      1375. +
    1375. Exercises
      1376. +
    1376. Exercise 1: Setting up various Python environments
      1377. +
    1377. Exercise 2: making your own data and exploring scikit-learn
      1378. +
    1378. Exercise 3: Split data in test and training data
      1379. @@ -382,9 +366,9 @@ topics and tools as well as showing the power of various Python libraries for machine learning and statistical data analysis.

      -

      Here, we will mainly focus on two +

      Although we have projects where you write your own codes, we will also focus on two specific Python packages for Machine Learning, Scikit-Learn and -Tensorflow (see below for links etc). Moreover, the examples we +Tensorflow with Keras (see below for links etc). Moreover, the examples we introduce will serve as inputs to many of our discussions later, as well as allowing you to set up models and produce your own data and get started with programming. @@ -415,7 +399,7 @@ get started with programming.

    1379. 25
    1380. 26
    1381. ...
      1382. -
    1382. 66
      1383. +
    1383. 62
    1384. »
      1385. diff --git a/doc/pub/week34/html/._week34-bs017.html b/doc/pub/week34/html/._week34-bs017.html index ab6dee720..731a1423d 100644 --- a/doc/pub/week34/html/._week34-bs017.html +++ b/doc/pub/week34/html/._week34-bs017.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1385. Installing R, C++, cython or Julia
    1386. Installing R, C++, cython, Numba etc
    1387. Numpy examples and Important Matrix and vector handling packages
      1388. -
    1388. Basic Matrix Features
      1389. -
    1389.    Some famous Matrices
      1390. -
    1390.    More Basic Matrix Features
      1391. -
    1391. Numpy and arrays
      1392. -
    1392. Matrices in Python
      1393. -
    1393. Meet the Pandas
      1394. -
    1394. Friday August 27
      1395. -
    1395.    Simple linear regression model using scikit-learn
      1396. -
    1396.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1397. -
    1397.    Organizing our data
      1398. -
    1398.    Seeing the wood for the trees
      1399. -
    1399.    And what about using neural networks?
      1400. -
    1400. A first summary
      1401. -
    1401. Why Linear Regression (aka Ordinary Least Squares and family)
      1402. -
    1402. Regression analysis, overarching aims
      1403. -
    1403. Regression analysis, overarching aims II
      1404. -
    1404. Examples
      1405. -
    1405. General linear models
      1406. -
    1406. Rewriting the fitting procedure as a linear algebra problem
      1407. -
    1407. Rewriting the fitting procedure as a linear algebra problem, more details
      1408. -
    1408. Generalizing the fitting procedure as a linear algebra problem
      1409. -
    1409. Generalizing the fitting procedure as a linear algebra problem
      1410. -
    1410. Optimizing our parameters
      1411. -
    1411. Our model for the nuclear binding energies
      1412. -
    1412. Optimizing our parameters, more details
      1413. -
    1413. Interpretations and optimizing our parameters
      1414. -
    1414. Interpretations and optimizing our parameters
      1415. -
    1415. Some useful matrix and vector expressions
      1416. -
    1416. Interpretations and optimizing our parameters
      1417. -
    1417. Own code for Ordinary Least Squares
      1418. -
    1418. Adding error analysis and training set up
      1419. -
    1419. The \( \chi^2 \) function
      1420. -
    1420. The \( \chi^2 \) function
      1421. -
    1421. The \( \chi^2 \) function
      1422. -
    1422. The \( \chi^2 \) function
      1423. -
    1423. The \( \chi^2 \) function
      1424. -
    1424. The \( \chi^2 \) function
      1425. -
    1425. Fitting an Equation of State for Dense Nuclear Matter
      1426. -
    1426. The code
      1427. -
    1427. Splitting our Data in Training and Test data
      1428. -
    1428. Exercises
      1429. -
    1429. Exercise 1: Setting up various Python environments
      1430. -
    1430. Exercise 2: making your own data and exploring scikit-learn
      1431. -
    1431. Exercise 3: Normalizing our data
      1432. +
    1432. Numpy and arrays
      1433. +
    1433. Matrices in Python
      1434. +
    1434. Meet the Pandas
      1435. +
    1435.    Simple linear regression model using scikit-learn
      1436. +
    1436.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1437. +
    1437.    Organizing our data
      1438. +
    1438.    And what about using neural networks?
      1439. +
    1439. A first summary
      1440. +
    1440. Why Linear Regression (aka Ordinary Least Squares and family)
      1441. +
    1441. Regression analysis, overarching aims
      1442. +
    1442. Regression analysis, overarching aims II
      1443. +
    1443. Examples
      1444. +
    1444. General linear models
      1445. +
    1445. Rewriting the fitting procedure as a linear algebra problem
      1446. +
    1446. Rewriting the fitting procedure as a linear algebra problem, more details
      1447. +
    1447. Generalizing the fitting procedure as a linear algebra problem
      1448. +
    1448. Generalizing the fitting procedure as a linear algebra problem
      1449. +
    1449. Optimizing our parameters
      1450. +
    1450. Our model for the nuclear binding energies
      1451. +
    1451. Optimizing our parameters, more details
      1452. +
    1452. Interpretations and optimizing our parameters
      1453. +
    1453. Interpretations and optimizing our parameters
      1454. +
    1454. Some useful matrix and vector expressions
      1455. +
    1455. Interpretations and optimizing our parameters
      1456. +
    1456. Own code for Ordinary Least Squares
      1457. +
    1457. Adding error analysis and training set up
      1458. +
    1458. The \( \chi^2 \) function
      1459. +
    1459. The \( \chi^2 \) function
      1460. +
    1460. The \( \chi^2 \) function
      1461. +
    1461. The \( \chi^2 \) function
      1462. +
    1462. The \( \chi^2 \) function
      1463. +
    1463. The \( \chi^2 \) function
      1464. +
    1464. Fitting an Equation of State for Dense Nuclear Matter
      1465. +
    1465. The code
      1466. +
    1466. Splitting our Data in Training and Test data
      1467. +
    1467. Exercises
      1468. +
    1468. Exercise 1: Setting up various Python environments
      1469. +
    1469. Exercise 2: making your own data and exploring scikit-learn
      1470. +
    1470. Exercise 3: Split data in test and training data
      1471. @@ -446,7 +430,7 @@ of algorithms and methods we will discuss.
    1471. 26
    1472. 27
    1473. ...
      1474. -
    1474. 66
      1475. +
    1475. 62
    1476. »
      1477. diff --git a/doc/pub/week34/html/._week34-bs018.html b/doc/pub/week34/html/._week34-bs018.html index 6ba6a4a4c..0fc707830 100644 --- a/doc/pub/week34/html/._week34-bs018.html +++ b/doc/pub/week34/html/._week34-bs018.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1477. Installing R, C++, cython or Julia
    1478. Installing R, C++, cython, Numba etc
    1479. Numpy examples and Important Matrix and vector handling packages
      1480. -
    1480. Basic Matrix Features
      1481. -
    1481.    Some famous Matrices
      1482. -
    1482.    More Basic Matrix Features
      1483. -
    1483. Numpy and arrays
      1484. -
    1484. Matrices in Python
      1485. -
    1485. Meet the Pandas
      1486. -
    1486. Friday August 27
      1487. -
    1487.    Simple linear regression model using scikit-learn
      1488. -
    1488.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1489. -
    1489.    Organizing our data
      1490. -
    1490.    Seeing the wood for the trees
      1491. -
    1491.    And what about using neural networks?
      1492. -
    1492. A first summary
      1493. -
    1493. Why Linear Regression (aka Ordinary Least Squares and family)
      1494. -
    1494. Regression analysis, overarching aims
      1495. -
    1495. Regression analysis, overarching aims II
      1496. -
    1496. Examples
      1497. -
    1497. General linear models
      1498. -
    1498. Rewriting the fitting procedure as a linear algebra problem
      1499. -
    1499. Rewriting the fitting procedure as a linear algebra problem, more details
      1500. -
    1500. Generalizing the fitting procedure as a linear algebra problem
      1501. -
    1501. Generalizing the fitting procedure as a linear algebra problem
      1502. -
    1502. Optimizing our parameters
      1503. -
    1503. Our model for the nuclear binding energies
      1504. -
    1504. Optimizing our parameters, more details
      1505. -
    1505. Interpretations and optimizing our parameters
      1506. -
    1506. Interpretations and optimizing our parameters
      1507. -
    1507. Some useful matrix and vector expressions
      1508. -
    1508. Interpretations and optimizing our parameters
      1509. -
    1509. Own code for Ordinary Least Squares
      1510. -
    1510. Adding error analysis and training set up
      1511. -
    1511. The \( \chi^2 \) function
      1512. -
    1512. The \( \chi^2 \) function
      1513. -
    1513. The \( \chi^2 \) function
      1514. -
    1514. The \( \chi^2 \) function
      1515. -
    1515. The \( \chi^2 \) function
      1516. -
    1516. The \( \chi^2 \) function
      1517. -
    1517. Fitting an Equation of State for Dense Nuclear Matter
      1518. -
    1518. The code
      1519. -
    1519. Splitting our Data in Training and Test data
      1520. -
    1520. Exercises
      1521. -
    1521. Exercise 1: Setting up various Python environments
      1522. -
    1522. Exercise 2: making your own data and exploring scikit-learn
      1523. -
    1523. Exercise 3: Normalizing our data
      1524. +
    1524. Numpy and arrays
      1525. +
    1525. Matrices in Python
      1526. +
    1526. Meet the Pandas
      1527. +
    1527.    Simple linear regression model using scikit-learn
      1528. +
    1528.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1529. +
    1529.    Organizing our data
      1530. +
    1530.    And what about using neural networks?
      1531. +
    1531. A first summary
      1532. +
    1532. Why Linear Regression (aka Ordinary Least Squares and family)
      1533. +
    1533. Regression analysis, overarching aims
      1534. +
    1534. Regression analysis, overarching aims II
      1535. +
    1535. Examples
      1536. +
    1536. General linear models
      1537. +
    1537. Rewriting the fitting procedure as a linear algebra problem
      1538. +
    1538. Rewriting the fitting procedure as a linear algebra problem, more details
      1539. +
    1539. Generalizing the fitting procedure as a linear algebra problem
      1540. +
    1540. Generalizing the fitting procedure as a linear algebra problem
      1541. +
    1541. Optimizing our parameters
      1542. +
    1542. Our model for the nuclear binding energies
      1543. +
    1543. Optimizing our parameters, more details
      1544. +
    1544. Interpretations and optimizing our parameters
      1545. +
    1545. Interpretations and optimizing our parameters
      1546. +
    1546. Some useful matrix and vector expressions
      1547. +
    1547. Interpretations and optimizing our parameters
      1548. +
    1548. Own code for Ordinary Least Squares
      1549. +
    1549. Adding error analysis and training set up
      1550. +
    1550. The \( \chi^2 \) function
      1551. +
    1551. The \( \chi^2 \) function
      1552. +
    1552. The \( \chi^2 \) function
      1553. +
    1553. The \( \chi^2 \) function
      1554. +
    1554. The \( \chi^2 \) function
      1555. +
    1555. The \( \chi^2 \) function
      1556. +
    1556. Fitting an Equation of State for Dense Nuclear Matter
      1557. +
    1557. The code
      1558. +
    1558. Splitting our Data in Training and Test data
      1559. +
    1559. Exercises
      1560. +
    1560. Exercise 1: Setting up various Python environments
      1561. +
    1561. Exercise 2: making your own data and exploring scikit-learn
      1562. +
    1562. Exercise 3: Split data in test and training data
      1563. @@ -398,7 +382,7 @@ desired output of a system. Some of the most common tasks are:
    1563. 27
    1564. 28
    1565. ...
      1566. -
    1566. 66
      1567. +
    1567. 62
    1568. »
      1569. diff --git a/doc/pub/week34/html/._week34-bs019.html b/doc/pub/week34/html/._week34-bs019.html index 0f1c39c1b..ecb4e968d 100644 --- a/doc/pub/week34/html/._week34-bs019.html +++ b/doc/pub/week34/html/._week34-bs019.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1569. Installing R, C++, cython or Julia
    1570. Installing R, C++, cython, Numba etc
    1571. Numpy examples and Important Matrix and vector handling packages
      1572. -
    1572. Basic Matrix Features
      1573. -
    1573.    Some famous Matrices
      1574. -
    1574.    More Basic Matrix Features
      1575. -
    1575. Numpy and arrays
      1576. -
    1576. Matrices in Python
      1577. -
    1577. Meet the Pandas
      1578. -
    1578. Friday August 27
      1579. -
    1579.    Simple linear regression model using scikit-learn
      1580. -
    1580.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1581. -
    1581.    Organizing our data
      1582. -
    1582.    Seeing the wood for the trees
      1583. -
    1583.    And what about using neural networks?
      1584. -
    1584. A first summary
      1585. -
    1585. Why Linear Regression (aka Ordinary Least Squares and family)
      1586. -
    1586. Regression analysis, overarching aims
      1587. -
    1587. Regression analysis, overarching aims II
      1588. -
    1588. Examples
      1589. -
    1589. General linear models
      1590. -
    1590. Rewriting the fitting procedure as a linear algebra problem
      1591. -
    1591. Rewriting the fitting procedure as a linear algebra problem, more details
      1592. -
    1592. Generalizing the fitting procedure as a linear algebra problem
      1593. -
    1593. Generalizing the fitting procedure as a linear algebra problem
      1594. -
    1594. Optimizing our parameters
      1595. -
    1595. Our model for the nuclear binding energies
      1596. -
    1596. Optimizing our parameters, more details
      1597. -
    1597. Interpretations and optimizing our parameters
      1598. -
    1598. Interpretations and optimizing our parameters
      1599. -
    1599. Some useful matrix and vector expressions
      1600. -
    1600. Interpretations and optimizing our parameters
      1601. -
    1601. Own code for Ordinary Least Squares
      1602. -
    1602. Adding error analysis and training set up
      1603. -
    1603. The \( \chi^2 \) function
      1604. -
    1604. The \( \chi^2 \) function
      1605. -
    1605. The \( \chi^2 \) function
      1606. -
    1606. The \( \chi^2 \) function
      1607. -
    1607. The \( \chi^2 \) function
      1608. -
    1608. The \( \chi^2 \) function
      1609. -
    1609. Fitting an Equation of State for Dense Nuclear Matter
      1610. -
    1610. The code
      1611. -
    1611. Splitting our Data in Training and Test data
      1612. -
    1612. Exercises
      1613. -
    1613. Exercise 1: Setting up various Python environments
      1614. -
    1614. Exercise 2: making your own data and exploring scikit-learn
      1615. -
    1615. Exercise 3: Normalizing our data
      1616. +
    1616. Numpy and arrays
      1617. +
    1617. Matrices in Python
      1618. +
    1618. Meet the Pandas
      1619. +
    1619.    Simple linear regression model using scikit-learn
      1620. +
    1620.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1621. +
    1621.    Organizing our data
      1622. +
    1622.    And what about using neural networks?
      1623. +
    1623. A first summary
      1624. +
    1624. Why Linear Regression (aka Ordinary Least Squares and family)
      1625. +
    1625. Regression analysis, overarching aims
      1626. +
    1626. Regression analysis, overarching aims II
      1627. +
    1627. Examples
      1628. +
    1628. General linear models
      1629. +
    1629. Rewriting the fitting procedure as a linear algebra problem
      1630. +
    1630. Rewriting the fitting procedure as a linear algebra problem, more details
      1631. +
    1631. Generalizing the fitting procedure as a linear algebra problem
      1632. +
    1632. Generalizing the fitting procedure as a linear algebra problem
      1633. +
    1633. Optimizing our parameters
      1634. +
    1634. Our model for the nuclear binding energies
      1635. +
    1635. Optimizing our parameters, more details
      1636. +
    1636. Interpretations and optimizing our parameters
      1637. +
    1637. Interpretations and optimizing our parameters
      1638. +
    1638. Some useful matrix and vector expressions
      1639. +
    1639. Interpretations and optimizing our parameters
      1640. +
    1640. Own code for Ordinary Least Squares
      1641. +
    1641. Adding error analysis and training set up
      1642. +
    1642. The \( \chi^2 \) function
      1643. +
    1643. The \( \chi^2 \) function
      1644. +
    1644. The \( \chi^2 \) function
      1645. +
    1645. The \( \chi^2 \) function
      1646. +
    1646. The \( \chi^2 \) function
      1647. +
    1647. The \( \chi^2 \) function
      1648. +
    1648. Fitting an Equation of State for Dense Nuclear Matter
      1649. +
    1649. The code
      1650. +
    1650. Splitting our Data in Training and Test data
      1651. +
    1651. Exercises
      1652. +
    1652. Exercise 1: Setting up various Python environments
      1653. +
    1653. Exercise 2: making your own data and exploring scikit-learn
      1654. +
    1654. Exercise 3: Split data in test and training data
      1655. @@ -389,7 +373,7 @@ whether we deal with supervised or unsupervised learning.
    1655. 28
    1656. 29
    1657. ...
      1658. -
    1658. 66
      1659. +
    1659. 62
    1660. »
      1661. diff --git a/doc/pub/week34/html/._week34-bs020.html b/doc/pub/week34/html/._week34-bs020.html index c3d562647..bbcbcae60 100644 --- a/doc/pub/week34/html/._week34-bs020.html +++ b/doc/pub/week34/html/._week34-bs020.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1661. Installing R, C++, cython or Julia
    1662. Installing R, C++, cython, Numba etc
    1663. Numpy examples and Important Matrix and vector handling packages
      1664. -
    1664. Basic Matrix Features
      1665. -
    1665.    Some famous Matrices
      1666. -
    1666.    More Basic Matrix Features
      1667. -
    1667. Numpy and arrays
      1668. -
    1668. Matrices in Python
      1669. -
    1669. Meet the Pandas
      1670. -
    1670. Friday August 27
      1671. -
    1671.    Simple linear regression model using scikit-learn
      1672. -
    1672.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1673. -
    1673.    Organizing our data
      1674. -
    1674.    Seeing the wood for the trees
      1675. -
    1675.    And what about using neural networks?
      1676. -
    1676. A first summary
      1677. -
    1677. Why Linear Regression (aka Ordinary Least Squares and family)
      1678. -
    1678. Regression analysis, overarching aims
      1679. -
    1679. Regression analysis, overarching aims II
      1680. -
    1680. Examples
      1681. -
    1681. General linear models
      1682. -
    1682. Rewriting the fitting procedure as a linear algebra problem
      1683. -
    1683. Rewriting the fitting procedure as a linear algebra problem, more details
      1684. -
    1684. Generalizing the fitting procedure as a linear algebra problem
      1685. -
    1685. Generalizing the fitting procedure as a linear algebra problem
      1686. -
    1686. Optimizing our parameters
      1687. -
    1687. Our model for the nuclear binding energies
      1688. -
    1688. Optimizing our parameters, more details
      1689. -
    1689. Interpretations and optimizing our parameters
      1690. -
    1690. Interpretations and optimizing our parameters
      1691. -
    1691. Some useful matrix and vector expressions
      1692. -
    1692. Interpretations and optimizing our parameters
      1693. -
    1693. Own code for Ordinary Least Squares
      1694. -
    1694. Adding error analysis and training set up
      1695. -
    1695. The \( \chi^2 \) function
      1696. -
    1696. The \( \chi^2 \) function
      1697. -
    1697. The \( \chi^2 \) function
      1698. -
    1698. The \( \chi^2 \) function
      1699. -
    1699. The \( \chi^2 \) function
      1700. -
    1700. The \( \chi^2 \) function
      1701. -
    1701. Fitting an Equation of State for Dense Nuclear Matter
      1702. -
    1702. The code
      1703. -
    1703. Splitting our Data in Training and Test data
      1704. -
    1704. Exercises
      1705. -
    1705. Exercise 1: Setting up various Python environments
      1706. -
    1706. Exercise 2: making your own data and exploring scikit-learn
      1707. -
    1707. Exercise 3: Normalizing our data
      1708. +
    1708. Numpy and arrays
      1709. +
    1709. Matrices in Python
      1710. +
    1710. Meet the Pandas
      1711. +
    1711.    Simple linear regression model using scikit-learn
      1712. +
    1712.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1713. +
    1713.    Organizing our data
      1714. +
    1714.    And what about using neural networks?
      1715. +
    1715. A first summary
      1716. +
    1716. Why Linear Regression (aka Ordinary Least Squares and family)
      1717. +
    1717. Regression analysis, overarching aims
      1718. +
    1718. Regression analysis, overarching aims II
      1719. +
    1719. Examples
      1720. +
    1720. General linear models
      1721. +
    1721. Rewriting the fitting procedure as a linear algebra problem
      1722. +
    1722. Rewriting the fitting procedure as a linear algebra problem, more details
      1723. +
    1723. Generalizing the fitting procedure as a linear algebra problem
      1724. +
    1724. Generalizing the fitting procedure as a linear algebra problem
      1725. +
    1725. Optimizing our parameters
      1726. +
    1726. Our model for the nuclear binding energies
      1727. +
    1727. Optimizing our parameters, more details
      1728. +
    1728. Interpretations and optimizing our parameters
      1729. +
    1729. Interpretations and optimizing our parameters
      1730. +
    1730. Some useful matrix and vector expressions
      1731. +
    1731. Interpretations and optimizing our parameters
      1732. +
    1732. Own code for Ordinary Least Squares
      1733. +
    1733. Adding error analysis and training set up
      1734. +
    1734. The \( \chi^2 \) function
      1735. +
    1735. The \( \chi^2 \) function
      1736. +
    1736. The \( \chi^2 \) function
      1737. +
    1737. The \( \chi^2 \) function
      1738. +
    1738. The \( \chi^2 \) function
      1739. +
    1739. The \( \chi^2 \) function
      1740. +
    1740. Fitting an Equation of State for Dense Nuclear Matter
      1741. +
    1741. The code
      1742. +
    1742. Splitting our Data in Training and Test data
      1743. +
    1743. Exercises
      1744. +
    1744. Exercise 1: Setting up various Python environments
      1745. +
    1745. Exercise 2: making your own data and exploring scikit-learn
      1746. +
    1746. Exercise 3: Split data in test and training data
      1747. @@ -380,7 +364,7 @@ MathJax.Hub.Config({
    1747. 29
    1748. 30
    1749. ...
      1750. -
    1750. 66
      1751. +
    1751. 62
    1752. »
      1753. diff --git a/doc/pub/week34/html/._week34-bs021.html b/doc/pub/week34/html/._week34-bs021.html index 09005b222..f7e277367 100644 --- a/doc/pub/week34/html/._week34-bs021.html +++ b/doc/pub/week34/html/._week34-bs021.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1753. Installing R, C++, cython or Julia
    1754. Installing R, C++, cython, Numba etc
    1755. Numpy examples and Important Matrix and vector handling packages
      1756. -
    1756. Basic Matrix Features
      1757. -
    1757.    Some famous Matrices
      1758. -
    1758.    More Basic Matrix Features
      1759. -
    1759. Numpy and arrays
      1760. -
    1760. Matrices in Python
      1761. -
    1761. Meet the Pandas
      1762. -
    1762. Friday August 27
      1763. -
    1763.    Simple linear regression model using scikit-learn
      1764. -
    1764.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1765. -
    1765.    Organizing our data
      1766. -
    1766.    Seeing the wood for the trees
      1767. -
    1767.    And what about using neural networks?
      1768. -
    1768. A first summary
      1769. -
    1769. Why Linear Regression (aka Ordinary Least Squares and family)
      1770. -
    1770. Regression analysis, overarching aims
      1771. -
    1771. Regression analysis, overarching aims II
      1772. -
    1772. Examples
      1773. -
    1773. General linear models
      1774. -
    1774. Rewriting the fitting procedure as a linear algebra problem
      1775. -
    1775. Rewriting the fitting procedure as a linear algebra problem, more details
      1776. -
    1776. Generalizing the fitting procedure as a linear algebra problem
      1777. -
    1777. Generalizing the fitting procedure as a linear algebra problem
      1778. -
    1778. Optimizing our parameters
      1779. -
    1779. Our model for the nuclear binding energies
      1780. -
    1780. Optimizing our parameters, more details
      1781. -
    1781. Interpretations and optimizing our parameters
      1782. -
    1782. Interpretations and optimizing our parameters
      1783. -
    1783. Some useful matrix and vector expressions
      1784. -
    1784. Interpretations and optimizing our parameters
      1785. -
    1785. Own code for Ordinary Least Squares
      1786. -
    1786. Adding error analysis and training set up
      1787. -
    1787. The \( \chi^2 \) function
      1788. -
    1788. The \( \chi^2 \) function
      1789. -
    1789. The \( \chi^2 \) function
      1790. -
    1790. The \( \chi^2 \) function
      1791. -
    1791. The \( \chi^2 \) function
      1792. -
    1792. The \( \chi^2 \) function
      1793. -
    1793. Fitting an Equation of State for Dense Nuclear Matter
      1794. -
    1794. The code
      1795. -
    1795. Splitting our Data in Training and Test data
      1796. -
    1796. Exercises
      1797. -
    1797. Exercise 1: Setting up various Python environments
      1798. -
    1798. Exercise 2: making your own data and exploring scikit-learn
      1799. -
    1799. Exercise 3: Normalizing our data
      1800. +
    1800. Numpy and arrays
      1801. +
    1801. Matrices in Python
      1802. +
    1802. Meet the Pandas
      1803. +
    1803.    Simple linear regression model using scikit-learn
      1804. +
    1804.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1805. +
    1805.    Organizing our data
      1806. +
    1806.    And what about using neural networks?
      1807. +
    1807. A first summary
      1808. +
    1808. Why Linear Regression (aka Ordinary Least Squares and family)
      1809. +
    1809. Regression analysis, overarching aims
      1810. +
    1810. Regression analysis, overarching aims II
      1811. +
    1811. Examples
      1812. +
    1812. General linear models
      1813. +
    1813. Rewriting the fitting procedure as a linear algebra problem
      1814. +
    1814. Rewriting the fitting procedure as a linear algebra problem, more details
      1815. +
    1815. Generalizing the fitting procedure as a linear algebra problem
      1816. +
    1816. Generalizing the fitting procedure as a linear algebra problem
      1817. +
    1817. Optimizing our parameters
      1818. +
    1818. Our model for the nuclear binding energies
      1819. +
    1819. Optimizing our parameters, more details
      1820. +
    1820. Interpretations and optimizing our parameters
      1821. +
    1821. Interpretations and optimizing our parameters
      1822. +
    1822. Some useful matrix and vector expressions
      1823. +
    1823. Interpretations and optimizing our parameters
      1824. +
    1824. Own code for Ordinary Least Squares
      1825. +
    1825. Adding error analysis and training set up
      1826. +
    1826. The \( \chi^2 \) function
      1827. +
    1827. The \( \chi^2 \) function
      1828. +
    1828. The \( \chi^2 \) function
      1829. +
    1829. The \( \chi^2 \) function
      1830. +
    1830. The \( \chi^2 \) function
      1831. +
    1831. The \( \chi^2 \) function
      1832. +
    1832. Fitting an Equation of State for Dense Nuclear Matter
      1833. +
    1833. The code
      1834. +
    1834. Splitting our Data in Training and Test data
      1835. +
    1835. Exercises
      1836. +
    1836. Exercise 1: Setting up various Python environments
      1837. +
    1837. Exercise 2: making your own data and exploring scikit-learn
      1838. +
    1838. Exercise 3: Split data in test and training data
      1839. @@ -406,7 +390,7 @@ what is the likelihood of finding \( B \).
    1839. 30
    1840. 31
    1841. ...
      1842. -
    1842. 66
      1843. +
    1843. 62
    1844. »
      1845. diff --git a/doc/pub/week34/html/._week34-bs022.html b/doc/pub/week34/html/._week34-bs022.html index e42624982..79bcf29c2 100644 --- a/doc/pub/week34/html/._week34-bs022.html +++ b/doc/pub/week34/html/._week34-bs022.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1845. Installing R, C++, cython or Julia
    1846. Installing R, C++, cython, Numba etc
    1847. Numpy examples and Important Matrix and vector handling packages
      1848. -
    1848. Basic Matrix Features
      1849. -
    1849.    Some famous Matrices
      1850. -
    1850.    More Basic Matrix Features
      1851. -
    1851. Numpy and arrays
      1852. -
    1852. Matrices in Python
      1853. -
    1853. Meet the Pandas
      1854. -
    1854. Friday August 27
      1855. -
    1855.    Simple linear regression model using scikit-learn
      1856. -
    1856.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1857. -
    1857.    Organizing our data
      1858. -
    1858.    Seeing the wood for the trees
      1859. -
    1859.    And what about using neural networks?
      1860. -
    1860. A first summary
      1861. -
    1861. Why Linear Regression (aka Ordinary Least Squares and family)
      1862. -
    1862. Regression analysis, overarching aims
      1863. -
    1863. Regression analysis, overarching aims II
      1864. -
    1864. Examples
      1865. -
    1865. General linear models
      1866. -
    1866. Rewriting the fitting procedure as a linear algebra problem
      1867. -
    1867. Rewriting the fitting procedure as a linear algebra problem, more details
      1868. -
    1868. Generalizing the fitting procedure as a linear algebra problem
      1869. -
    1869. Generalizing the fitting procedure as a linear algebra problem
      1870. -
    1870. Optimizing our parameters
      1871. -
    1871. Our model for the nuclear binding energies
      1872. -
    1872. Optimizing our parameters, more details
      1873. -
    1873. Interpretations and optimizing our parameters
      1874. -
    1874. Interpretations and optimizing our parameters
      1875. -
    1875. Some useful matrix and vector expressions
      1876. -
    1876. Interpretations and optimizing our parameters
      1877. -
    1877. Own code for Ordinary Least Squares
      1878. -
    1878. Adding error analysis and training set up
      1879. -
    1879. The \( \chi^2 \) function
      1880. -
    1880. The \( \chi^2 \) function
      1881. -
    1881. The \( \chi^2 \) function
      1882. -
    1882. The \( \chi^2 \) function
      1883. -
    1883. The \( \chi^2 \) function
      1884. -
    1884. The \( \chi^2 \) function
      1885. -
    1885. Fitting an Equation of State for Dense Nuclear Matter
      1886. -
    1886. The code
      1887. -
    1887. Splitting our Data in Training and Test data
      1888. -
    1888. Exercises
      1889. -
    1889. Exercise 1: Setting up various Python environments
      1890. -
    1890. Exercise 2: making your own data and exploring scikit-learn
      1891. -
    1891. Exercise 3: Normalizing our data
      1892. +
    1892. Numpy and arrays
      1893. +
    1893. Matrices in Python
      1894. +
    1894. Meet the Pandas
      1895. +
    1895.    Simple linear regression model using scikit-learn
      1896. +
    1896.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1897. +
    1897.    Organizing our data
      1898. +
    1898.    And what about using neural networks?
      1899. +
    1899. A first summary
      1900. +
    1900. Why Linear Regression (aka Ordinary Least Squares and family)
      1901. +
    1901. Regression analysis, overarching aims
      1902. +
    1902. Regression analysis, overarching aims II
      1903. +
    1903. Examples
      1904. +
    1904. General linear models
      1905. +
    1905. Rewriting the fitting procedure as a linear algebra problem
      1906. +
    1906. Rewriting the fitting procedure as a linear algebra problem, more details
      1907. +
    1907. Generalizing the fitting procedure as a linear algebra problem
      1908. +
    1908. Generalizing the fitting procedure as a linear algebra problem
      1909. +
    1909. Optimizing our parameters
      1910. +
    1910. Our model for the nuclear binding energies
      1911. +
    1911. Optimizing our parameters, more details
      1912. +
    1912. Interpretations and optimizing our parameters
      1913. +
    1913. Interpretations and optimizing our parameters
      1914. +
    1914. Some useful matrix and vector expressions
      1915. +
    1915. Interpretations and optimizing our parameters
      1916. +
    1916. Own code for Ordinary Least Squares
      1917. +
    1917. Adding error analysis and training set up
      1918. +
    1918. The \( \chi^2 \) function
      1919. +
    1919. The \( \chi^2 \) function
      1920. +
    1920. The \( \chi^2 \) function
      1921. +
    1921. The \( \chi^2 \) function
      1922. +
    1922. The \( \chi^2 \) function
      1923. +
    1923. The \( \chi^2 \) function
      1924. +
    1924. Fitting an Equation of State for Dense Nuclear Matter
      1925. +
    1925. The code
      1926. +
    1926. Splitting our Data in Training and Test data
      1927. +
    1927. Exercises
      1928. +
    1928. Exercise 1: Setting up various Python environments
      1929. +
    1929. Exercise 2: making your own data and exploring scikit-learn
      1930. +
    1930. Exercise 3: Split data in test and training data
      1931. @@ -403,7 +387,7 @@ could easily be many different models that fit the given data set equally we
    1931. 31
    1932. 32
    1933. ...
      1934. -
    1934. 66
      1935. +
    1935. 62
    1936. »
      1937. diff --git a/doc/pub/week34/html/._week34-bs023.html b/doc/pub/week34/html/._week34-bs023.html index 0ea4198fe..1549569bc 100644 --- a/doc/pub/week34/html/._week34-bs023.html +++ b/doc/pub/week34/html/._week34-bs023.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    1937. Installing R, C++, cython or Julia
    1938. Installing R, C++, cython, Numba etc
    1939. Numpy examples and Important Matrix and vector handling packages
      1940. -
    1940. Basic Matrix Features
      1941. -
    1941.    Some famous Matrices
      1942. -
    1942.    More Basic Matrix Features
      1943. -
    1943. Numpy and arrays
      1944. -
    1944. Matrices in Python
      1945. -
    1945. Meet the Pandas
      1946. -
    1946. Friday August 27
      1947. -
    1947.    Simple linear regression model using scikit-learn
      1948. -
    1948.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1949. -
    1949.    Organizing our data
      1950. -
    1950.    Seeing the wood for the trees
      1951. -
    1951.    And what about using neural networks?
      1952. -
    1952. A first summary
      1953. -
    1953. Why Linear Regression (aka Ordinary Least Squares and family)
      1954. -
    1954. Regression analysis, overarching aims
      1955. -
    1955. Regression analysis, overarching aims II
      1956. -
    1956. Examples
      1957. -
    1957. General linear models
      1958. -
    1958. Rewriting the fitting procedure as a linear algebra problem
      1959. -
    1959. Rewriting the fitting procedure as a linear algebra problem, more details
      1960. -
    1960. Generalizing the fitting procedure as a linear algebra problem
      1961. -
    1961. Generalizing the fitting procedure as a linear algebra problem
      1962. -
    1962. Optimizing our parameters
      1963. -
    1963. Our model for the nuclear binding energies
      1964. -
    1964. Optimizing our parameters, more details
      1965. -
    1965. Interpretations and optimizing our parameters
      1966. -
    1966. Interpretations and optimizing our parameters
      1967. -
    1967. Some useful matrix and vector expressions
      1968. -
    1968. Interpretations and optimizing our parameters
      1969. -
    1969. Own code for Ordinary Least Squares
      1970. -
    1970. Adding error analysis and training set up
      1971. -
    1971. The \( \chi^2 \) function
      1972. -
    1972. The \( \chi^2 \) function
      1973. -
    1973. The \( \chi^2 \) function
      1974. -
    1974. The \( \chi^2 \) function
      1975. -
    1975. The \( \chi^2 \) function
      1976. -
    1976. The \( \chi^2 \) function
      1977. -
    1977. Fitting an Equation of State for Dense Nuclear Matter
      1978. -
    1978. The code
      1979. -
    1979. Splitting our Data in Training and Test data
      1980. -
    1980. Exercises
      1981. -
    1981. Exercise 1: Setting up various Python environments
      1982. -
    1982. Exercise 2: making your own data and exploring scikit-learn
      1983. -
    1983. Exercise 3: Normalizing our data
      1984. +
    1984. Numpy and arrays
      1985. +
    1985. Matrices in Python
      1986. +
    1986. Meet the Pandas
      1987. +
    1987.    Simple linear regression model using scikit-learn
      1988. +
    1988.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      1989. +
    1989.    Organizing our data
      1990. +
    1990.    And what about using neural networks?
      1991. +
    1991. A first summary
      1992. +
    1992. Why Linear Regression (aka Ordinary Least Squares and family)
      1993. +
    1993. Regression analysis, overarching aims
      1994. +
    1994. Regression analysis, overarching aims II
      1995. +
    1995. Examples
      1996. +
    1996. General linear models
      1997. +
    1997. Rewriting the fitting procedure as a linear algebra problem
      1998. +
    1998. Rewriting the fitting procedure as a linear algebra problem, more details
      1999. +
    1999. Generalizing the fitting procedure as a linear algebra problem
      2000. +
    2000. Generalizing the fitting procedure as a linear algebra problem
      2001. +
    2001. Optimizing our parameters
      2002. +
    2002. Our model for the nuclear binding energies
      2003. +
    2003. Optimizing our parameters, more details
      2004. +
    2004. Interpretations and optimizing our parameters
      2005. +
    2005. Interpretations and optimizing our parameters
      2006. +
    2006. Some useful matrix and vector expressions
      2007. +
    2007. Interpretations and optimizing our parameters
      2008. +
    2008. Own code for Ordinary Least Squares
      2009. +
    2009. Adding error analysis and training set up
      2010. +
    2010. The \( \chi^2 \) function
      2011. +
    2011. The \( \chi^2 \) function
      2012. +
    2012. The \( \chi^2 \) function
      2013. +
    2013. The \( \chi^2 \) function
      2014. +
    2014. The \( \chi^2 \) function
      2015. +
    2015. The \( \chi^2 \) function
      2016. +
    2016. Fitting an Equation of State for Dense Nuclear Matter
      2017. +
    2017. The code
      2018. +
    2018. Splitting our Data in Training and Test data
      2019. +
    2019. Exercises
      2020. +
    2020. Exercise 1: Setting up various Python environments
      2021. +
    2021. Exercise 2: making your own data and exploring scikit-learn
      2022. +
    2022. Exercise 3: Split data in test and training data
      2023. @@ -401,7 +385,7 @@ may first try the simplest class of models, namely linear models, followed obvio
    2023. 32
    2024. 33
    2025. ...
      2026. -
    2026. 66
      2027. +
    2027. 62
    2028. »
      2029. diff --git a/doc/pub/week34/html/._week34-bs024.html b/doc/pub/week34/html/._week34-bs024.html index 163cccaeb..8a8526070 100644 --- a/doc/pub/week34/html/._week34-bs024.html +++ b/doc/pub/week34/html/._week34-bs024.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2029. Installing R, C++, cython or Julia
    2030. Installing R, C++, cython, Numba etc
    2031. Numpy examples and Important Matrix and vector handling packages
      2032. -
    2032. Basic Matrix Features
      2033. -
    2033.    Some famous Matrices
      2034. -
    2034.    More Basic Matrix Features
      2035. -
    2035. Numpy and arrays
      2036. -
    2036. Matrices in Python
      2037. -
    2037. Meet the Pandas
      2038. -
    2038. Friday August 27
      2039. -
    2039.    Simple linear regression model using scikit-learn
      2040. -
    2040.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2041. -
    2041.    Organizing our data
      2042. -
    2042.    Seeing the wood for the trees
      2043. -
    2043.    And what about using neural networks?
      2044. -
    2044. A first summary
      2045. -
    2045. Why Linear Regression (aka Ordinary Least Squares and family)
      2046. -
    2046. Regression analysis, overarching aims
      2047. -
    2047. Regression analysis, overarching aims II
      2048. -
    2048. Examples
      2049. -
    2049. General linear models
      2050. -
    2050. Rewriting the fitting procedure as a linear algebra problem
      2051. -
    2051. Rewriting the fitting procedure as a linear algebra problem, more details
      2052. -
    2052. Generalizing the fitting procedure as a linear algebra problem
      2053. -
    2053. Generalizing the fitting procedure as a linear algebra problem
      2054. -
    2054. Optimizing our parameters
      2055. -
    2055. Our model for the nuclear binding energies
      2056. -
    2056. Optimizing our parameters, more details
      2057. -
    2057. Interpretations and optimizing our parameters
      2058. -
    2058. Interpretations and optimizing our parameters
      2059. -
    2059. Some useful matrix and vector expressions
      2060. -
    2060. Interpretations and optimizing our parameters
      2061. -
    2061. Own code for Ordinary Least Squares
      2062. -
    2062. Adding error analysis and training set up
      2063. -
    2063. The \( \chi^2 \) function
      2064. -
    2064. The \( \chi^2 \) function
      2065. -
    2065. The \( \chi^2 \) function
      2066. -
    2066. The \( \chi^2 \) function
      2067. -
    2067. The \( \chi^2 \) function
      2068. -
    2068. The \( \chi^2 \) function
      2069. -
    2069. Fitting an Equation of State for Dense Nuclear Matter
      2070. -
    2070. The code
      2071. -
    2071. Splitting our Data in Training and Test data
      2072. -
    2072. Exercises
      2073. -
    2073. Exercise 1: Setting up various Python environments
      2074. -
    2074. Exercise 2: making your own data and exploring scikit-learn
      2075. -
    2075. Exercise 3: Normalizing our data
      2076. +
    2076. Numpy and arrays
      2077. +
    2077. Matrices in Python
      2078. +
    2078. Meet the Pandas
      2079. +
    2079.    Simple linear regression model using scikit-learn
      2080. +
    2080.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2081. +
    2081.    Organizing our data
      2082. +
    2082.    And what about using neural networks?
      2083. +
    2083. A first summary
      2084. +
    2084. Why Linear Regression (aka Ordinary Least Squares and family)
      2085. +
    2085. Regression analysis, overarching aims
      2086. +
    2086. Regression analysis, overarching aims II
      2087. +
    2087. Examples
      2088. +
    2088. General linear models
      2089. +
    2089. Rewriting the fitting procedure as a linear algebra problem
      2090. +
    2090. Rewriting the fitting procedure as a linear algebra problem, more details
      2091. +
    2091. Generalizing the fitting procedure as a linear algebra problem
      2092. +
    2092. Generalizing the fitting procedure as a linear algebra problem
      2093. +
    2093. Optimizing our parameters
      2094. +
    2094. Our model for the nuclear binding energies
      2095. +
    2095. Optimizing our parameters, more details
      2096. +
    2096. Interpretations and optimizing our parameters
      2097. +
    2097. Interpretations and optimizing our parameters
      2098. +
    2098. Some useful matrix and vector expressions
      2099. +
    2099. Interpretations and optimizing our parameters
      2100. +
    2100. Own code for Ordinary Least Squares
      2101. +
    2101. Adding error analysis and training set up
      2102. +
    2102. The \( \chi^2 \) function
      2103. +
    2103. The \( \chi^2 \) function
      2104. +
    2104. The \( \chi^2 \) function
      2105. +
    2105. The \( \chi^2 \) function
      2106. +
    2106. The \( \chi^2 \) function
      2107. +
    2107. The \( \chi^2 \) function
      2108. +
    2108. Fitting an Equation of State for Dense Nuclear Matter
      2109. +
    2109. The code
      2110. +
    2110. Splitting our Data in Training and Test data
      2111. +
    2111. Exercises
      2112. +
    2112. Exercise 1: Setting up various Python environments
      2113. +
    2113. Exercise 2: making your own data and exploring scikit-learn
      2114. +
    2114. Exercise 3: Split data in test and training data
      2115. @@ -414,7 +398,7 @@ you can use pip as well and simply install Python as
    2115. 33
    2116. 34
    2117. ...
      2118. -
    2118. 66
      2119. +
    2119. 62
    2120. »
      2121. diff --git a/doc/pub/week34/html/._week34-bs025.html b/doc/pub/week34/html/._week34-bs025.html index 0cedc4673..f25a18b9e 100644 --- a/doc/pub/week34/html/._week34-bs025.html +++ b/doc/pub/week34/html/._week34-bs025.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2121. Installing R, C++, cython or Julia
    2122. Installing R, C++, cython, Numba etc
    2123. Numpy examples and Important Matrix and vector handling packages
      2124. -
    2124. Basic Matrix Features
      2125. -
    2125.    Some famous Matrices
      2126. -
    2126.    More Basic Matrix Features
      2127. -
    2127. Numpy and arrays
      2128. -
    2128. Matrices in Python
      2129. -
    2129. Meet the Pandas
      2130. -
    2130. Friday August 27
      2131. -
    2131.    Simple linear regression model using scikit-learn
      2132. -
    2132.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2133. -
    2133.    Organizing our data
      2134. -
    2134.    Seeing the wood for the trees
      2135. -
    2135.    And what about using neural networks?
      2136. -
    2136. A first summary
      2137. -
    2137. Why Linear Regression (aka Ordinary Least Squares and family)
      2138. -
    2138. Regression analysis, overarching aims
      2139. -
    2139. Regression analysis, overarching aims II
      2140. -
    2140. Examples
      2141. -
    2141. General linear models
      2142. -
    2142. Rewriting the fitting procedure as a linear algebra problem
      2143. -
    2143. Rewriting the fitting procedure as a linear algebra problem, more details
      2144. -
    2144. Generalizing the fitting procedure as a linear algebra problem
      2145. -
    2145. Generalizing the fitting procedure as a linear algebra problem
      2146. -
    2146. Optimizing our parameters
      2147. -
    2147. Our model for the nuclear binding energies
      2148. -
    2148. Optimizing our parameters, more details
      2149. -
    2149. Interpretations and optimizing our parameters
      2150. -
    2150. Interpretations and optimizing our parameters
      2151. -
    2151. Some useful matrix and vector expressions
      2152. -
    2152. Interpretations and optimizing our parameters
      2153. -
    2153. Own code for Ordinary Least Squares
      2154. -
    2154. Adding error analysis and training set up
      2155. -
    2155. The \( \chi^2 \) function
      2156. -
    2156. The \( \chi^2 \) function
      2157. -
    2157. The \( \chi^2 \) function
      2158. -
    2158. The \( \chi^2 \) function
      2159. -
    2159. The \( \chi^2 \) function
      2160. -
    2160. The \( \chi^2 \) function
      2161. -
    2161. Fitting an Equation of State for Dense Nuclear Matter
      2162. -
    2162. The code
      2163. -
    2163. Splitting our Data in Training and Test data
      2164. -
    2164. Exercises
      2165. -
    2165. Exercise 1: Setting up various Python environments
      2166. -
    2166. Exercise 2: making your own data and exploring scikit-learn
      2167. -
    2167. Exercise 3: Normalizing our data
      2168. +
    2168. Numpy and arrays
      2169. +
    2169. Matrices in Python
      2170. +
    2170. Meet the Pandas
      2171. +
    2171.    Simple linear regression model using scikit-learn
      2172. +
    2172.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2173. +
    2173.    Organizing our data
      2174. +
    2174.    And what about using neural networks?
      2175. +
    2175. A first summary
      2176. +
    2176. Why Linear Regression (aka Ordinary Least Squares and family)
      2177. +
    2177. Regression analysis, overarching aims
      2178. +
    2178. Regression analysis, overarching aims II
      2179. +
    2179. Examples
      2180. +
    2180. General linear models
      2181. +
    2181. Rewriting the fitting procedure as a linear algebra problem
      2182. +
    2182. Rewriting the fitting procedure as a linear algebra problem, more details
      2183. +
    2183. Generalizing the fitting procedure as a linear algebra problem
      2184. +
    2184. Generalizing the fitting procedure as a linear algebra problem
      2185. +
    2185. Optimizing our parameters
      2186. +
    2186. Our model for the nuclear binding energies
      2187. +
    2187. Optimizing our parameters, more details
      2188. +
    2188. Interpretations and optimizing our parameters
      2189. +
    2189. Interpretations and optimizing our parameters
      2190. +
    2190. Some useful matrix and vector expressions
      2191. +
    2191. Interpretations and optimizing our parameters
      2192. +
    2192. Own code for Ordinary Least Squares
      2193. +
    2193. Adding error analysis and training set up
      2194. +
    2194. The \( \chi^2 \) function
      2195. +
    2195. The \( \chi^2 \) function
      2196. +
    2196. The \( \chi^2 \) function
      2197. +
    2197. The \( \chi^2 \) function
      2198. +
    2198. The \( \chi^2 \) function
      2199. +
    2199. The \( \chi^2 \) function
      2200. +
    2200. Fitting an Equation of State for Dense Nuclear Matter
      2201. +
    2201. The code
      2202. +
    2202. Splitting our Data in Training and Test data
      2203. +
    2203. Exercises
      2204. +
    2204. Exercise 1: Setting up various Python environments
      2205. +
    2205. Exercise 2: making your own data and exploring scikit-learn
      2206. +
    2206. Exercise 3: Split data in test and training data
      2207. @@ -407,7 +391,7 @@ no setup and runs entirely in the cloud. Try it out!
    2207. 34
    2208. 35
    2209. ...
      2210. -
    2210. 66
      2211. +
    2211. 62
    2212. »
      2213. diff --git a/doc/pub/week34/html/._week34-bs026.html b/doc/pub/week34/html/._week34-bs026.html index af49df0fe..d1ed46eeb 100644 --- a/doc/pub/week34/html/._week34-bs026.html +++ b/doc/pub/week34/html/._week34-bs026.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2213. Installing R, C++, cython or Julia
    2214. Installing R, C++, cython, Numba etc
    2215. Numpy examples and Important Matrix and vector handling packages
      2216. -
    2216. Basic Matrix Features
      2217. -
    2217.    Some famous Matrices
      2218. -
    2218.    More Basic Matrix Features
      2219. -
    2219. Numpy and arrays
      2220. -
    2220. Matrices in Python
      2221. -
    2221. Meet the Pandas
      2222. -
    2222. Friday August 27
      2223. -
    2223.    Simple linear regression model using scikit-learn
      2224. -
    2224.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2225. -
    2225.    Organizing our data
      2226. -
    2226.    Seeing the wood for the trees
      2227. -
    2227.    And what about using neural networks?
      2228. -
    2228. A first summary
      2229. -
    2229. Why Linear Regression (aka Ordinary Least Squares and family)
      2230. -
    2230. Regression analysis, overarching aims
      2231. -
    2231. Regression analysis, overarching aims II
      2232. -
    2232. Examples
      2233. -
    2233. General linear models
      2234. -
    2234. Rewriting the fitting procedure as a linear algebra problem
      2235. -
    2235. Rewriting the fitting procedure as a linear algebra problem, more details
      2236. -
    2236. Generalizing the fitting procedure as a linear algebra problem
      2237. -
    2237. Generalizing the fitting procedure as a linear algebra problem
      2238. -
    2238. Optimizing our parameters
      2239. -
    2239. Our model for the nuclear binding energies
      2240. -
    2240. Optimizing our parameters, more details
      2241. -
    2241. Interpretations and optimizing our parameters
      2242. -
    2242. Interpretations and optimizing our parameters
      2243. -
    2243. Some useful matrix and vector expressions
      2244. -
    2244. Interpretations and optimizing our parameters
      2245. -
    2245. Own code for Ordinary Least Squares
      2246. -
    2246. Adding error analysis and training set up
      2247. -
    2247. The \( \chi^2 \) function
      2248. -
    2248. The \( \chi^2 \) function
      2249. -
    2249. The \( \chi^2 \) function
      2250. -
    2250. The \( \chi^2 \) function
      2251. -
    2251. The \( \chi^2 \) function
      2252. -
    2252. The \( \chi^2 \) function
      2253. -
    2253. Fitting an Equation of State for Dense Nuclear Matter
      2254. -
    2254. The code
      2255. -
    2255. Splitting our Data in Training and Test data
      2256. -
    2256. Exercises
      2257. -
    2257. Exercise 1: Setting up various Python environments
      2258. -
    2258. Exercise 2: making your own data and exploring scikit-learn
      2259. -
    2259. Exercise 3: Normalizing our data
      2260. +
    2260. Numpy and arrays
      2261. +
    2261. Matrices in Python
      2262. +
    2262. Meet the Pandas
      2263. +
    2263.    Simple linear regression model using scikit-learn
      2264. +
    2264.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2265. +
    2265.    Organizing our data
      2266. +
    2266.    And what about using neural networks?
      2267. +
    2267. A first summary
      2268. +
    2268. Why Linear Regression (aka Ordinary Least Squares and family)
      2269. +
    2269. Regression analysis, overarching aims
      2270. +
    2270. Regression analysis, overarching aims II
      2271. +
    2271. Examples
      2272. +
    2272. General linear models
      2273. +
    2273. Rewriting the fitting procedure as a linear algebra problem
      2274. +
    2274. Rewriting the fitting procedure as a linear algebra problem, more details
      2275. +
    2275. Generalizing the fitting procedure as a linear algebra problem
      2276. +
    2276. Generalizing the fitting procedure as a linear algebra problem
      2277. +
    2277. Optimizing our parameters
      2278. +
    2278. Our model for the nuclear binding energies
      2279. +
    2279. Optimizing our parameters, more details
      2280. +
    2280. Interpretations and optimizing our parameters
      2281. +
    2281. Interpretations and optimizing our parameters
      2282. +
    2282. Some useful matrix and vector expressions
      2283. +
    2283. Interpretations and optimizing our parameters
      2284. +
    2284. Own code for Ordinary Least Squares
      2285. +
    2285. Adding error analysis and training set up
      2286. +
    2286. The \( \chi^2 \) function
      2287. +
    2287. The \( \chi^2 \) function
      2288. +
    2288. The \( \chi^2 \) function
      2289. +
    2289. The \( \chi^2 \) function
      2290. +
    2290. The \( \chi^2 \) function
      2291. +
    2291. The \( \chi^2 \) function
      2292. +
    2292. Fitting an Equation of State for Dense Nuclear Matter
      2293. +
    2293. The code
      2294. +
    2294. Splitting our Data in Training and Test data
      2295. +
    2295. Exercises
      2296. +
    2296. Exercise 1: Setting up various Python environments
      2297. +
    2297. Exercise 2: making your own data and exploring scikit-learn
      2298. +
    2298. Exercise 3: Split data in test and training data
      2299. @@ -393,7 +377,7 @@ MathJax.Hub.Config({
    2299. 35
    2300. 36
    2301. ...
      2302. -
    2302. 66
      2303. +
    2303. 62
    2304. »
      2305. diff --git a/doc/pub/week34/html/._week34-bs027.html b/doc/pub/week34/html/._week34-bs027.html index c91007472..d1aff13da 100644 --- a/doc/pub/week34/html/._week34-bs027.html +++ b/doc/pub/week34/html/._week34-bs027.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2305. Installing R, C++, cython or Julia
    2306. Installing R, C++, cython, Numba etc
    2307. Numpy examples and Important Matrix and vector handling packages
      2308. -
    2308. Basic Matrix Features
      2309. -
    2309.    Some famous Matrices
      2310. -
    2310.    More Basic Matrix Features
      2311. -
    2311. Numpy and arrays
      2312. -
    2312. Matrices in Python
      2313. -
    2313. Meet the Pandas
      2314. -
    2314. Friday August 27
      2315. -
    2315.    Simple linear regression model using scikit-learn
      2316. -
    2316.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2317. -
    2317.    Organizing our data
      2318. -
    2318.    Seeing the wood for the trees
      2319. -
    2319.    And what about using neural networks?
      2320. -
    2320. A first summary
      2321. -
    2321. Why Linear Regression (aka Ordinary Least Squares and family)
      2322. -
    2322. Regression analysis, overarching aims
      2323. -
    2323. Regression analysis, overarching aims II
      2324. -
    2324. Examples
      2325. -
    2325. General linear models
      2326. -
    2326. Rewriting the fitting procedure as a linear algebra problem
      2327. -
    2327. Rewriting the fitting procedure as a linear algebra problem, more details
      2328. -
    2328. Generalizing the fitting procedure as a linear algebra problem
      2329. -
    2329. Generalizing the fitting procedure as a linear algebra problem
      2330. -
    2330. Optimizing our parameters
      2331. -
    2331. Our model for the nuclear binding energies
      2332. -
    2332. Optimizing our parameters, more details
      2333. -
    2333. Interpretations and optimizing our parameters
      2334. -
    2334. Interpretations and optimizing our parameters
      2335. -
    2335. Some useful matrix and vector expressions
      2336. -
    2336. Interpretations and optimizing our parameters
      2337. -
    2337. Own code for Ordinary Least Squares
      2338. -
    2338. Adding error analysis and training set up
      2339. -
    2339. The \( \chi^2 \) function
      2340. -
    2340. The \( \chi^2 \) function
      2341. -
    2341. The \( \chi^2 \) function
      2342. -
    2342. The \( \chi^2 \) function
      2343. -
    2343. The \( \chi^2 \) function
      2344. -
    2344. The \( \chi^2 \) function
      2345. -
    2345. Fitting an Equation of State for Dense Nuclear Matter
      2346. -
    2346. The code
      2347. -
    2347. Splitting our Data in Training and Test data
      2348. -
    2348. Exercises
      2349. -
    2349. Exercise 1: Setting up various Python environments
      2350. -
    2350. Exercise 2: making your own data and exploring scikit-learn
      2351. -
    2351. Exercise 3: Normalizing our data
      2352. +
    2352. Numpy and arrays
      2353. +
    2353. Matrices in Python
      2354. +
    2354. Meet the Pandas
      2355. +
    2355.    Simple linear regression model using scikit-learn
      2356. +
    2356.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2357. +
    2357.    Organizing our data
      2358. +
    2358.    And what about using neural networks?
      2359. +
    2359. A first summary
      2360. +
    2360. Why Linear Regression (aka Ordinary Least Squares and family)
      2361. +
    2361. Regression analysis, overarching aims
      2362. +
    2362. Regression analysis, overarching aims II
      2363. +
    2363. Examples
      2364. +
    2364. General linear models
      2365. +
    2365. Rewriting the fitting procedure as a linear algebra problem
      2366. +
    2366. Rewriting the fitting procedure as a linear algebra problem, more details
      2367. +
    2367. Generalizing the fitting procedure as a linear algebra problem
      2368. +
    2368. Generalizing the fitting procedure as a linear algebra problem
      2369. +
    2369. Optimizing our parameters
      2370. +
    2370. Our model for the nuclear binding energies
      2371. +
    2371. Optimizing our parameters, more details
      2372. +
    2372. Interpretations and optimizing our parameters
      2373. +
    2373. Interpretations and optimizing our parameters
      2374. +
    2374. Some useful matrix and vector expressions
      2375. +
    2375. Interpretations and optimizing our parameters
      2376. +
    2376. Own code for Ordinary Least Squares
      2377. +
    2377. Adding error analysis and training set up
      2378. +
    2378. The \( \chi^2 \) function
      2379. +
    2379. The \( \chi^2 \) function
      2380. +
    2380. The \( \chi^2 \) function
      2381. +
    2381. The \( \chi^2 \) function
      2382. +
    2382. The \( \chi^2 \) function
      2383. +
    2383. The \( \chi^2 \) function
      2384. +
    2384. Fitting an Equation of State for Dense Nuclear Matter
      2385. +
    2385. The code
      2386. +
    2386. Splitting our Data in Training and Test data
      2387. +
    2387. Exercises
      2388. +
    2388. Exercise 1: Setting up various Python environments
      2389. +
    2389. Exercise 2: making your own data and exploring scikit-learn
      2390. +
    2390. Exercise 3: Split data in test and training data
      2391. @@ -358,8 +342,8 @@ use Python during our lectures and in various projects and exercises. Those of you already familiar with R should feel free to continue using R, keeping however an eye on the parallel Python set ups. Similarly, if you are a -Python afecionado, feel free to explore R as well. Jupyter/Ipython -notebook allows you to run R codes interactively in your +Python afecionado, feel free to explore R as well. Jupyter(Julia, Python and R) /Ipython +notebook allows you to run R codes and Julia codes interactively in your browser. The software library R is really tailored for statistical data analysis and allows for an easy usage of the tools and algorithms we will discuss in these lectures. @@ -394,7 +378,7 @@ lectures.
    2391. 36
    2392. 37
    2393. ...
      2394. -
    2394. 66
      2395. +
    2395. 62
    2396. »
      2397. diff --git a/doc/pub/week34/html/._week34-bs028.html b/doc/pub/week34/html/._week34-bs028.html index 7eb6e2136..5d204ba8a 100644 --- a/doc/pub/week34/html/._week34-bs028.html +++ b/doc/pub/week34/html/._week34-bs028.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2397. Installing R, C++, cython or Julia
    2398. Installing R, C++, cython, Numba etc
    2399. Numpy examples and Important Matrix and vector handling packages
      2400. -
    2400. Basic Matrix Features
      2401. -
    2401.    Some famous Matrices
      2402. -
    2402.    More Basic Matrix Features
      2403. -
    2403. Numpy and arrays
      2404. -
    2404. Matrices in Python
      2405. -
    2405. Meet the Pandas
      2406. -
    2406. Friday August 27
      2407. -
    2407.    Simple linear regression model using scikit-learn
      2408. -
    2408.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2409. -
    2409.    Organizing our data
      2410. -
    2410.    Seeing the wood for the trees
      2411. -
    2411.    And what about using neural networks?
      2412. -
    2412. A first summary
      2413. -
    2413. Why Linear Regression (aka Ordinary Least Squares and family)
      2414. -
    2414. Regression analysis, overarching aims
      2415. -
    2415. Regression analysis, overarching aims II
      2416. -
    2416. Examples
      2417. -
    2417. General linear models
      2418. -
    2418. Rewriting the fitting procedure as a linear algebra problem
      2419. -
    2419. Rewriting the fitting procedure as a linear algebra problem, more details
      2420. -
    2420. Generalizing the fitting procedure as a linear algebra problem
      2421. -
    2421. Generalizing the fitting procedure as a linear algebra problem
      2422. -
    2422. Optimizing our parameters
      2423. -
    2423. Our model for the nuclear binding energies
      2424. -
    2424. Optimizing our parameters, more details
      2425. -
    2425. Interpretations and optimizing our parameters
      2426. -
    2426. Interpretations and optimizing our parameters
      2427. -
    2427. Some useful matrix and vector expressions
      2428. -
    2428. Interpretations and optimizing our parameters
      2429. -
    2429. Own code for Ordinary Least Squares
      2430. -
    2430. Adding error analysis and training set up
      2431. -
    2431. The \( \chi^2 \) function
      2432. -
    2432. The \( \chi^2 \) function
      2433. -
    2433. The \( \chi^2 \) function
      2434. -
    2434. The \( \chi^2 \) function
      2435. -
    2435. The \( \chi^2 \) function
      2436. -
    2436. The \( \chi^2 \) function
      2437. -
    2437. Fitting an Equation of State for Dense Nuclear Matter
      2438. -
    2438. The code
      2439. -
    2439. Splitting our Data in Training and Test data
      2440. -
    2440. Exercises
      2441. -
    2441. Exercise 1: Setting up various Python environments
      2442. -
    2442. Exercise 2: making your own data and exploring scikit-learn
      2443. -
    2443. Exercise 3: Normalizing our data
      2444. +
    2444. Numpy and arrays
      2445. +
    2445. Matrices in Python
      2446. +
    2446. Meet the Pandas
      2447. +
    2447.    Simple linear regression model using scikit-learn
      2448. +
    2448.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2449. +
    2449.    Organizing our data
      2450. +
    2450.    And what about using neural networks?
      2451. +
    2451. A first summary
      2452. +
    2452. Why Linear Regression (aka Ordinary Least Squares and family)
      2453. +
    2453. Regression analysis, overarching aims
      2454. +
    2454. Regression analysis, overarching aims II
      2455. +
    2455. Examples
      2456. +
    2456. General linear models
      2457. +
    2457. Rewriting the fitting procedure as a linear algebra problem
      2458. +
    2458. Rewriting the fitting procedure as a linear algebra problem, more details
      2459. +
    2459. Generalizing the fitting procedure as a linear algebra problem
      2460. +
    2460. Generalizing the fitting procedure as a linear algebra problem
      2461. +
    2461. Optimizing our parameters
      2462. +
    2462. Our model for the nuclear binding energies
      2463. +
    2463. Optimizing our parameters, more details
      2464. +
    2464. Interpretations and optimizing our parameters
      2465. +
    2465. Interpretations and optimizing our parameters
      2466. +
    2466. Some useful matrix and vector expressions
      2467. +
    2467. Interpretations and optimizing our parameters
      2468. +
    2468. Own code for Ordinary Least Squares
      2469. +
    2469. Adding error analysis and training set up
      2470. +
    2470. The \( \chi^2 \) function
      2471. +
    2471. The \( \chi^2 \) function
      2472. +
    2472. The \( \chi^2 \) function
      2473. +
    2473. The \( \chi^2 \) function
      2474. +
    2474. The \( \chi^2 \) function
      2475. +
    2475. The \( \chi^2 \) function
      2476. +
    2476. Fitting an Equation of State for Dense Nuclear Matter
      2477. +
    2477. The code
      2478. +
    2478. Splitting our Data in Training and Test data
      2479. +
    2479. Exercises
      2480. +
    2480. Exercise 1: Setting up various Python environments
      2481. +
    2481. Exercise 2: making your own data and exploring scikit-learn
      2482. +
    2482. Exercise 3: Split data in test and training data
      2483. @@ -397,10 +381,7 @@ further processing. For example, convert to latex as

      And to add more versatility, the Python package SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python.

      -

      Finally, if you wish to use the light mark-up language -doconce you can convert a standard ascii text file into various HTML -formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using doconce. -

      +

      Finally, we recommend strongly using Autograd or JAX for automatic differentiation.

      @@ -427,7 +408,7 @@ formats, ipython notebooks, latex files, pdf files etc with minimal edits. These

    2483. 37
    2484. 38
    2485. ...
      2486. -
    2486. 66
      2487. +
    2487. 62
    2488. »
      2489. diff --git a/doc/pub/week34/html/._week34-bs029.html b/doc/pub/week34/html/._week34-bs029.html index 167881953..375a1c0c6 100644 --- a/doc/pub/week34/html/._week34-bs029.html +++ b/doc/pub/week34/html/._week34-bs029.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2489. Installing R, C++, cython or Julia
    2490. Installing R, C++, cython, Numba etc
    2491. Numpy examples and Important Matrix and vector handling packages
      2492. -
    2492. Basic Matrix Features
      2493. -
    2493.    Some famous Matrices
      2494. -
    2494.    More Basic Matrix Features
      2495. -
    2495. Numpy and arrays
      2496. -
    2496. Matrices in Python
      2497. -
    2497. Meet the Pandas
      2498. -
    2498. Friday August 27
      2499. -
    2499.    Simple linear regression model using scikit-learn
      2500. -
    2500.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2501. -
    2501.    Organizing our data
      2502. -
    2502.    Seeing the wood for the trees
      2503. -
    2503.    And what about using neural networks?
      2504. -
    2504. A first summary
      2505. -
    2505. Why Linear Regression (aka Ordinary Least Squares and family)
      2506. -
    2506. Regression analysis, overarching aims
      2507. -
    2507. Regression analysis, overarching aims II
      2508. -
    2508. Examples
      2509. -
    2509. General linear models
      2510. -
    2510. Rewriting the fitting procedure as a linear algebra problem
      2511. -
    2511. Rewriting the fitting procedure as a linear algebra problem, more details
      2512. -
    2512. Generalizing the fitting procedure as a linear algebra problem
      2513. -
    2513. Generalizing the fitting procedure as a linear algebra problem
      2514. -
    2514. Optimizing our parameters
      2515. -
    2515. Our model for the nuclear binding energies
      2516. -
    2516. Optimizing our parameters, more details
      2517. -
    2517. Interpretations and optimizing our parameters
      2518. -
    2518. Interpretations and optimizing our parameters
      2519. -
    2519. Some useful matrix and vector expressions
      2520. -
    2520. Interpretations and optimizing our parameters
      2521. -
    2521. Own code for Ordinary Least Squares
      2522. -
    2522. Adding error analysis and training set up
      2523. -
    2523. The \( \chi^2 \) function
      2524. -
    2524. The \( \chi^2 \) function
      2525. -
    2525. The \( \chi^2 \) function
      2526. -
    2526. The \( \chi^2 \) function
      2527. -
    2527. The \( \chi^2 \) function
      2528. -
    2528. The \( \chi^2 \) function
      2529. -
    2529. Fitting an Equation of State for Dense Nuclear Matter
      2530. -
    2530. The code
      2531. -
    2531. Splitting our Data in Training and Test data
      2532. -
    2532. Exercises
      2533. -
    2533. Exercise 1: Setting up various Python environments
      2534. -
    2534. Exercise 2: making your own data and exploring scikit-learn
      2535. -
    2535. Exercise 3: Normalizing our data
      2536. +
    2536. Numpy and arrays
      2537. +
    2537. Matrices in Python
      2538. +
    2538. Meet the Pandas
      2539. +
    2539.    Simple linear regression model using scikit-learn
      2540. +
    2540.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2541. +
    2541.    Organizing our data
      2542. +
    2542.    And what about using neural networks?
      2543. +
    2543. A first summary
      2544. +
    2544. Why Linear Regression (aka Ordinary Least Squares and family)
      2545. +
    2545. Regression analysis, overarching aims
      2546. +
    2546. Regression analysis, overarching aims II
      2547. +
    2547. Examples
      2548. +
    2548. General linear models
      2549. +
    2549. Rewriting the fitting procedure as a linear algebra problem
      2550. +
    2550. Rewriting the fitting procedure as a linear algebra problem, more details
      2551. +
    2551. Generalizing the fitting procedure as a linear algebra problem
      2552. +
    2552. Generalizing the fitting procedure as a linear algebra problem
      2553. +
    2553. Optimizing our parameters
      2554. +
    2554. Our model for the nuclear binding energies
      2555. +
    2555. Optimizing our parameters, more details
      2556. +
    2556. Interpretations and optimizing our parameters
      2557. +
    2557. Interpretations and optimizing our parameters
      2558. +
    2558. Some useful matrix and vector expressions
      2559. +
    2559. Interpretations and optimizing our parameters
      2560. +
    2560. Own code for Ordinary Least Squares
      2561. +
    2561. Adding error analysis and training set up
      2562. +
    2562. The \( \chi^2 \) function
      2563. +
    2563. The \( \chi^2 \) function
      2564. +
    2564. The \( \chi^2 \) function
      2565. +
    2565. The \( \chi^2 \) function
      2566. +
    2566. The \( \chi^2 \) function
      2567. +
    2567. The \( \chi^2 \) function
      2568. +
    2568. Fitting an Equation of State for Dense Nuclear Matter
      2569. +
    2569. The code
      2570. +
    2570. Splitting our Data in Training and Test data
      2571. +
    2571. Exercises
      2572. +
    2572. Exercise 1: Setting up various Python environments
      2573. +
    2573. Exercise 2: making your own data and exploring scikit-learn
      2574. +
    2574. Exercise 3: Split data in test and training data
      2575. @@ -389,7 +373,7 @@ developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly he
    2575. 38
    2576. 39
    2577. ...
      2578. -
    2578. 66
      2579. +
    2579. 62
    2580. »
      2581. diff --git a/doc/pub/week34/html/._week34-bs030.html b/doc/pub/week34/html/._week34-bs030.html index 72f73c715..6430d9ef8 100644 --- a/doc/pub/week34/html/._week34-bs030.html +++ b/doc/pub/week34/html/._week34-bs030.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2581. Installing R, C++, cython or Julia
    2582. Installing R, C++, cython, Numba etc
    2583. Numpy examples and Important Matrix and vector handling packages
      2584. -
    2584. Basic Matrix Features
      2585. -
    2585.    Some famous Matrices
      2586. -
    2586.    More Basic Matrix Features
      2587. -
    2587. Numpy and arrays
      2588. -
    2588. Matrices in Python
      2589. -
    2589. Meet the Pandas
      2590. -
    2590. Friday August 27
      2591. -
    2591.    Simple linear regression model using scikit-learn
      2592. -
    2592.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2593. -
    2593.    Organizing our data
      2594. -
    2594.    Seeing the wood for the trees
      2595. -
    2595.    And what about using neural networks?
      2596. -
    2596. A first summary
      2597. -
    2597. Why Linear Regression (aka Ordinary Least Squares and family)
      2598. -
    2598. Regression analysis, overarching aims
      2599. -
    2599. Regression analysis, overarching aims II
      2600. -
    2600. Examples
      2601. -
    2601. General linear models
      2602. -
    2602. Rewriting the fitting procedure as a linear algebra problem
      2603. -
    2603. Rewriting the fitting procedure as a linear algebra problem, more details
      2604. -
    2604. Generalizing the fitting procedure as a linear algebra problem
      2605. -
    2605. Generalizing the fitting procedure as a linear algebra problem
      2606. -
    2606. Optimizing our parameters
      2607. -
    2607. Our model for the nuclear binding energies
      2608. -
    2608. Optimizing our parameters, more details
      2609. -
    2609. Interpretations and optimizing our parameters
      2610. -
    2610. Interpretations and optimizing our parameters
      2611. -
    2611. Some useful matrix and vector expressions
      2612. -
    2612. Interpretations and optimizing our parameters
      2613. -
    2613. Own code for Ordinary Least Squares
      2614. -
    2614. Adding error analysis and training set up
      2615. -
    2615. The \( \chi^2 \) function
      2616. -
    2616. The \( \chi^2 \) function
      2617. -
    2617. The \( \chi^2 \) function
      2618. -
    2618. The \( \chi^2 \) function
      2619. -
    2619. The \( \chi^2 \) function
      2620. -
    2620. The \( \chi^2 \) function
      2621. -
    2621. Fitting an Equation of State for Dense Nuclear Matter
      2622. -
    2622. The code
      2623. -
    2623. Splitting our Data in Training and Test data
      2624. -
    2624. Exercises
      2625. -
    2625. Exercise 1: Setting up various Python environments
      2626. -
    2626. Exercise 2: making your own data and exploring scikit-learn
      2627. -
    2627. Exercise 3: Normalizing our data
      2628. +
    2628. Numpy and arrays
      2629. +
    2629. Matrices in Python
      2630. +
    2630. Meet the Pandas
      2631. +
    2631.    Simple linear regression model using scikit-learn
      2632. +
    2632.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2633. +
    2633.    Organizing our data
      2634. +
    2634.    And what about using neural networks?
      2635. +
    2635. A first summary
      2636. +
    2636. Why Linear Regression (aka Ordinary Least Squares and family)
      2637. +
    2637. Regression analysis, overarching aims
      2638. +
    2638. Regression analysis, overarching aims II
      2639. +
    2639. Examples
      2640. +
    2640. General linear models
      2641. +
    2641. Rewriting the fitting procedure as a linear algebra problem
      2642. +
    2642. Rewriting the fitting procedure as a linear algebra problem, more details
      2643. +
    2643. Generalizing the fitting procedure as a linear algebra problem
      2644. +
    2644. Generalizing the fitting procedure as a linear algebra problem
      2645. +
    2645. Optimizing our parameters
      2646. +
    2646. Our model for the nuclear binding energies
      2647. +
    2647. Optimizing our parameters, more details
      2648. +
    2648. Interpretations and optimizing our parameters
      2649. +
    2649. Interpretations and optimizing our parameters
      2650. +
    2650. Some useful matrix and vector expressions
      2651. +
    2651. Interpretations and optimizing our parameters
      2652. +
    2652. Own code for Ordinary Least Squares
      2653. +
    2653. Adding error analysis and training set up
      2654. +
    2654. The \( \chi^2 \) function
      2655. +
    2655. The \( \chi^2 \) function
      2656. +
    2656. The \( \chi^2 \) function
      2657. +
    2657. The \( \chi^2 \) function
      2658. +
    2658. The \( \chi^2 \) function
      2659. +
    2659. The \( \chi^2 \) function
      2660. +
    2660. Fitting an Equation of State for Dense Nuclear Matter
      2661. +
    2661. The code
      2662. +
    2662. Splitting our Data in Training and Test data
      2663. +
    2663. Exercises
      2664. +
    2664. Exercise 1: Setting up various Python environments
      2665. +
    2665. Exercise 2: making your own data and exploring scikit-learn
      2666. +
    2666. Exercise 3: Split data in test and training data
      2667. @@ -351,50 +335,229 @@ MathJax.Hub.Config({

       

       

       

      -

      Basic Matrix Features

      - -
      -
      - -$$ - \mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -$$ - -

      The inverse of a matrix is defined by

      - -$$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -$$ +

      Numpy and arrays

      +

      Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

      -
      -
      - - - - - - - - - - - -
      Relations Name matrix elements
      \( A=A^{T} \) symmetric \( a_{ij}=a_{ji} \)
      \( A=\left (A^{T}\right )^{-1} \) real orthogonal \( \sum_k a_{ik}a_{jk}=\sum_k a_{ki} a_{kj}=\delta_{ij} \)
      \( A=A^* \) real matrix \( a_{ij}=a_{ij}^* \)
      \( A=A^{\dagger} \) hermitian \( a_{ij}=a_{ji}^* \)
      \( A=\left(A^{\dagger}\right )^{-1} \) unitary \( \sum_k a_{ik}a_{jk}^*=\sum_k a_{ki}^* a_{kj}=\delta_{ij} \)
      -
      -
      + +
      +
      +
      +
      +
      + 
      import numpy as np
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,

      + + +
      +
      +
      +
      +
      + 
      n = 10
+x = np.random.normal(size=n)
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We defined a vector \( x \) with \( n=10 \) elements with its values given by the Normal distribution \( N(0,1) \). +Another alternative is to declare a vector as follows +

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+x = np.array([1, 2, 3])
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Here we have defined a vector with three elements, with \( x_0=1 \), \( x_1=2 \) and \( x_2=3 \). Note that both Python and C++ +start numbering array elements from \( 0 \) and on. This means that a vector with \( n \) elements has a sequence of entities \( x_0, x_1, x_2, \dots, x_{n-1} \). We could also let (recommended) Numpy to compute the logarithms of a specific array as +

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+x = np.log(np.array([4, 7, 8]))
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      In the last example we used Numpy's unary function \( np.log \). This function is +highly tuned to compute array elements since the code is vectorized +and does not require looping. We normaly recommend that you use the +Numpy intrinsic functions instead of the corresponding log function +from Python's math module. The looping is done explicitely by the +np.log function. The alternative, and slower way to compute the +logarithms of a vector would be to write +

      + + + +
      +
      +
      +
      +
      + 
      import numpy as np
+from math import log
+x = np.array([4, 7, 8])
+for i in range(0, len(x)):
+    x[i] = log(x[i])
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We note that our code is much longer already and we need to import the log function from the math module. +The attentive reader will also notice that the output is \( [1, 1, 2] \). Python interprets automagically our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as +

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+x = np.log(np.array([4, 7, 8], dtype = np.float64))
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0]))
+print(x)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the itemsize functionality (the array \( x \) is actually an object which inherits the functionalities defined in Numpy) as

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0]))
+print(x.itemsize)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      @@ -423,7 +586,7 @@ $$
    2667. 39
    2668. 40
    2669. ...
      2670. -
    2670. 66
      2671. +
    2671. 62
    2672. »
      2673. diff --git a/doc/pub/week34/html/._week34-bs031.html b/doc/pub/week34/html/._week34-bs031.html index 988bbf56d..84bd312b1 100644 --- a/doc/pub/week34/html/._week34-bs031.html +++ b/doc/pub/week34/html/._week34-bs031.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2673. Installing R, C++, cython or Julia
    2674. Installing R, C++, cython, Numba etc
    2675. Numpy examples and Important Matrix and vector handling packages
      2676. -
    2676. Basic Matrix Features
      2677. -
    2677.    Some famous Matrices
      2678. -
    2678.    More Basic Matrix Features
      2679. -
    2679. Numpy and arrays
      2680. -
    2680. Matrices in Python
      2681. -
    2681. Meet the Pandas
      2682. -
    2682. Friday August 27
      2683. -
    2683.    Simple linear regression model using scikit-learn
      2684. -
    2684.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2685. -
    2685.    Organizing our data
      2686. -
    2686.    Seeing the wood for the trees
      2687. -
    2687.    And what about using neural networks?
      2688. -
    2688. A first summary
      2689. -
    2689. Why Linear Regression (aka Ordinary Least Squares and family)
      2690. -
    2690. Regression analysis, overarching aims
      2691. -
    2691. Regression analysis, overarching aims II
      2692. -
    2692. Examples
      2693. -
    2693. General linear models
      2694. -
    2694. Rewriting the fitting procedure as a linear algebra problem
      2695. -
    2695. Rewriting the fitting procedure as a linear algebra problem, more details
      2696. -
    2696. Generalizing the fitting procedure as a linear algebra problem
      2697. -
    2697. Generalizing the fitting procedure as a linear algebra problem
      2698. -
    2698. Optimizing our parameters
      2699. -
    2699. Our model for the nuclear binding energies
      2700. -
    2700. Optimizing our parameters, more details
      2701. -
    2701. Interpretations and optimizing our parameters
      2702. -
    2702. Interpretations and optimizing our parameters
      2703. -
    2703. Some useful matrix and vector expressions
      2704. -
    2704. Interpretations and optimizing our parameters
      2705. -
    2705. Own code for Ordinary Least Squares
      2706. -
    2706. Adding error analysis and training set up
      2707. -
    2707. The \( \chi^2 \) function
      2708. -
    2708. The \( \chi^2 \) function
      2709. -
    2709. The \( \chi^2 \) function
      2710. -
    2710. The \( \chi^2 \) function
      2711. -
    2711. The \( \chi^2 \) function
      2712. -
    2712. The \( \chi^2 \) function
      2713. -
    2713. Fitting an Equation of State for Dense Nuclear Matter
      2714. -
    2714. The code
      2715. -
    2715. Splitting our Data in Training and Test data
      2716. -
    2716. Exercises
      2717. -
    2717. Exercise 1: Setting up various Python environments
      2718. -
    2718. Exercise 2: making your own data and exploring scikit-learn
      2719. -
    2719. Exercise 3: Normalizing our data
      2720. +
    2720. Numpy and arrays
      2721. +
    2721. Matrices in Python
      2722. +
    2722. Meet the Pandas
      2723. +
    2723.    Simple linear regression model using scikit-learn
      2724. +
    2724.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2725. +
    2725.    Organizing our data
      2726. +
    2726.    And what about using neural networks?
      2727. +
    2727. A first summary
      2728. +
    2728. Why Linear Regression (aka Ordinary Least Squares and family)
      2729. +
    2729. Regression analysis, overarching aims
      2730. +
    2730. Regression analysis, overarching aims II
      2731. +
    2731. Examples
      2732. +
    2732. General linear models
      2733. +
    2733. Rewriting the fitting procedure as a linear algebra problem
      2734. +
    2734. Rewriting the fitting procedure as a linear algebra problem, more details
      2735. +
    2735. Generalizing the fitting procedure as a linear algebra problem
      2736. +
    2736. Generalizing the fitting procedure as a linear algebra problem
      2737. +
    2737. Optimizing our parameters
      2738. +
    2738. Our model for the nuclear binding energies
      2739. +
    2739. Optimizing our parameters, more details
      2740. +
    2740. Interpretations and optimizing our parameters
      2741. +
    2741. Interpretations and optimizing our parameters
      2742. +
    2742. Some useful matrix and vector expressions
      2743. +
    2743. Interpretations and optimizing our parameters
      2744. +
    2744. Own code for Ordinary Least Squares
      2745. +
    2745. Adding error analysis and training set up
      2746. +
    2746. The \( \chi^2 \) function
      2747. +
    2747. The \( \chi^2 \) function
      2748. +
    2748. The \( \chi^2 \) function
      2749. +
    2749. The \( \chi^2 \) function
      2750. +
    2750. The \( \chi^2 \) function
      2751. +
    2751. The \( \chi^2 \) function
      2752. +
    2752. Fitting an Equation of State for Dense Nuclear Matter
      2753. +
    2753. The code
      2754. +
    2754. Splitting our Data in Training and Test data
      2755. +
    2755. Exercises
      2756. +
    2756. Exercise 1: Setting up various Python environments
      2757. +
    2757. Exercise 2: making your own data and exploring scikit-learn
      2758. +
    2758. Exercise 3: Split data in test and training data
      2759. @@ -351,19 +335,277 @@ MathJax.Hub.Config({

       

       

       

      -

      Some famous Matrices

      +

      Matrices in Python

      + +

      Having defined vectors, we are now ready to try out matrices. We can +define a \( 3 \times 3 \) real matrix \( \boldsymbol{A} \) as (recall that we user +lowercase letters for vectors and uppercase letters for matrices) +

      + + + +
      +
      +
      +
      +
      + 
      import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+print(A)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      If we use the shape function we would get \( (3, 3) \) as output, that is verifying that our matrix is a \( 3\times 3 \) matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+# print the first column, row-major order and elements start with 0
+print(A[:,0]) 
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We can continue this was by printing out other columns or rows. The example here prints out the second column

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+# print the first column, row-major order and elements start with 0
+print(A[1,:]) 
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to zero
+A = np.zeros( (n, n) )
+print(A) 
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      or initializing all elements to

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to one
+A = np.ones( (n, n) )
+print(A) 
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      or as unitarily distributed random numbers (see the material on random number generators in the statistics part)

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
+A = np.random.rand(n, n)
+print(A) 
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. +As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors +\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as +

      +$$ +\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ + \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ + \sigma_{zx} & \sigma_{zy} & \sigma_{zz} + \end{bmatrix}, +$$ + +

      where for example

      +$$ +\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). +$$ + +

      The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. +The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \) +

      +$$ +\boldsymbol{W} = \begin{bmatrix} x_0 & x_1 & x_2 & \dots & x_{n-2} & x_{n-1} \\ + y_0 & y_1 & y_2 & \dots & y_{n-2} & y_{n-1} \\ + z_0 & z_1 & z_2 & \dots & z_{n-2} & z_{n-1} \\ + \end{bmatrix}, +$$ + +

      which in turn is converted into into the \( 3\times 3 \) covariance matrix +\( \boldsymbol{\Sigma} \) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy +function np.mean(x). We can also extract the eigenvalues of the +covariance matrix through the np.linalg.eig() function. +

      + + + +
      +
      +
      +
      +
      + 
      # Importing various packages
+import numpy as np
+
+n = 100
+x = np.random.normal(size=n)
+print(np.mean(x))
+y = 4+3*x+np.random.normal(size=n)
+print(np.mean(y))
+z = x**3+np.random.normal(size=n)
+print(np.mean(z))
+W = np.vstack((x, y, z))
+Sigma = np.cov(W)
+print(Sigma)
+Eigvals, Eigvecs = np.linalg.eig(Sigma)
+print(Eigvals)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +
      +
      +
      +
      +
      + 
      import numpy as np
+import matplotlib.pyplot as plt
+from scipy import sparse
+eye = np.eye(4)
+print(eye)
+sparse_mtx = sparse.csr_matrix(eye)
+print(sparse_mtx)
+x = np.linspace(-10,10,100)
+y = np.sin(x)
+plt.plot(x,y,marker='x')
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + -
        -
      • Diagonal if \( a_{ij}=0 \) for \( i\ne j \)
        • -
      • Upper triangular if \( a_{ij}=0 \) for \( i>j \)
        • -
      • Lower triangular if \( a_{ij}=0 \) for \( i < j \)
        • -
      • Upper Hessenberg if \( a_{ij}=0 \) for \( i>j+1 \)
        • -
      • Lower Hessenberg if \( a_{ij}=0 \) for \( i < j+1 \)
        • -
      • Tridiagonal if \( a_{ij}=0 \) for \( |i -j|>1 \)
        • -
      • Lower banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i>j+p \)
        • -
      • Upper banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i < j+p \)
        • -
      • Banded, block upper triangular, block lower triangular....
        • -

        @@ -389,7 +631,7 @@ MathJax.Hub.Config({
      • 40
      • 41
      • ...
        • -
      • 66
        • +
      • 62
      • »
      diff --git a/doc/pub/week34/html/._week34-bs032.html b/doc/pub/week34/html/._week34-bs032.html index f53225b17..cfaccfe40 100644 --- a/doc/pub/week34/html/._week34-bs032.html +++ b/doc/pub/week34/html/._week34-bs032.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2759. Installing R, C++, cython or Julia
    2760. Installing R, C++, cython, Numba etc
    2761. Numpy examples and Important Matrix and vector handling packages
      2762. -
    2762. Basic Matrix Features
      2763. -
    2763.    Some famous Matrices
      2764. -
    2764.    More Basic Matrix Features
      2765. -
    2765. Numpy and arrays
      2766. -
    2766. Matrices in Python
      2767. -
    2767. Meet the Pandas
      2768. -
    2768. Friday August 27
      2769. -
    2769.    Simple linear regression model using scikit-learn
      2770. -
    2770.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2771. -
    2771.    Organizing our data
      2772. -
    2772.    Seeing the wood for the trees
      2773. -
    2773.    And what about using neural networks?
      2774. -
    2774. A first summary
      2775. -
    2775. Why Linear Regression (aka Ordinary Least Squares and family)
      2776. -
    2776. Regression analysis, overarching aims
      2777. -
    2777. Regression analysis, overarching aims II
      2778. -
    2778. Examples
      2779. -
    2779. General linear models
      2780. -
    2780. Rewriting the fitting procedure as a linear algebra problem
      2781. -
    2781. Rewriting the fitting procedure as a linear algebra problem, more details
      2782. -
    2782. Generalizing the fitting procedure as a linear algebra problem
      2783. -
    2783. Generalizing the fitting procedure as a linear algebra problem
      2784. -
    2784. Optimizing our parameters
      2785. -
    2785. Our model for the nuclear binding energies
      2786. -
    2786. Optimizing our parameters, more details
      2787. -
    2787. Interpretations and optimizing our parameters
      2788. -
    2788. Interpretations and optimizing our parameters
      2789. -
    2789. Some useful matrix and vector expressions
      2790. -
    2790. Interpretations and optimizing our parameters
      2791. -
    2791. Own code for Ordinary Least Squares
      2792. -
    2792. Adding error analysis and training set up
      2793. -
    2793. The \( \chi^2 \) function
      2794. -
    2794. The \( \chi^2 \) function
      2795. -
    2795. The \( \chi^2 \) function
      2796. -
    2796. The \( \chi^2 \) function
      2797. -
    2797. The \( \chi^2 \) function
      2798. -
    2798. The \( \chi^2 \) function
      2799. -
    2799. Fitting an Equation of State for Dense Nuclear Matter
      2800. -
    2800. The code
      2801. -
    2801. Splitting our Data in Training and Test data
      2802. -
    2802. Exercises
      2803. -
    2803. Exercise 1: Setting up various Python environments
      2804. -
    2804. Exercise 2: making your own data and exploring scikit-learn
      2805. -
    2805. Exercise 3: Normalizing our data
      2806. +
    2806. Numpy and arrays
      2807. +
    2807. Matrices in Python
      2808. +
    2808. Meet the Pandas
      2809. +
    2809.    Simple linear regression model using scikit-learn
      2810. +
    2810.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2811. +
    2811.    Organizing our data
      2812. +
    2812.    And what about using neural networks?
      2813. +
    2813. A first summary
      2814. +
    2814. Why Linear Regression (aka Ordinary Least Squares and family)
      2815. +
    2815. Regression analysis, overarching aims
      2816. +
    2816. Regression analysis, overarching aims II
      2817. +
    2817. Examples
      2818. +
    2818. General linear models
      2819. +
    2819. Rewriting the fitting procedure as a linear algebra problem
      2820. +
    2820. Rewriting the fitting procedure as a linear algebra problem, more details
      2821. +
    2821. Generalizing the fitting procedure as a linear algebra problem
      2822. +
    2822. Generalizing the fitting procedure as a linear algebra problem
      2823. +
    2823. Optimizing our parameters
      2824. +
    2824. Our model for the nuclear binding energies
      2825. +
    2825. Optimizing our parameters, more details
      2826. +
    2826. Interpretations and optimizing our parameters
      2827. +
    2827. Interpretations and optimizing our parameters
      2828. +
    2828. Some useful matrix and vector expressions
      2829. +
    2829. Interpretations and optimizing our parameters
      2830. +
    2830. Own code for Ordinary Least Squares
      2831. +
    2831. Adding error analysis and training set up
      2832. +
    2832. The \( \chi^2 \) function
      2833. +
    2833. The \( \chi^2 \) function
      2834. +
    2834. The \( \chi^2 \) function
      2835. +
    2835. The \( \chi^2 \) function
      2836. +
    2836. The \( \chi^2 \) function
      2837. +
    2837. The \( \chi^2 \) function
      2838. +
    2838. Fitting an Equation of State for Dense Nuclear Matter
      2839. +
    2839. The code
      2840. +
    2840. Splitting our Data in Training and Test data
      2841. +
    2841. Exercises
      2842. +
    2842. Exercise 1: Setting up various Python environments
      2843. +
    2843. Exercise 2: making your own data and exploring scikit-learn
      2844. +
    2844. Exercise 3: Split data in test and training data
      2845. @@ -351,24 +335,253 @@ MathJax.Hub.Config({

       

       

       

      -

      More Basic Matrix Features

      +

      Meet the Pandas

      -
      -
      - -

      For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

      +

      +
      +

      +
      +

      -
        -
      • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
        • -
      • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
        • -
      • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
        • -
      • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
        • -
      • \( \mathbf{A} \) is a product of elementary matrices.
        • -
      • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
        • -
      +

      Another useful Python package is +pandas, which is an open source library +providing high-performance, easy-to-use data structures and data +analysis tools for Python. pandas stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. +pandas has two major classes, the DataFrame class with two-dimensional data objects and tabular data organized in columns and the class Series with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. +pandas allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. +

      + +

      The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of pandas, in particular in connection with classification of data.

      + + + +
      +
      +
      +
      +
      + 
      import pandas as pd
+from IPython.display import display
+data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
+        'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
+        'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
+        'Date of Birth T.A.': [2968, 2890, 2931, 2980]
+        }
+data_pandas = pd.DataFrame(data)
+display(data_pandas)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      In the above we have imported pandas with the shorthand pd, the latter has become the standard way we import pandas. We make then a list of various variables +and reorganize the aboves lists into a DataFrame and then print out a neat table with specific column labels as Name, place of birth and date of birth. +Displaying these results, we see that the indices are given by the default numbers from zero to three. +pandas is extremely flexible and we can easily change the above indices by defining a new type of indexing as +

      + + +
      +
      +
      +
      +
      + 
      data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
+display(data_pandas)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Thereafter we display the content of the row which begins with the index Aragorn

      + + +
      +
      +
      +
      +
      + 
      display(data_pandas.loc['Aragorn'])
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We can easily append data to this, for example

      + + +
      +
      +
      +
      +
      + 
      new_hobbit = {'First Name': ["Peregrin"],
+              'Last Name': ["Took"],
+              'Place of birth': ["Shire"],
+              'Date of Birth T.A.': [2990]
+              }
+data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
+display(data_pandas)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix +of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. +

      + + +
      +
      +
      +
      +
      + 
      import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 matrix
+rows = 10
+cols = 5
+a = np.random.randn(rows,cols)
+df = pd.DataFrame(a)
+display(df)
+print(df.mean())
+print(df.std())
+display(df**2)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Thereafter we can select specific columns only and plot final results

      + + +
      +
      +
      +
      +
      + 
      df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
+df.index = np.arange(10)
+
+display(df)
+print(df['Second'].mean() )
+print(df.info())
+print(df.describe())
+
+from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+df.cumsum().plot(lw=2.0, figsize=(10,6))
+plt.show()
+
+
+df.plot.bar(figsize=(10,6), rot=15)
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We can produce a \( 4\times 4 \) matrix

      + + +
      +
      +
      +
      +
      + 
      b = np.arange(16).reshape((4,4))
+print(b)
+df1 = pd.DataFrame(b)
+print(df1)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      and many other operations.

      + +

      The Series class is another important class included in +pandas. You can view it as a specialization of DataFrame but where +we have just a single column of data. It shares many of the same features as DataFrame. As with DataFrame, +most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays. +As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in. +For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray. +

      @@ -395,7 +608,7 @@ MathJax.Hub.Config({

    2845. 41
    2846. 42
    2847. ...
      2848. -
    2848. 66
      2849. +
    2849. 62
    2850. »
      2851. diff --git a/doc/pub/week34/html/._week34-bs033.html b/doc/pub/week34/html/._week34-bs033.html index 0914f3c4f..b9d74b0c5 100644 --- a/doc/pub/week34/html/._week34-bs033.html +++ b/doc/pub/week34/html/._week34-bs033.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2851. Installing R, C++, cython or Julia
    2852. Installing R, C++, cython, Numba etc
    2853. Numpy examples and Important Matrix and vector handling packages
      2854. -
    2854. Basic Matrix Features
      2855. -
    2855.    Some famous Matrices
      2856. -
    2856.    More Basic Matrix Features
      2857. -
    2857. Numpy and arrays
      2858. -
    2858. Matrices in Python
      2859. -
    2859. Meet the Pandas
      2860. -
    2860. Friday August 27
      2861. -
    2861.    Simple linear regression model using scikit-learn
      2862. -
    2862.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2863. -
    2863.    Organizing our data
      2864. -
    2864.    Seeing the wood for the trees
      2865. -
    2865.    And what about using neural networks?
      2866. -
    2866. A first summary
      2867. -
    2867. Why Linear Regression (aka Ordinary Least Squares and family)
      2868. -
    2868. Regression analysis, overarching aims
      2869. -
    2869. Regression analysis, overarching aims II
      2870. -
    2870. Examples
      2871. -
    2871. General linear models
      2872. -
    2872. Rewriting the fitting procedure as a linear algebra problem
      2873. -
    2873. Rewriting the fitting procedure as a linear algebra problem, more details
      2874. -
    2874. Generalizing the fitting procedure as a linear algebra problem
      2875. -
    2875. Generalizing the fitting procedure as a linear algebra problem
      2876. -
    2876. Optimizing our parameters
      2877. -
    2877. Our model for the nuclear binding energies
      2878. -
    2878. Optimizing our parameters, more details
      2879. -
    2879. Interpretations and optimizing our parameters
      2880. -
    2880. Interpretations and optimizing our parameters
      2881. -
    2881. Some useful matrix and vector expressions
      2882. -
    2882. Interpretations and optimizing our parameters
      2883. -
    2883. Own code for Ordinary Least Squares
      2884. -
    2884. Adding error analysis and training set up
      2885. -
    2885. The \( \chi^2 \) function
      2886. -
    2886. The \( \chi^2 \) function
      2887. -
    2887. The \( \chi^2 \) function
      2888. -
    2888. The \( \chi^2 \) function
      2889. -
    2889. The \( \chi^2 \) function
      2890. -
    2890. The \( \chi^2 \) function
      2891. -
    2891. Fitting an Equation of State for Dense Nuclear Matter
      2892. -
    2892. The code
      2893. -
    2893. Splitting our Data in Training and Test data
      2894. -
    2894. Exercises
      2895. -
    2895. Exercise 1: Setting up various Python environments
      2896. -
    2896. Exercise 2: making your own data and exploring scikit-learn
      2897. -
    2897. Exercise 3: Normalizing our data
      2898. +
    2898. Numpy and arrays
      2899. +
    2899. Matrices in Python
      2900. +
    2900. Meet the Pandas
      2901. +
    2901.    Simple linear regression model using scikit-learn
      2902. +
    2902.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2903. +
    2903.    Organizing our data
      2904. +
    2904.    And what about using neural networks?
      2905. +
    2905. A first summary
      2906. +
    2906. Why Linear Regression (aka Ordinary Least Squares and family)
      2907. +
    2907. Regression analysis, overarching aims
      2908. +
    2908. Regression analysis, overarching aims II
      2909. +
    2909. Examples
      2910. +
    2910. General linear models
      2911. +
    2911. Rewriting the fitting procedure as a linear algebra problem
      2912. +
    2912. Rewriting the fitting procedure as a linear algebra problem, more details
      2913. +
    2913. Generalizing the fitting procedure as a linear algebra problem
      2914. +
    2914. Generalizing the fitting procedure as a linear algebra problem
      2915. +
    2915. Optimizing our parameters
      2916. +
    2916. Our model for the nuclear binding energies
      2917. +
    2917. Optimizing our parameters, more details
      2918. +
    2918. Interpretations and optimizing our parameters
      2919. +
    2919. Interpretations and optimizing our parameters
      2920. +
    2920. Some useful matrix and vector expressions
      2921. +
    2921. Interpretations and optimizing our parameters
      2922. +
    2922. Own code for Ordinary Least Squares
      2923. +
    2923. Adding error analysis and training set up
      2924. +
    2924. The \( \chi^2 \) function
      2925. +
    2925. The \( \chi^2 \) function
      2926. +
    2926. The \( \chi^2 \) function
      2927. +
    2927. The \( \chi^2 \) function
      2928. +
    2928. The \( \chi^2 \) function
      2929. +
    2929. The \( \chi^2 \) function
      2930. +
    2930. Fitting an Equation of State for Dense Nuclear Matter
      2931. +
    2931. The code
      2932. +
    2932. Splitting our Data in Training and Test data
      2933. +
    2933. Exercises
      2934. +
    2934. Exercise 1: Setting up various Python environments
      2935. +
    2935. Exercise 2: making your own data and exploring scikit-learn
      2936. +
    2936. Exercise 3: Split data in test and training data
      2937. @@ -351,9 +335,168 @@ MathJax.Hub.Config({

       

       

       

      -

      Numpy and arrays

      -

      Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

      +

      Simple linear regression model using scikit-learn

      +

      We start with perhaps our simplest possible example, using Scikit-Learn to perform linear regression analysis on a data set produced by us.

      + +

      What follows is a simple Python code where we have defined a function +\( y \) in terms of the variable \( x \). Both are defined as vectors with \( 100 \) entries. +The numbers in the vector \( \boldsymbol{x} \) are given +by random numbers generated with a uniform distribution with entries +\( x_i \in [0,1] \) (more about probability distribution functions +later). These values are then used to define a function \( y(x) \) +(tabulated again as a vector) with a linear dependence on \( x \) plus a +random noise added via the normal distribution. +

      + +

      The Numpy functions are imported used the import numpy as np +statement and the random number generator for the uniform distribution +is called using the function np.random.rand(), where we specificy +that we want \( 100 \) random variables. Using Numpy we define +automatically an array with the specified number of elements, \( 100 \) in +our case. With the Numpy function randn() we can compute random +numbers with the normal distribution (mean value \( \mu \) equal to zero and +variance \( \sigma^2 \) set to one) and produce the values of \( y \) assuming a linear +dependence as function of \( x \) +

      + +$$ +y = 2x+N(0,1), +$$ + +

      where \( N(0,1) \) represents random numbers generated by the normal +distribution. From Scikit-Learn we import then the +LinearRegression functionality and make a prediction \( \tilde{y} = +\alpha + \beta x \) using the function fit(x,y). We call the set of +data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package +scikit-learn has also a functionality which extracts the above +fitting parameters \( \alpha \) and \( \beta \) (see below). Later we will +distinguish between training data and test data. +

      + +

      For plotting we use the Python package +matplotlib which produces publication +quality figures. Feel free to explore the extensive +gallery of examples. In +this example we plot our original values of \( x \) and \( y \) as well as the +prediction ypredict (\( \tilde{y} \)), which attempts at fitting our +data with a straight line. +

      + +

      The Python code follows here.

      + + +
      +
      +
      +
      +
      + 
      # Importing various packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression
+
+x = np.random.rand(100,1)
+y = 2*x+np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+xnew = np.array([[0],[1]])
+ypredict = linreg.predict(xnew)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(x, y ,'ro')
+plt.axis([0,1.0,0, 5.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Simple Linear Regression')
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      This example serves several aims. It allows us to demonstrate several +aspects of data analysis and later machine learning algorithms. The +immediate visualization shows that our linear fit is not +impressive. It goes through the data points, but there are many +outliers which are not reproduced by our linear regression. We could +now play around with this small program and change for example the +factor in front of \( x \) and the normal distribution. Try to change the +function \( y \) to +

      + +$$ +y = 10x+0.01 \times N(0,1), +$$ + +

      where \( x \) is defined as before. Does the fit look better? Indeed, by +reducing the role of the noise given by the normal distribution we see immediately that +our linear prediction seemingly reproduces better the training +set. However, this testing 'by the eye' is obviouly not satisfactory in the +long run. Here we have only defined the training data and our model, and +have not discussed a more rigorous approach to the cost function. +

      + +

      We need more rigorous criteria in defining whether we have succeeded or +not in modeling our training data. You will be surprised to see that +many scientists seldomly venture beyond this 'by the eye' approach. A +standard approach for the cost function is the so-called \( \chi^2 \) +function (a variant of the mean-squared error (MSE)) +

      + +$$ \chi^2 = \frac{1}{n} +\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, +$$ + +

      where \( \sigma_i^2 \) is the variance (to be defined later) of the entry +\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves +however the aim of scaling the equations and make the cost function +dimensionless. +

      + +

      Minimizing the cost function is a central aspect of +our discussions to come. Finding its minima as function of the model +parameters (\( \alpha \) and \( \beta \) in our case) will be a recurring +theme in these series of lectures. Essentially all machine learning +algorithms we will discuss center around the minimization of the +chosen cost function. This depends in turn on our specific +model for describing the data, a typical situation in supervised +learning. Automatizing the search for the minima of the cost function is a +central ingredient in all algorithms. Typical methods which are +employed are various variants of gradient methods. These will be +discussed in more detail later. Again, you'll be surprised to hear that +many practitioners minimize the above function ''by the eye', popularly dubbed as +'chi by the eye'. That is, change a parameter and see (visually and numerically) that +the \( \chi^2 \) function becomes smaller. +

      + +

      There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define +the relative error (why would we prefer the MSE instead of the relative error?) as +

      + +$$ +\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}. +$$ + +

      The squared cost function results in an arithmetic mean-unbiased +estimator, and the absolute-value cost function results in a +median-unbiased estimator (in the one-dimensional case, and a +geometric median-unbiased estimator for the multi-dimensional +case). The squared cost function has the disadvantage that it has the tendency +to be dominated by outliers. +

      + +

      We can modify easily the above Python code and plot the relative error instead

      @@ -362,6 +505,21 @@ MathJax.Hub.Config({ 
      import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression
+
+x = np.random.rand(100,1)
+y = 5*x+0.01*np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+ypredict = linreg.predict(x)
+
+plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
+plt.axis([0,1.0,0.0, 0.5])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
+plt.title(r'Relative error')
+plt.show()
      @@ -377,34 +535,20 @@ MathJax.Hub.Config({
      -

      Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,

      +

      Depending on the parameter in front of the normal distribution, we may +have a small or larger relative error. Try to play around with +different training data sets and study (graphically) the value of the +relative error. +

      - -
      -
      -
      -
      -
      - 
      n = 10
-x = np.random.normal(size=n)
-print(x)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      +

      As mentioned above, Scikit-Learn has an impressive functionality. +We can for example extract the values of \( \alpha \) and \( \beta \) and +their error estimates, or the variance and standard deviation and many +other properties from the statistical data analysis. +

      -

      We defined a vector \( x \) with \( n=10 \) elements with its values given by the Normal distribution \( N(0,1) \). -Another alternative is to declare a vector as follows +

      Here we show an +example of the functionality of Scikit-Learn.

      @@ -413,9 +557,33 @@ Another alternative is to declare a vector as follows
      - 
      import numpy as np
-x = np.array([1, 2, 3])
-print(x)
+  
      import numpy as np 
+import matplotlib.pyplot as plt 
+from sklearn.linear_model import LinearRegression 
+from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
+
+x = np.random.rand(100,1)
+y = 2.0+ 5*x+0.5*np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+ypredict = linreg.predict(x)
+print('The intercept alpha: \n', linreg.intercept_)
+print('Coefficient beta : \n', linreg.coef_)
+# The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(y, ypredict))
+# Mean squared log error                                                        
+print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
+plt.plot(x, ypredict, "r-")
+plt.plot(x, y ,'ro')
+plt.axis([0.0,1.0,1.5, 7.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Linear Regression fit ')
+plt.show()
      @@ -431,9 +599,159 @@ x = np.a
      -

      Here we have defined a vector with three elements, with \( x_0=1 \), \( x_1=2 \) and \( x_2=3 \). Note that both Python and C++ -start numbering array elements from \( 0 \) and on. This means that a vector with \( n \) elements has a sequence of entities \( x_0, x_1, x_2, \dots, x_{n-1} \). We could also let (recommended) Numpy to compute the logarithms of a specific array as +

      The function coef gives us the parameter \( \beta \) of our fit while intercept yields +\( \alpha \). Depending on the constant in front of the normal distribution, we get values near or far from \( \alpha =2 \) and \( \beta =5 \). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as

      +$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +$$ + +

      The smaller the value, the better the fit. Ideally we would like to +have an MSE equal zero. The attentive reader has probably recognized +this function as being similar to the \( \chi^2 \) function defined above. +

      + +

      The r2score function computes \( R^2 \), the coefficient of +determination. It provides a measure of how well future samples are +likely to be predicted by the model. Best possible score is 1.0 and it +can be negative (because the model can be arbitrarily worse). A +constant model that always predicts the expected value of \( \boldsymbol{y} \), +disregarding the input features, would get a \( R^2 \) score of \( 0.0 \). +

      + +

      If \( \tilde{\boldsymbol{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as

      +$$ +R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +$$ + +

      where we have defined the mean value of \( \boldsymbol{y} \) as

      +$$ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +$$ + +

      Another quantity taht we will meet again in our discussions of regression analysis is + the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error. +The MAE is defined as follows +

      +$$ +\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. +$$ + +

      We present the +squared logarithmic (quadratic) error +

      +$$ +\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, +$$ + +

      where \( \log_e (x) \) stands for the natural logarithm of \( x \). This error +estimate is best to use when targets having exponential growth, such +as population counts, average sales of a commodity over a span of +years etc. +

      + +

      Finally, another cost function is the Huber cost function used in robust regression.

      + +

      The rationale behind this possible cost function is its reduced +sensitivity to outliers in the data set. In our discussions on +dimensionality reduction and normalization of data we will meet other +ways of dealing with outliers. +

      + +

      The Huber cost function is defined as

      +$$ +H_{\delta}(\boldsymbol{a})=\left\{\begin{array}{cc}\frac{1}{2} \boldsymbol{a}^{2}& \text{for }|\boldsymbol{a}|\leq \delta\\ \delta (|\boldsymbol{a}|-\frac{1}{2}\delta ),&\text{otherwise}.\end{array}\right. +$$ + +

      Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).

      + +

      We will discuss in more detail these and other functions in the +various lectures and lab sessions. +

      +

      To our real data: nuclear binding energies. Brief reminder on masses and binding energies

      + +

      Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding +energies. A basic quantity which can be measured for the ground +states of nuclei is the atomic mass \( M(N, Z) \) of the neutral atom with +atomic mass number \( A \) and charge \( Z \). The number of neutrons is \( N \). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). +

      + +

      Atomic masses are usually tabulated in terms of the mass excess defined by

      +$$ +\Delta M(N, Z) = M(N, Z) - uA, +$$ + +

      where \( u \) is the Atomic Mass Unit

      +$$ +u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. +$$ + +

      The nucleon masses are

      +$$ +m_p = 1.00727646693(9)u, +$$ + +

      and

      +$$ +m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. +$$ + +

      In the 2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu +there are data on masses and decays of 3437 nuclei. +

      + +

      The nuclear binding energy is defined as the energy required to break +up a given nucleus into its constituent parts of \( N \) neutrons and \( Z \) +protons. In terms of the atomic masses \( M(N, Z) \) the binding energy is +defined by +

      + +$$ +BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , +$$ + +

      where \( M_H \) is the mass of the hydrogen atom and \( m_n \) is the mass of the neutron. +In terms of the mass excess the binding energy is given by +

      +$$ +BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , +$$ + +

      where \( \Delta_H c^2 = 7.2890 \) MeV and \( \Delta_n c^2 = 8.0713 \) MeV.

      + +

      A popular and physically intuitive model which can be used to parametrize +the experimental binding energies as function of \( A \), is the so-called +liquid drop model. The ansatz is based on the following expression +

      + +$$ +BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A}, +$$ + +

      where \( A \) stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit +to the experimental data. +

      + +

      To arrive at the above expression we have assumed that we can make the following assumptions:

      + +
        +
      • There is a volume term \( a_1A \) proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.
        • +
      • There is a surface energy term \( a_2A^{2/3} \). The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
        • +
      • There is a Coulomb energy term \( a_3\frac{Z^2}{A^{1/3}} \). The electric repulsion between each pair of protons in a nucleus yields less binding.
        • +
      • There is an asymmetry term \( a_4\frac{(N-Z)^2}{A} \). This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.
        • +
      +

      We could also add a so-called pairing term, which is a correction term that +arises from the tendency of proton pairs and neutron pairs to +occur. An even number of particles is more stable than an odd number. +

      +

      Organizing our data

      + +

      Let us start with reading and organizing our data. +We start with the compilation of masses and binding energies from 2016. +After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data. +

      + +

      We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of scikit-learn.

      @@ -441,9 +759,39 @@ start numbering array elements from \( 0 \) and on. This means that a vector wit
      - 
      import numpy as np
-x = np.log(np.array([4, 7, 8]))
-print(x)
+  
      # Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import sklearn.linear_model as skl
+from sklearn.model_selection import train_test_split
+from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("MassEval2016.dat"),'r')
      @@ -459,13 +807,84 @@ x = np.l
      -

      In the last example we used Numpy's unary function \( np.log \). This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding log function -from Python's math module. The looping is done explicitely by the -np.log function. The alternative, and slower way to compute the -logarithms of a vector would be to write +

      Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various matplotlib commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.

      + + +
      +
      +
      +
      +
      + 
      from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+def MakePlot(x,y, styles, labels, axlabels):
+    plt.figure(figsize=(10,6))
+    for i in range(len(x)):
+        plt.plot(x[i], y[i], styles[i], label = labels[i])
+        plt.xlabel(axlabels[0])
+        plt.ylabel(axlabels[1])
+    plt.legend(loc=0)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Our next step is to read the data on experimental binding energies and +reorganize them as functions of the mass number \( A \), the number of +protons \( Z \) and neutrons \( N \) using pandas. Before we do this it is +always useful (unless you have a binary file or other types of compressed +data) to actually open the file and simply take a look at it! +

      + +

      In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with pandas. The file begins with some basic format information.

      + + +
      +
      +
      +
      +
      + 
      """                                                                                                                         
+This is taken from the data file of the mass 2016 evaluation.                                                               
+All files are 3436 lines long with 124 character per line.                                                                  
+       Headers are 39 lines long.                                                                                           
+   col 1     :  Fortran character control: 1 = page feed  0 = line feed                                                     
+   format    :  a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5                     
+   These formats are reflected in the pandas widths variable below, see the statement                                       
+   widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),                                                            
+   Pandas has also a variable header, with length 39 in this case.                                                          
+"""
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      The data we are interested in are in columns 2, 3, 4 and 11, giving us +the number of neutrons, protons, mass numbers and binding energies, +respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will +covert them into the pandas DataFrame structure.

      @@ -475,12 +894,24 @@ logarithms of a vector would be to write
      - 
      import numpy as np
-from math import log
-x = np.array([4, 7, 8])
-for i in range(0, len(x)):
-    x[i] = log(x[i])
-print(x)
+  
      # Read the experimental data with Pandas
+Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
+              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
+              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
+              header=39,
+              index_col=False)
+
+# Extrapolated values are indicated by '#' in place of the decimal place, so
+# the Ebinding column won't be numeric. Coerce to float and drop these entries.
+Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
+Masses = Masses.dropna()
+# Convert from keV to MeV.
+Masses['Ebinding'] /= 1000
+
+# Group the DataFrame by nucleon number, A.
+Masses = Masses.groupby('A')
+# Find the rows of the grouped DataFrame with the maximum binding energy.
+Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
      @@ -496,8 +927,17 @@ x = np.a
      -

      We note that our code is much longer already and we need to import the log function from the math module. -The attentive reader will also notice that the output is \( [1, 1, 2] \). Python interprets automagically our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as +

      We have now read in the data, grouped them according to the variables we are interested in. +We see how easy it is to reorganize the data using pandas. If we +were to do these operations in C/C++ or Fortran, we would have had to +write various functions/subroutines which perform the above +reorganizations for us. Having reorganized the data, we can now start +to make some simple fits using both the functionalities in numpy and +Scikit-Learn afterwards. +

      + +

      Now we define five variables which contain +the number of nucleons \( A \), the number of protons \( Z \) and the number of neutrons \( N \), the element name and finally the energies themselves.

      @@ -506,9 +946,12 @@ The attentive reader will also notice that the output is \( [1, 1, 2] \). Python
      - 
      import numpy as np
-x = np.log(np.array([4, 7, 8], dtype = np.float64))
-print(x)
+  
      A = Masses['A']
+Z = Masses['Z']
+N = Masses['N']
+Element = Masses['Element']
+Energies = Masses['Ebinding']
+print(Masses)
      @@ -524,7 +967,9 @@ x = np.l
      -

      or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is

      +

      The next step, and we will define this mathematically later, is to set up the so-called design matrix. We will throughout call this matrix \( \boldsymbol{X} \). +It has dimensionality \( p\times n \), where \( n \) is the number of data points and \( p \) are the so-called predictors. In our case here they are given by the number of polynomials in \( A \) we wish to include in the fit. +

      @@ -532,9 +977,13 @@ x = np.l
      - 
      import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0]))
-print(x)
+  
      # Now we set up the design matrix X
+X = np.zeros((len(A),5))
+X[:,0] = 1
+X[:,1] = A
+X[:,2] = A**(2.0/3.0)
+X[:,3] = A**(-1.0/3.0)
+X[:,4] = A**(-1.0)
      @@ -550,7 +999,7 @@ x = np.l
      -

      To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the itemsize functionality (the array \( x \) is actually an object which inherits the functionalities defined in Numpy) as

      +

      With scikitlearn we are now ready to use linear regression and fit our data.

      @@ -558,9 +1007,8 @@ x = np.l
      - 
      import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0]))
-print(x.itemsize)
+  
      clf = skl.LinearRegression().fit(X, Energies)
+fity = clf.predict(X)
      @@ -576,6 +1024,117 @@ x = np.l
      +

      Pretty simple! +Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data. +

      + + +
      +
      +
      +
      +
      + 
      # The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(Energies, fity))
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
+print(clf.coef_, clf.intercept_)
+
+Masses['Eapprox']  = fity
+# Generate a plot comparing the experimental with the fitted values values.
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$A = N + Z$')
+ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
+ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
+            label='Ame2016')
+ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
+            label='Fit')
+ax.legend()
+save_fig("Masses2016")
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      And what about using neural networks?

      +

      The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn's multi-layer perceptron (or feed forward neural network) +functionality. +

      + + +
      +
      +
      +
      +
      + 
      from sklearn.neural_network import MLPRegressor
+from sklearn.metrics import accuracy_score
+import seaborn as sns
+
+X_train = X
+Y_train = Energies
+n_hidden_neurons = 100
+epochs = 100
+# store models for later use
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+sns.set()
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X_train, Y_train)
+        DNN_scikit[i][j] = dnn
+        train_accuracy[i][j] = dnn.score(X_train, Y_train)
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      A first summary

      + +

      The aim behind these introductory words was to present to you various +Python libraries and their functionalities, in particular libraries like +numpy, pandas, xarray and matplotlib and other that make our life much easier +in handling various data sets and visualizing data. +

      + +

      Furthermore, +Scikit-Learn allows us with few lines of code to implement popular +Machine Learning algorithms for supervised learning. Later we will meet Tensorflow, a powerful library for deep learning. +Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails. +

      @@ -602,7 +1161,7 @@ x = np.l

    2937. 42
    2938. 43
    2939. ...
      2940. -
    2940. 66
      2941. +
    2941. 62
    2942. »
      2943. diff --git a/doc/pub/week34/html/._week34-bs034.html b/doc/pub/week34/html/._week34-bs034.html index dce995f67..ce6e498af 100644 --- a/doc/pub/week34/html/._week34-bs034.html +++ b/doc/pub/week34/html/._week34-bs034.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    2943. Installing R, C++, cython or Julia
    2944. Installing R, C++, cython, Numba etc
    2945. Numpy examples and Important Matrix and vector handling packages
      2946. -
    2946. Basic Matrix Features
      2947. -
    2947.    Some famous Matrices
      2948. -
    2948.    More Basic Matrix Features
      2949. -
    2949. Numpy and arrays
      2950. -
    2950. Matrices in Python
      2951. -
    2951. Meet the Pandas
      2952. -
    2952. Friday August 27
      2953. -
    2953.    Simple linear regression model using scikit-learn
      2954. -
    2954.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2955. -
    2955.    Organizing our data
      2956. -
    2956.    Seeing the wood for the trees
      2957. -
    2957.    And what about using neural networks?
      2958. -
    2958. A first summary
      2959. -
    2959. Why Linear Regression (aka Ordinary Least Squares and family)
      2960. -
    2960. Regression analysis, overarching aims
      2961. -
    2961. Regression analysis, overarching aims II
      2962. -
    2962. Examples
      2963. -
    2963. General linear models
      2964. -
    2964. Rewriting the fitting procedure as a linear algebra problem
      2965. -
    2965. Rewriting the fitting procedure as a linear algebra problem, more details
      2966. -
    2966. Generalizing the fitting procedure as a linear algebra problem
      2967. -
    2967. Generalizing the fitting procedure as a linear algebra problem
      2968. -
    2968. Optimizing our parameters
      2969. -
    2969. Our model for the nuclear binding energies
      2970. -
    2970. Optimizing our parameters, more details
      2971. -
    2971. Interpretations and optimizing our parameters
      2972. -
    2972. Interpretations and optimizing our parameters
      2973. -
    2973. Some useful matrix and vector expressions
      2974. -
    2974. Interpretations and optimizing our parameters
      2975. -
    2975. Own code for Ordinary Least Squares
      2976. -
    2976. Adding error analysis and training set up
      2977. -
    2977. The \( \chi^2 \) function
      2978. -
    2978. The \( \chi^2 \) function
      2979. -
    2979. The \( \chi^2 \) function
      2980. -
    2980. The \( \chi^2 \) function
      2981. -
    2981. The \( \chi^2 \) function
      2982. -
    2982. The \( \chi^2 \) function
      2983. -
    2983. Fitting an Equation of State for Dense Nuclear Matter
      2984. -
    2984. The code
      2985. -
    2985. Splitting our Data in Training and Test data
      2986. -
    2986. Exercises
      2987. -
    2987. Exercise 1: Setting up various Python environments
      2988. -
    2988. Exercise 2: making your own data and exploring scikit-learn
      2989. -
    2989. Exercise 3: Normalizing our data
      2990. +
    2990. Numpy and arrays
      2991. +
    2991. Matrices in Python
      2992. +
    2992. Meet the Pandas
      2993. +
    2993.    Simple linear regression model using scikit-learn
      2994. +
    2994.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      2995. +
    2995.    Organizing our data
      2996. +
    2996.    And what about using neural networks?
      2997. +
    2997. A first summary
      2998. +
    2998. Why Linear Regression (aka Ordinary Least Squares and family)
      2999. +
    2999. Regression analysis, overarching aims
      3000. +
    3000. Regression analysis, overarching aims II
      3001. +
    3001. Examples
      3002. +
    3002. General linear models
      3003. +
    3003. Rewriting the fitting procedure as a linear algebra problem
      3004. +
    3004. Rewriting the fitting procedure as a linear algebra problem, more details
      3005. +
    3005. Generalizing the fitting procedure as a linear algebra problem
      3006. +
    3006. Generalizing the fitting procedure as a linear algebra problem
      3007. +
    3007. Optimizing our parameters
      3008. +
    3008. Our model for the nuclear binding energies
      3009. +
    3009. Optimizing our parameters, more details
      3010. +
    3010. Interpretations and optimizing our parameters
      3011. +
    3011. Interpretations and optimizing our parameters
      3012. +
    3012. Some useful matrix and vector expressions
      3013. +
    3013. Interpretations and optimizing our parameters
      3014. +
    3014. Own code for Ordinary Least Squares
      3015. +
    3015. Adding error analysis and training set up
      3016. +
    3016. The \( \chi^2 \) function
      3017. +
    3017. The \( \chi^2 \) function
      3018. +
    3018. The \( \chi^2 \) function
      3019. +
    3019. The \( \chi^2 \) function
      3020. +
    3020. The \( \chi^2 \) function
      3021. +
    3021. The \( \chi^2 \) function
      3022. +
    3022. Fitting an Equation of State for Dense Nuclear Matter
      3023. +
    3023. The code
      3024. +
    3024. Splitting our Data in Training and Test data
      3025. +
    3025. Exercises
      3026. +
    3026. Exercise 1: Setting up various Python environments
      3027. +
    3027. Exercise 2: making your own data and exploring scikit-learn
      3028. +
    3028. Exercise 3: Split data in test and training data
      3029. @@ -351,277 +335,24 @@ MathJax.Hub.Config({

       

       

       

      -

      Matrices in Python

      +

      Why Linear Regression (aka Ordinary Least Squares and family)

      -

      Having defined vectors, we are now ready to try out matrices. We can -define a \( 3 \times 3 \) real matrix \( \boldsymbol{A} \) as (recall that we user -lowercase letters for vectors and uppercase letters for matrices) +

      Fitting a continuous function with linear parameterization in terms of the parameters \( \boldsymbol{\beta} \).

      +
        +
      • Method of choice for fitting a continuous function!
        • +
      • Gives an excellent introduction to central Machine Learning features with understandable pedagogical links to other methods like Neural Networks, Support Vector Machines etc
        • +
      • Analytical expression for the fitting parameters \( \boldsymbol{\beta} \)
        • +
      • Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more
        • +
      • Analytical relation with probabilistic interpretations
        • +
      • Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics
        • +
      • Easy to code! And links well with classification problems and logistic regression and neural networks
        • +
      • Allows for easy hands-on understanding of gradient descent methods
        • +
      • and many more features
        • +
      +

      For more discussions of Ridge and Lasso regression, Wessel van Wieringen's article is highly recommended. +Similarly, Mehta et al's article is also recommended.

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-print(A)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      If we use the shape function we would get \( (3, 3) \) as output, that is verifying that our matrix is a \( 3\times 3 \) matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[:,0]) 
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      We can continue this was by printing out other columns or rows. The example here prints out the second column

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[1,:]) 
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to zero
-A = np.zeros( (n, n) )
-print(A) 
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      or initializing all elements to

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to one
-A = np.ones( (n, n) )
-print(A) 
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      or as unitarily distributed random numbers (see the material on random number generators in the statistics part)

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
-A = np.random.rand(n, n)
-print(A) 
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as -

      -$$ -\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -$$ - -

      where for example

      -$$ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -

      The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. -The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \) -

      -$$ -\boldsymbol{W} = \begin{bmatrix} x_0 & x_1 & x_2 & \dots & x_{n-2} & x_{n-1} \\ - y_0 & y_1 & y_2 & \dots & y_{n-2} & y_{n-1} \\ - z_0 & z_1 & z_2 & \dots & z_{n-2} & z_{n-1} \\ - \end{bmatrix}, -$$ - -

      which in turn is converted into into the \( 3\times 3 \) covariance matrix -\( \boldsymbol{\Sigma} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function. -

      - - - -
      -
      -
      -
      -
      - 
      # Importing various packages
-import numpy as np
-
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -
      -
      -
      -
      -
      - 
      import numpy as np
-import matplotlib.pyplot as plt
-from scipy import sparse
-eye = np.eye(4)
-print(eye)
-sparse_mtx = sparse.csr_matrix(eye)
-print(sparse_mtx)
-x = np.linspace(-10,10,100)
-y = np.sin(x)
-plt.plot(x,y,marker='x')
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      diff --git a/doc/pub/week34/html/._week34-bs035.html b/doc/pub/week34/html/._week34-bs035.html index 8b950918b..31dbbae4d 100644 --- a/doc/pub/week34/html/._week34-bs035.html +++ b/doc/pub/week34/html/._week34-bs035.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3029. Installing R, C++, cython or Julia
    3030. Installing R, C++, cython, Numba etc
    3031. Numpy examples and Important Matrix and vector handling packages
      3032. -
    3032. Basic Matrix Features
      3033. -
    3033.    Some famous Matrices
      3034. -
    3034.    More Basic Matrix Features
      3035. -
    3035. Numpy and arrays
      3036. -
    3036. Matrices in Python
      3037. -
    3037. Meet the Pandas
      3038. -
    3038. Friday August 27
      3039. -
    3039.    Simple linear regression model using scikit-learn
      3040. -
    3040.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3041. -
    3041.    Organizing our data
      3042. -
    3042.    Seeing the wood for the trees
      3043. -
    3043.    And what about using neural networks?
      3044. -
    3044. A first summary
      3045. -
    3045. Why Linear Regression (aka Ordinary Least Squares and family)
      3046. -
    3046. Regression analysis, overarching aims
      3047. -
    3047. Regression analysis, overarching aims II
      3048. -
    3048. Examples
      3049. -
    3049. General linear models
      3050. -
    3050. Rewriting the fitting procedure as a linear algebra problem
      3051. -
    3051. Rewriting the fitting procedure as a linear algebra problem, more details
      3052. -
    3052. Generalizing the fitting procedure as a linear algebra problem
      3053. -
    3053. Generalizing the fitting procedure as a linear algebra problem
      3054. -
    3054. Optimizing our parameters
      3055. -
    3055. Our model for the nuclear binding energies
      3056. -
    3056. Optimizing our parameters, more details
      3057. -
    3057. Interpretations and optimizing our parameters
      3058. -
    3058. Interpretations and optimizing our parameters
      3059. -
    3059. Some useful matrix and vector expressions
      3060. -
    3060. Interpretations and optimizing our parameters
      3061. -
    3061. Own code for Ordinary Least Squares
      3062. -
    3062. Adding error analysis and training set up
      3063. -
    3063. The \( \chi^2 \) function
      3064. -
    3064. The \( \chi^2 \) function
      3065. -
    3065. The \( \chi^2 \) function
      3066. -
    3066. The \( \chi^2 \) function
      3067. -
    3067. The \( \chi^2 \) function
      3068. -
    3068. The \( \chi^2 \) function
      3069. -
    3069. Fitting an Equation of State for Dense Nuclear Matter
      3070. -
    3070. The code
      3071. -
    3071. Splitting our Data in Training and Test data
      3072. -
    3072. Exercises
      3073. -
    3073. Exercise 1: Setting up various Python environments
      3074. -
    3074. Exercise 2: making your own data and exploring scikit-learn
      3075. -
    3075. Exercise 3: Normalizing our data
      3076. +
    3076. Numpy and arrays
      3077. +
    3077. Matrices in Python
      3078. +
    3078. Meet the Pandas
      3079. +
    3079.    Simple linear regression model using scikit-learn
      3080. +
    3080.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3081. +
    3081.    Organizing our data
      3082. +
    3082.    And what about using neural networks?
      3083. +
    3083. A first summary
      3084. +
    3084. Why Linear Regression (aka Ordinary Least Squares and family)
      3085. +
    3085. Regression analysis, overarching aims
      3086. +
    3086. Regression analysis, overarching aims II
      3087. +
    3087. Examples
      3088. +
    3088. General linear models
      3089. +
    3089. Rewriting the fitting procedure as a linear algebra problem
      3090. +
    3090. Rewriting the fitting procedure as a linear algebra problem, more details
      3091. +
    3091. Generalizing the fitting procedure as a linear algebra problem
      3092. +
    3092. Generalizing the fitting procedure as a linear algebra problem
      3093. +
    3093. Optimizing our parameters
      3094. +
    3094. Our model for the nuclear binding energies
      3095. +
    3095. Optimizing our parameters, more details
      3096. +
    3096. Interpretations and optimizing our parameters
      3097. +
    3097. Interpretations and optimizing our parameters
      3098. +
    3098. Some useful matrix and vector expressions
      3099. +
    3099. Interpretations and optimizing our parameters
      3100. +
    3100. Own code for Ordinary Least Squares
      3101. +
    3101. Adding error analysis and training set up
      3102. +
    3102. The \( \chi^2 \) function
      3103. +
    3103. The \( \chi^2 \) function
      3104. +
    3104. The \( \chi^2 \) function
      3105. +
    3105. The \( \chi^2 \) function
      3106. +
    3106. The \( \chi^2 \) function
      3107. +
    3107. The \( \chi^2 \) function
      3108. +
    3108. Fitting an Equation of State for Dense Nuclear Matter
      3109. +
    3109. The code
      3110. +
    3110. Splitting our Data in Training and Test data
      3111. +
    3111. Exercises
      3112. +
    3112. Exercise 1: Setting up various Python environments
      3113. +
    3113. Exercise 2: making your own data and exploring scikit-learn
      3114. +
    3114. Exercise 3: Split data in test and training data
      3115. @@ -351,253 +335,25 @@ MathJax.Hub.Config({

       

       

       

      -

      Meet the Pandas

      +

      Regression analysis, overarching aims

      +
      +
      + -

      -
      -

      -
      -

      - -

      Another useful Python package is -pandas, which is an open source library -providing high-performance, easy-to-use data structures and data -analysis tools for Python. pandas stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. -pandas has two major classes, the DataFrame class with two-dimensional data objects and tabular data organized in columns and the class Series with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. -pandas allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. +

      Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) and how it varies as function of another variable or a set of such variables \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). +The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) is called the independent variable, or the predictor variable or the explanatory variable.

      -

      The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of pandas, in particular in connection with classification of data.

      - - - -
      -
      -
      -
      -
      - 
      import pandas as pd
-from IPython.display import display
-data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
-        'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
-        'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
-        'Date of Birth T.A.': [2968, 2890, 2931, 2980]
-        }
-data_pandas = pd.DataFrame(data)
-display(data_pandas)
-
      +

      A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with

      +
        +
      • \( n \) cases \( i = 0, 1, 2, \dots, n-1 \)
        • +
      • Response (target, dependent or outcome) variable \( y_i \) with \( i = 0, 1, 2, \dots, n-1 \)
        • +
      • \( p \) so-called explanatory (independent or predictor) variables \( \boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}] \) with \( i = 0, 1, 2, \dots, n-1 \) and explanatory variables running from \( 0 \) to \( p-1 \). See below for more explicit examples.
        • +
      +

      The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.

      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      In the above we have imported pandas with the shorthand pd, the latter has become the standard way we import pandas. We make then a list of various variables -and reorganize the aboves lists into a DataFrame and then print out a neat table with specific column labels as Name, place of birth and date of birth. -Displaying these results, we see that the indices are given by the default numbers from zero to three. -pandas is extremely flexible and we can easily change the above indices by defining a new type of indexing as -

      - - -
      -
      -
      -
      -
      - 
      data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
-display(data_pandas)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Thereafter we display the content of the row which begins with the index Aragorn

      - - -
      -
      -
      -
      -
      - 
      display(data_pandas.loc['Aragorn'])
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      We can easily append data to this, for example

      - - -
      -
      -
      -
      -
      - 
      new_hobbit = {'First Name': ["Peregrin"],
-              'Last Name': ["Took"],
-              'Place of birth': ["Shire"],
-              'Date of Birth T.A.': [2990]
-              }
-data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
-display(data_pandas)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix -of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. -

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 matrix
-rows = 10
-cols = 5
-a = np.random.randn(rows,cols)
-df = pd.DataFrame(a)
-display(df)
-print(df.mean())
-print(df.std())
-display(df**2)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Thereafter we can select specific columns only and plot final results

      - - -
      -
      -
      -
      -
      - 
      df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
-df.index = np.arange(10)
-
-display(df)
-print(df['Second'].mean() )
-print(df.info())
-print(df.describe())
-
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-df.cumsum().plot(lw=2.0, figsize=(10,6))
-plt.show()
-
-
-df.plot.bar(figsize=(10,6), rot=15)
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      We can produce a \( 4\times 4 \) matrix

      - - -
      -
      -
      -
      -
      - 
      b = np.arange(16).reshape((4,4))
-print(b)
-df1 = pd.DataFrame(b)
-print(df1)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      and many other operations.

      - -

      The Series class is another important class included in -pandas. You can view it as a specialization of DataFrame but where -we have just a single column of data. It shares many of the same features as DataFrame. As with DataFrame, -most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays. -As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in. -For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray. -

      @@ -624,7 +380,7 @@ For multidimensional arrays, we recommend strongly 44

    3115. 45
    3116. ...
      3117. -
    3117. 66
      3118. +
    3118. 62
    3119. »
      3120. diff --git a/doc/pub/week34/html/._week34-bs036.html b/doc/pub/week34/html/._week34-bs036.html index e16415e53..cbf6191f6 100644 --- a/doc/pub/week34/html/._week34-bs036.html +++ b/doc/pub/week34/html/._week34-bs036.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3120. Installing R, C++, cython or Julia
    3121. Installing R, C++, cython, Numba etc
    3122. Numpy examples and Important Matrix and vector handling packages
      3123. -
    3123. Basic Matrix Features
      3124. -
    3124.    Some famous Matrices
      3125. -
    3125.    More Basic Matrix Features
      3126. -
    3126. Numpy and arrays
      3127. -
    3127. Matrices in Python
      3128. -
    3128. Meet the Pandas
      3129. -
    3129. Friday August 27
      3130. -
    3130.    Simple linear regression model using scikit-learn
      3131. -
    3131.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3132. -
    3132.    Organizing our data
      3133. -
    3133.    Seeing the wood for the trees
      3134. -
    3134.    And what about using neural networks?
      3135. -
    3135. A first summary
      3136. -
    3136. Why Linear Regression (aka Ordinary Least Squares and family)
      3137. -
    3137. Regression analysis, overarching aims
      3138. -
    3138. Regression analysis, overarching aims II
      3139. -
    3139. Examples
      3140. -
    3140. General linear models
      3141. -
    3141. Rewriting the fitting procedure as a linear algebra problem
      3142. -
    3142. Rewriting the fitting procedure as a linear algebra problem, more details
      3143. -
    3143. Generalizing the fitting procedure as a linear algebra problem
      3144. -
    3144. Generalizing the fitting procedure as a linear algebra problem
      3145. -
    3145. Optimizing our parameters
      3146. -
    3146. Our model for the nuclear binding energies
      3147. -
    3147. Optimizing our parameters, more details
      3148. -
    3148. Interpretations and optimizing our parameters
      3149. -
    3149. Interpretations and optimizing our parameters
      3150. -
    3150. Some useful matrix and vector expressions
      3151. -
    3151. Interpretations and optimizing our parameters
      3152. -
    3152. Own code for Ordinary Least Squares
      3153. -
    3153. Adding error analysis and training set up
      3154. -
    3154. The \( \chi^2 \) function
      3155. -
    3155. The \( \chi^2 \) function
      3156. -
    3156. The \( \chi^2 \) function
      3157. -
    3157. The \( \chi^2 \) function
      3158. -
    3158. The \( \chi^2 \) function
      3159. -
    3159. The \( \chi^2 \) function
      3160. -
    3160. Fitting an Equation of State for Dense Nuclear Matter
      3161. -
    3161. The code
      3162. -
    3162. Splitting our Data in Training and Test data
      3163. -
    3163. Exercises
      3164. -
    3164. Exercise 1: Setting up various Python environments
      3165. -
    3165. Exercise 2: making your own data and exploring scikit-learn
      3166. -
    3166. Exercise 3: Normalizing our data
      3167. +
    3167. Numpy and arrays
      3168. +
    3168. Matrices in Python
      3169. +
    3169. Meet the Pandas
      3170. +
    3170.    Simple linear regression model using scikit-learn
      3171. +
    3171.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3172. +
    3172.    Organizing our data
      3173. +
    3173.    And what about using neural networks?
      3174. +
    3174. A first summary
      3175. +
    3175. Why Linear Regression (aka Ordinary Least Squares and family)
      3176. +
    3176. Regression analysis, overarching aims
      3177. +
    3177. Regression analysis, overarching aims II
      3178. +
    3178. Examples
      3179. +
    3179. General linear models
      3180. +
    3180. Rewriting the fitting procedure as a linear algebra problem
      3181. +
    3181. Rewriting the fitting procedure as a linear algebra problem, more details
      3182. +
    3182. Generalizing the fitting procedure as a linear algebra problem
      3183. +
    3183. Generalizing the fitting procedure as a linear algebra problem
      3184. +
    3184. Optimizing our parameters
      3185. +
    3185. Our model for the nuclear binding energies
      3186. +
    3186. Optimizing our parameters, more details
      3187. +
    3187. Interpretations and optimizing our parameters
      3188. +
    3188. Interpretations and optimizing our parameters
      3189. +
    3189. Some useful matrix and vector expressions
      3190. +
    3190. Interpretations and optimizing our parameters
      3191. +
    3191. Own code for Ordinary Least Squares
      3192. +
    3192. Adding error analysis and training set up
      3193. +
    3193. The \( \chi^2 \) function
      3194. +
    3194. The \( \chi^2 \) function
      3195. +
    3195. The \( \chi^2 \) function
      3196. +
    3196. The \( \chi^2 \) function
      3197. +
    3197. The \( \chi^2 \) function
      3198. +
    3198. The \( \chi^2 \) function
      3199. +
    3199. Fitting an Equation of State for Dense Nuclear Matter
      3200. +
    3200. The code
      3201. +
    3201. Splitting our Data in Training and Test data
      3202. +
    3202. Exercises
      3203. +
    3203. Exercise 1: Setting up various Python environments
      3204. +
    3204. Exercise 2: making your own data and exploring scikit-learn
      3205. +
    3205. Exercise 3: Split data in test and training data
      3206. @@ -351,11 +335,33 @@ MathJax.Hub.Config({

       

       

       

      -

      Friday August 27

      +

      Regression analysis, overarching aims II

      +
      +
      + -

      "Video of Lecture August 27, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust27.mp4?vrtx=view-as-webpage

      +

      Consider an experiment in which \( p \) characteristics of \( n \) samples are +measured. The data from this experiment, for various explanatory variables \( p \) are normally represented by a matrix +\( \mathbf{X} \). +

      + +

      The matrix \( \mathbf{X} \) is called the design +matrix. Additional information of the samples is available in the +form of \( \boldsymbol{y} \) (also as above). The variable \( \boldsymbol{y} \) is +generally referred to as the response variable. The aim of +regression analysis is to explain \( \boldsymbol{y} \) in terms of +\( \boldsymbol{X} \) through a functional relationship like \( y_i = +f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of +\( f(\cdot) \) is available, it is common to assume a linear relationship +between \( \boldsymbol{X} \) and \( \boldsymbol{y} \). This assumption gives rise to +the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ldots, +\beta_{p-1}]^{T} \) are the regression parameters. +

      + +

      Linear regression gives us a set of analytical equations for the parameters \( \beta_j \).

      +
      +
      -

      Video of Lecture from fall 2020 and Handwritten notes

      @@ -382,7 +388,7 @@ MathJax.Hub.Config({

    3206. 45
    3207. 46
    3208. ...
      3209. -
    3209. 66
      3210. +
    3210. 62
    3211. »
      3212. diff --git a/doc/pub/week34/html/._week34-bs037.html b/doc/pub/week34/html/._week34-bs037.html index 7e4fcd790..d4b7b24fd 100644 --- a/doc/pub/week34/html/._week34-bs037.html +++ b/doc/pub/week34/html/._week34-bs037.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3212. Installing R, C++, cython or Julia
    3213. Installing R, C++, cython, Numba etc
    3214. Numpy examples and Important Matrix and vector handling packages
      3215. -
    3215. Basic Matrix Features
      3216. -
    3216.    Some famous Matrices
      3217. -
    3217.    More Basic Matrix Features
      3218. -
    3218. Numpy and arrays
      3219. -
    3219. Matrices in Python
      3220. -
    3220. Meet the Pandas
      3221. -
    3221. Friday August 27
      3222. -
    3222.    Simple linear regression model using scikit-learn
      3223. -
    3223.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3224. -
    3224.    Organizing our data
      3225. -
    3225.    Seeing the wood for the trees
      3226. -
    3226.    And what about using neural networks?
      3227. -
    3227. A first summary
      3228. -
    3228. Why Linear Regression (aka Ordinary Least Squares and family)
      3229. -
    3229. Regression analysis, overarching aims
      3230. -
    3230. Regression analysis, overarching aims II
      3231. -
    3231. Examples
      3232. -
    3232. General linear models
      3233. -
    3233. Rewriting the fitting procedure as a linear algebra problem
      3234. -
    3234. Rewriting the fitting procedure as a linear algebra problem, more details
      3235. -
    3235. Generalizing the fitting procedure as a linear algebra problem
      3236. -
    3236. Generalizing the fitting procedure as a linear algebra problem
      3237. -
    3237. Optimizing our parameters
      3238. -
    3238. Our model for the nuclear binding energies
      3239. -
    3239. Optimizing our parameters, more details
      3240. -
    3240. Interpretations and optimizing our parameters
      3241. -
    3241. Interpretations and optimizing our parameters
      3242. -
    3242. Some useful matrix and vector expressions
      3243. -
    3243. Interpretations and optimizing our parameters
      3244. -
    3244. Own code for Ordinary Least Squares
      3245. -
    3245. Adding error analysis and training set up
      3246. -
    3246. The \( \chi^2 \) function
      3247. -
    3247. The \( \chi^2 \) function
      3248. -
    3248. The \( \chi^2 \) function
      3249. -
    3249. The \( \chi^2 \) function
      3250. -
    3250. The \( \chi^2 \) function
      3251. -
    3251. The \( \chi^2 \) function
      3252. -
    3252. Fitting an Equation of State for Dense Nuclear Matter
      3253. -
    3253. The code
      3254. -
    3254. Splitting our Data in Training and Test data
      3255. -
    3255. Exercises
      3256. -
    3256. Exercise 1: Setting up various Python environments
      3257. -
    3257. Exercise 2: making your own data and exploring scikit-learn
      3258. -
    3258. Exercise 3: Normalizing our data
      3259. +
    3259. Numpy and arrays
      3260. +
    3260. Matrices in Python
      3261. +
    3261. Meet the Pandas
      3262. +
    3262.    Simple linear regression model using scikit-learn
      3263. +
    3263.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3264. +
    3264.    Organizing our data
      3265. +
    3265.    And what about using neural networks?
      3266. +
    3266. A first summary
      3267. +
    3267. Why Linear Regression (aka Ordinary Least Squares and family)
      3268. +
    3268. Regression analysis, overarching aims
      3269. +
    3269. Regression analysis, overarching aims II
      3270. +
    3270. Examples
      3271. +
    3271. General linear models
      3272. +
    3272. Rewriting the fitting procedure as a linear algebra problem
      3273. +
    3273. Rewriting the fitting procedure as a linear algebra problem, more details
      3274. +
    3274. Generalizing the fitting procedure as a linear algebra problem
      3275. +
    3275. Generalizing the fitting procedure as a linear algebra problem
      3276. +
    3276. Optimizing our parameters
      3277. +
    3277. Our model for the nuclear binding energies
      3278. +
    3278. Optimizing our parameters, more details
      3279. +
    3279. Interpretations and optimizing our parameters
      3280. +
    3280. Interpretations and optimizing our parameters
      3281. +
    3281. Some useful matrix and vector expressions
      3282. +
    3282. Interpretations and optimizing our parameters
      3283. +
    3283. Own code for Ordinary Least Squares
      3284. +
    3284. Adding error analysis and training set up
      3285. +
    3285. The \( \chi^2 \) function
      3286. +
    3286. The \( \chi^2 \) function
      3287. +
    3287. The \( \chi^2 \) function
      3288. +
    3288. The \( \chi^2 \) function
      3289. +
    3289. The \( \chi^2 \) function
      3290. +
    3290. The \( \chi^2 \) function
      3291. +
    3291. Fitting an Equation of State for Dense Nuclear Matter
      3292. +
    3292. The code
      3293. +
    3293. Splitting our Data in Training and Test data
      3294. +
    3294. Exercises
      3295. +
    3295. Exercise 1: Setting up various Python environments
      3296. +
    3296. Exercise 2: making your own data and exploring scikit-learn
      3297. +
    3297. Exercise 3: Split data in test and training data
      3298. @@ -351,914 +335,32 @@ MathJax.Hub.Config({

       

       

       

      -

      Simple linear regression model using scikit-learn

      - -

      We start with perhaps our simplest possible example, using Scikit-Learn to perform linear regression analysis on a data set produced by us.

      - -

      What follows is a simple Python code where we have defined a function -\( y \) in terms of the variable \( x \). Both are defined as vectors with \( 100 \) entries. -The numbers in the vector \( \boldsymbol{x} \) are given -by random numbers generated with a uniform distribution with entries -\( x_i \in [0,1] \) (more about probability distribution functions -later). These values are then used to define a function \( y(x) \) -(tabulated again as a vector) with a linear dependence on \( x \) plus a -random noise added via the normal distribution. +

      Examples

      +
      +
      + +

      In order to understand the relation among the predictors (or features or properties) \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), +consider the model we discussed for describing nuclear binding energies.

      -

      The Numpy functions are imported used the import numpy as np -statement and the random number generator for the uniform distribution -is called using the function np.random.rand(), where we specificy -that we want \( 100 \) random variables. Using Numpy we define -automatically an array with the specified number of elements, \( 100 \) in -our case. With the Numpy function randn() we can compute random -numbers with the normal distribution (mean value \( \mu \) equal to zero and -variance \( \sigma^2 \) set to one) and produce the values of \( y \) assuming a linear -dependence as function of \( x \) +

      There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. +Assuming

      - $$ -y = 2x+N(0,1), +BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, $$ -

      where \( N(0,1) \) represents random numbers generated by the normal -distribution. From Scikit-Learn we import then the -LinearRegression functionality and make a prediction \( \tilde{y} = -\alpha + \beta x \) using the function fit(x,y). We call the set of -data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package -scikit-learn has also a functionality which extracts the above -fitting parameters \( \alpha \) and \( \beta \) (see below). Later we will -distinguish between training data and test data. +

      we have five predictors, that is the intercept, the \( A \) dependent term, the \( A^{2/3} \) term and the \( A^{-1/3} \) and \( A^{-1} \) terms. +This gives \( p=0,1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is a +\( p\times n \) matrix \( \boldsymbol{X} \).

      -

      For plotting we use the Python package -matplotlib which produces publication -quality figures. Feel free to explore the extensive -gallery of examples. In -this example we plot our original values of \( x \) and \( y \) as well as the -prediction ypredict (\( \tilde{y} \)), which attempts at fitting our -data with a straight line. +

      Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the +so-called credit card default data from Taiwan. The data set contains data on \( n=30000 \) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \( 24 \) such predictors or attributes leading to a design matrix of dimensionality \( 24 \times 30000 \). This is however a classification problem and we will come back to it when we discuss Logistic Regression.

      - -

      The Python code follows here.

      - - -
      -
      -
      -
      -
      - 
      # Importing various packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression
-
-x = np.random.rand(100,1)
-y = 2*x+np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-xnew = np.array([[0],[1]])
-ypredict = linreg.predict(xnew)
-
-plt.plot(xnew, ypredict, "r-")
-plt.plot(x, y ,'ro')
-plt.axis([0,1.0,0, 5.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Simple Linear Regression')
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      This example serves several aims. It allows us to demonstrate several -aspects of data analysis and later machine learning algorithms. The -immediate visualization shows that our linear fit is not -impressive. It goes through the data points, but there are many -outliers which are not reproduced by our linear regression. We could -now play around with this small program and change for example the -factor in front of \( x \) and the normal distribution. Try to change the -function \( y \) to -

      - -$$ -y = 10x+0.01 \times N(0,1), -$$ - -

      where \( x \) is defined as before. Does the fit look better? Indeed, by -reducing the role of the noise given by the normal distribution we see immediately that -our linear prediction seemingly reproduces better the training -set. However, this testing 'by the eye' is obviouly not satisfactory in the -long run. Here we have only defined the training data and our model, and -have not discussed a more rigorous approach to the cost function. -

      - -

      We need more rigorous criteria in defining whether we have succeeded or -not in modeling our training data. You will be surprised to see that -many scientists seldomly venture beyond this 'by the eye' approach. A -standard approach for the cost function is the so-called \( \chi^2 \) -function (a variant of the mean-squared error (MSE)) -

      - -$$ \chi^2 = \frac{1}{n} -\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, -$$ - -

      where \( \sigma_i^2 \) is the variance (to be defined later) of the entry -\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves -however the aim of scaling the equations and make the cost function -dimensionless. -

      - -

      Minimizing the cost function is a central aspect of -our discussions to come. Finding its minima as function of the model -parameters (\( \alpha \) and \( \beta \) in our case) will be a recurring -theme in these series of lectures. Essentially all machine learning -algorithms we will discuss center around the minimization of the -chosen cost function. This depends in turn on our specific -model for describing the data, a typical situation in supervised -learning. Automatizing the search for the minima of the cost function is a -central ingredient in all algorithms. Typical methods which are -employed are various variants of gradient methods. These will be -discussed in more detail later. Again, you'll be surprised to hear that -many practitioners minimize the above function ''by the eye', popularly dubbed as -'chi by the eye'. That is, change a parameter and see (visually and numerically) that -the \( \chi^2 \) function becomes smaller. -

      - -

      There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define -the relative error (why would we prefer the MSE instead of the relative error?) as -

      - -$$ -\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}. -$$ - -

      The squared cost function results in an arithmetic mean-unbiased -estimator, and the absolute-value cost function results in a -median-unbiased estimator (in the one-dimensional case, and a -geometric median-unbiased estimator for the multi-dimensional -case). The squared cost function has the disadvantage that it has the tendency -to be dominated by outliers. -

      - -

      We can modify easily the above Python code and plot the relative error instead

      - - -
      -
      -
      -
      -
      - 
      import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression
-
-x = np.random.rand(100,1)
-y = 5*x+0.01*np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-ypredict = linreg.predict(x)
-
-plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
-plt.axis([0,1.0,0.0, 0.5])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
-plt.title(r'Relative error')
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Depending on the parameter in front of the normal distribution, we may -have a small or larger relative error. Try to play around with -different training data sets and study (graphically) the value of the -relative error. -

      - -

      As mentioned above, Scikit-Learn has an impressive functionality. -We can for example extract the values of \( \alpha \) and \( \beta \) and -their error estimates, or the variance and standard deviation and many -other properties from the statistical data analysis. -

      - -

      Here we show an -example of the functionality of Scikit-Learn. -

      - - -
      -
      -
      -
      -
      - 
      import numpy as np 
-import matplotlib.pyplot as plt 
-from sklearn.linear_model import LinearRegression 
-from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
-
-x = np.random.rand(100,1)
-y = 2.0+ 5*x+0.5*np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-ypredict = linreg.predict(x)
-print('The intercept alpha: \n', linreg.intercept_)
-print('Coefficient beta : \n', linreg.coef_)
-# The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(y, ypredict))
-# Mean squared log error                                                        
-print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
-plt.plot(x, ypredict, "r-")
-plt.plot(x, y ,'ro')
-plt.axis([0.0,1.0,1.5, 7.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Linear Regression fit ')
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      The function coef gives us the parameter \( \beta \) of our fit while intercept yields -\( \alpha \). Depending on the constant in front of the normal distribution, we get values near or far from \( \alpha =2 \) and \( \beta =5 \). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as -

      -$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -$$ - -

      The smaller the value, the better the fit. Ideally we would like to -have an MSE equal zero. The attentive reader has probably recognized -this function as being similar to the \( \chi^2 \) function defined above. -

      - -

      The r2score function computes \( R^2 \), the coefficient of -determination. It provides a measure of how well future samples are -likely to be predicted by the model. Best possible score is 1.0 and it -can be negative (because the model can be arbitrarily worse). A -constant model that always predicts the expected value of \( \boldsymbol{y} \), -disregarding the input features, would get a \( R^2 \) score of \( 0.0 \). -

      - -

      If \( \tilde{\boldsymbol{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as

      -$$ -R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -$$ - -

      where we have defined the mean value of \( \boldsymbol{y} \) as

      -$$ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -$$ - -

      Another quantity taht we will meet again in our discussions of regression analysis is - the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error. -The MAE is defined as follows -

      -$$ -\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. -$$ - -

      We present the -squared logarithmic (quadratic) error -

      -$$ -\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, -$$ - -

      where \( \log_e (x) \) stands for the natural logarithm of \( x \). This error -estimate is best to use when targets having exponential growth, such -as population counts, average sales of a commodity over a span of -years etc. -

      - -

      Finally, another cost function is the Huber cost function used in robust regression.

      - -

      The rationale behind this possible cost function is its reduced -sensitivity to outliers in the data set. In our discussions on -dimensionality reduction and normalization of data we will meet other -ways of dealing with outliers. -

      - -

      The Huber cost function is defined as

      -$$ -H_{\delta}(\boldsymbol{a})=\left\{\begin{array}{cc}\frac{1}{2} \boldsymbol{a}^{2}& \text{for }|\boldsymbol{a}|\leq \delta\\ \delta (|\boldsymbol{a}|-\frac{1}{2}\delta ),&\text{otherwise}.\end{array}\right. -$$ - -

      Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).

      - -

      We will discuss in more detail these and other functions in the -various lectures. We conclude this part with another example. Instead -of a linear \( x \)-dependence we study now a cubic polynomial and use the -polynomial regression analysis tools of scikit-learn. -

      - - - -
      -
      -
      -
      -
      - 
      import matplotlib.pyplot as plt
-import numpy as np
-import random
-from sklearn.linear_model import Ridge
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.pipeline import make_pipeline
-from sklearn.linear_model import LinearRegression
-
-x=np.linspace(0.02,0.98,200)
-noise = np.asarray(random.sample((range(200)),200))
-y=x**3*noise
-yn=x**3*100
-poly3 = PolynomialFeatures(degree=3)
-X = poly3.fit_transform(x[:,np.newaxis])
-clf3 = LinearRegression()
-clf3.fit(X,y)
-
-Xplot=poly3.fit_transform(x[:,np.newaxis])
-poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
-plt.plot(x,yn, color='red', label="True Cubic")
-plt.scatter(x, y, label='Data', color='orange', s=15)
-plt.legend()
-plt.show()
-
-def error(a):
-    for i in y:
-        err=(y-yn)/yn
-    return abs(np.sum(err))/len(err)
-
-print (error(y))
-
      -
      -
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      -
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      -
      -
      -
      -

      To our real data: nuclear binding energies. Brief reminder on masses and binding energies

      - -

      Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding -energies. A basic quantity which can be measured for the ground -states of nuclei is the atomic mass \( M(N, Z) \) of the neutral atom with -atomic mass number \( A \) and charge \( Z \). The number of neutrons is \( N \). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). -

      - -

      Atomic masses are usually tabulated in terms of the mass excess defined by

      -$$ -\Delta M(N, Z) = M(N, Z) - uA, -$$ - -

      where \( u \) is the Atomic Mass Unit

      -$$ -u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. -$$ - -

      The nucleon masses are

      -$$ -m_p = 1.00727646693(9)u, -$$ - -

      and

      -$$ -m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. -$$ - -

      In the 2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu -there are data on masses and decays of 3437 nuclei. -

      - -

      The nuclear binding energy is defined as the energy required to break -up a given nucleus into its constituent parts of \( N \) neutrons and \( Z \) -protons. In terms of the atomic masses \( M(N, Z) \) the binding energy is -defined by -

      - -$$ -BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , -$$ - -

      where \( M_H \) is the mass of the hydrogen atom and \( m_n \) is the mass of the neutron. -In terms of the mass excess the binding energy is given by -

      -$$ -BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , -$$ - -

      where \( \Delta_H c^2 = 7.2890 \) MeV and \( \Delta_n c^2 = 8.0713 \) MeV.

      - -

      A popular and physically intuitive model which can be used to parametrize -the experimental binding energies as function of \( A \), is the so-called -liquid drop model. The ansatz is based on the following expression -

      - -$$ -BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A}, -$$ - -

      where \( A \) stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit -to the experimental data. -

      - -

      To arrive at the above expression we have assumed that we can make the following assumptions:

      - -
        -
      • There is a volume term \( a_1A \) proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.
        • -
      • There is a surface energy term \( a_2A^{2/3} \). The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
        • -
      • There is a Coulomb energy term \( a_3\frac{Z^2}{A^{1/3}} \). The electric repulsion between each pair of protons in a nucleus yields less binding.
        • -
      • There is an asymmetry term \( a_4\frac{(N-Z)^2}{A} \). This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.
        • -
      -

      We could also add a so-called pairing term, which is a correction term that -arises from the tendency of proton pairs and neutron pairs to -occur. An even number of particles is more stable than an odd number. -

      -

      Organizing our data

      - -

      Let us start with reading and organizing our data. -We start with the compilation of masses and binding energies from 2016. -After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data. -

      - -

      We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of scikit-learn.

      - - -
      -
      -
      -
      -
      - 
      # Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-import sklearn.linear_model as skl
-from sklearn.model_selection import train_test_split
-from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
-import os
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("MassEval2016.dat"),'r')
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various matplotlib commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.

      - - -
      -
      -
      -
      -
      - 
      from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-def MakePlot(x,y, styles, labels, axlabels):
-    plt.figure(figsize=(10,6))
-    for i in range(len(x)):
-        plt.plot(x[i], y[i], styles[i], label = labels[i])
-        plt.xlabel(axlabels[0])
-        plt.ylabel(axlabels[1])
-    plt.legend(loc=0)
-
      -
      -
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      - -

      Our next step is to read the data on experimental binding energies and -reorganize them as functions of the mass number \( A \), the number of -protons \( Z \) and neutrons \( N \) using pandas. Before we do this it is -always useful (unless you have a binary file or other types of compressed -data) to actually open the file and simply take a look at it! -

      - -

      In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with pandas. The file begins with some basic format information.

      - - -
      -
      -
      -
      -
      - 
      """                                                                                                                         
-This is taken from the data file of the mass 2016 evaluation.                                                               
-All files are 3436 lines long with 124 character per line.                                                                  
-       Headers are 39 lines long.                                                                                           
-   col 1     :  Fortran character control: 1 = page feed  0 = line feed                                                     
-   format    :  a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5                     
-   These formats are reflected in the pandas widths variable below, see the statement                                       
-   widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),                                                            
-   Pandas has also a variable header, with length 39 in this case.                                                          
-"""
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      The data we are interested in are in columns 2, 3, 4 and 11, giving us -the number of neutrons, protons, mass numbers and binding energies, -respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will -covert them into the pandas DataFrame structure. -

      - - - -
      -
      -
      -
      -
      - 
      # Read the experimental data with Pandas
-Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
-              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
-              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
-              header=39,
-              index_col=False)
-
-# Extrapolated values are indicated by '#' in place of the decimal place, so
-# the Ebinding column won't be numeric. Coerce to float and drop these entries.
-Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
-Masses = Masses.dropna()
-# Convert from keV to MeV.
-Masses['Ebinding'] /= 1000
-
-# Group the DataFrame by nucleon number, A.
-Masses = Masses.groupby('A')
-# Find the rows of the grouped DataFrame with the maximum binding energy.
-Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
-
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      -
      - -

      We have now read in the data, grouped them according to the variables we are interested in. -We see how easy it is to reorganize the data using pandas. If we -were to do these operations in C/C++ or Fortran, we would have had to -write various functions/subroutines which perform the above -reorganizations for us. Having reorganized the data, we can now start -to make some simple fits using both the functionalities in numpy and -Scikit-Learn afterwards. -

      - -

      Now we define five variables which contain -the number of nucleons \( A \), the number of protons \( Z \) and the number of neutrons \( N \), the element name and finally the energies themselves. -

      - - -
      -
      -
      -
      -
      - 
      A = Masses['A']
-Z = Masses['Z']
-N = Masses['N']
-Element = Masses['Element']
-Energies = Masses['Ebinding']
-print(Masses)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      The next step, and we will define this mathematically later, is to set up the so-called design matrix. We will throughout call this matrix \( \boldsymbol{X} \). -It has dimensionality \( p\times n \), where \( n \) is the number of data points and \( p \) are the so-called predictors. In our case here they are given by the number of polynomials in \( A \) we wish to include in the fit. -

      - - -
      -
      -
      -
      -
      - 
      # Now we set up the design matrix X
-X = np.zeros((len(A),5))
-X[:,0] = 1
-X[:,1] = A
-X[:,2] = A**(2.0/3.0)
-X[:,3] = A**(-1.0/3.0)
-X[:,4] = A**(-1.0)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      With scikitlearn we are now ready to use linear regression and fit our data.

      - - -
      -
      -
      -
      -
      - 
      clf = skl.LinearRegression().fit(X, Energies)
-fity = clf.predict(X)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Pretty simple! -Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data. -

      - - -
      -
      -
      -
      -
      - 
      # The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(Energies, fity))
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
-print(clf.coef_, clf.intercept_)
-
-Masses['Eapprox']  = fity
-# Generate a plot comparing the experimental with the fitted values values.
-fig, ax = plt.subplots()
-ax.set_xlabel(r'$A = N + Z$')
-ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
-ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
-            label='Ame2016')
-ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
-            label='Fit')
-ax.legend()
-save_fig("Masses2016")
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      Seeing the wood for the trees

      - -

      As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!

      - - - -
      -
      -
      -
      -
      - 
      #Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=5)
-regr_2=DecisionTreeRegressor(max_depth=7)
-regr_3=DecisionTreeRegressor(max_depth=9)
-regr_1.fit(X, Energies)
-regr_2.fit(X, Energies)
-regr_3.fit(X, Energies)
-
-
-y_1 = regr_1.predict(X)
-y_2 = regr_2.predict(X)
-y_3=regr_3.predict(X)
-Masses['Eapprox'] = y_3
-# Plot the results
-plt.figure()
-plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
-plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
-plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
-plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
-
-plt.xlabel("$A$")
-plt.ylabel("$E$[MeV]")
-plt.title("Decision Tree Regression")
-plt.legend()
-save_fig("Masses2016Trees")
-plt.show()
-print(Masses)
-print(np.mean( (Energies-y_1)**2))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      And what about using neural networks?

      -

      The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn's multi-layer perceptron (or feed forward neural network) -functionality. -

      - - -
      -
      -
      -
      -
      - 
      from sklearn.neural_network import MLPRegressor
-from sklearn.metrics import accuracy_score
-import seaborn as sns
-
-X_train = X
-Y_train = Energies
-n_hidden_neurons = 100
-epochs = 100
-# store models for later use
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-# store the models for later use
-DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-sns.set()
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
-                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
-        dnn.fit(X_train, Y_train)
-        DNN_scikit[i][j] = dnn
-        train_accuracy[i][j] = dnn.score(X_train, Y_train)
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      A first summary

      - -

      The aim behind these introductory words was to present to you various -Python libraries and their functionalities, in particular libraries like -numpy, pandas, xarray and matplotlib and other that make our life much easier -in handling various data sets and visualizing data. -

      - -

      Furthermore, -Scikit-Learn allows us with few lines of code to implement popular -Machine Learning algorithms for supervised learning. Later we will meet Tensorflow, a powerful library for deep learning. -Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails. -

      @@ -1285,7 +387,7 @@ Now it is time to dive more into the details of various methods. We will start w

    3298. 46
    3299. 47
    3300. ...
      3301. -
    3301. 66
      3302. +
    3302. 62
    3303. »
      3304. diff --git a/doc/pub/week34/html/._week34-bs038.html b/doc/pub/week34/html/._week34-bs038.html index f3b259e59..1c16ab3e8 100644 --- a/doc/pub/week34/html/._week34-bs038.html +++ b/doc/pub/week34/html/._week34-bs038.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3304. Installing R, C++, cython or Julia
    3305. Installing R, C++, cython, Numba etc
    3306. Numpy examples and Important Matrix and vector handling packages
      3307. -
    3307. Basic Matrix Features
      3308. -
    3308.    Some famous Matrices
      3309. -
    3309.    More Basic Matrix Features
      3310. -
    3310. Numpy and arrays
      3311. -
    3311. Matrices in Python
      3312. -
    3312. Meet the Pandas
      3313. -
    3313. Friday August 27
      3314. -
    3314.    Simple linear regression model using scikit-learn
      3315. -
    3315.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3316. -
    3316.    Organizing our data
      3317. -
    3317.    Seeing the wood for the trees
      3318. -
    3318.    And what about using neural networks?
      3319. -
    3319. A first summary
      3320. -
    3320. Why Linear Regression (aka Ordinary Least Squares and family)
      3321. -
    3321. Regression analysis, overarching aims
      3322. -
    3322. Regression analysis, overarching aims II
      3323. -
    3323. Examples
      3324. -
    3324. General linear models
      3325. -
    3325. Rewriting the fitting procedure as a linear algebra problem
      3326. -
    3326. Rewriting the fitting procedure as a linear algebra problem, more details
      3327. -
    3327. Generalizing the fitting procedure as a linear algebra problem
      3328. -
    3328. Generalizing the fitting procedure as a linear algebra problem
      3329. -
    3329. Optimizing our parameters
      3330. -
    3330. Our model for the nuclear binding energies
      3331. -
    3331. Optimizing our parameters, more details
      3332. -
    3332. Interpretations and optimizing our parameters
      3333. -
    3333. Interpretations and optimizing our parameters
      3334. -
    3334. Some useful matrix and vector expressions
      3335. -
    3335. Interpretations and optimizing our parameters
      3336. -
    3336. Own code for Ordinary Least Squares
      3337. -
    3337. Adding error analysis and training set up
      3338. -
    3338. The \( \chi^2 \) function
      3339. -
    3339. The \( \chi^2 \) function
      3340. -
    3340. The \( \chi^2 \) function
      3341. -
    3341. The \( \chi^2 \) function
      3342. -
    3342. The \( \chi^2 \) function
      3343. -
    3343. The \( \chi^2 \) function
      3344. -
    3344. Fitting an Equation of State for Dense Nuclear Matter
      3345. -
    3345. The code
      3346. -
    3346. Splitting our Data in Training and Test data
      3347. -
    3347. Exercises
      3348. -
    3348. Exercise 1: Setting up various Python environments
      3349. -
    3349. Exercise 2: making your own data and exploring scikit-learn
      3350. -
    3350. Exercise 3: Normalizing our data
      3351. +
    3351. Numpy and arrays
      3352. +
    3352. Matrices in Python
      3353. +
    3353. Meet the Pandas
      3354. +
    3354.    Simple linear regression model using scikit-learn
      3355. +
    3355.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3356. +
    3356.    Organizing our data
      3357. +
    3357.    And what about using neural networks?
      3358. +
    3358. A first summary
      3359. +
    3359. Why Linear Regression (aka Ordinary Least Squares and family)
      3360. +
    3360. Regression analysis, overarching aims
      3361. +
    3361. Regression analysis, overarching aims II
      3362. +
    3362. Examples
      3363. +
    3363. General linear models
      3364. +
    3364. Rewriting the fitting procedure as a linear algebra problem
      3365. +
    3365. Rewriting the fitting procedure as a linear algebra problem, more details
      3366. +
    3366. Generalizing the fitting procedure as a linear algebra problem
      3367. +
    3367. Generalizing the fitting procedure as a linear algebra problem
      3368. +
    3368. Optimizing our parameters
      3369. +
    3369. Our model for the nuclear binding energies
      3370. +
    3370. Optimizing our parameters, more details
      3371. +
    3371. Interpretations and optimizing our parameters
      3372. +
    3372. Interpretations and optimizing our parameters
      3373. +
    3373. Some useful matrix and vector expressions
      3374. +
    3374. Interpretations and optimizing our parameters
      3375. +
    3375. Own code for Ordinary Least Squares
      3376. +
    3376. Adding error analysis and training set up
      3377. +
    3377. The \( \chi^2 \) function
      3378. +
    3378. The \( \chi^2 \) function
      3379. +
    3379. The \( \chi^2 \) function
      3380. +
    3380. The \( \chi^2 \) function
      3381. +
    3381. The \( \chi^2 \) function
      3382. +
    3382. The \( \chi^2 \) function
      3383. +
    3383. Fitting an Equation of State for Dense Nuclear Matter
      3384. +
    3384. The code
      3385. +
    3385. Splitting our Data in Training and Test data
      3386. +
    3386. Exercises
      3387. +
    3387. Exercise 1: Setting up various Python environments
      3388. +
    3388. Exercise 2: making your own data and exploring scikit-learn
      3389. +
    3389. Exercise 3: Split data in test and training data
      3390. @@ -351,23 +335,21 @@ MathJax.Hub.Config({

       

       

       

      -

      Why Linear Regression (aka Ordinary Least Squares and family)

      +

      General linear models

      +
      +
      + +

      Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.

      + +

      Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \( y \) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \( n-1 \) with \( n \) points, that is

      +$$ +y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, +$$ + +

      where \( \epsilon_i \) is the error in our approximation.

      +
      +
      -

      Fitting a continuous function with linear parameterization in terms of the parameters \( \boldsymbol{\beta} \).

      -
        -
      • Method of choice for fitting a continuous function!
        • -
      • Gives an excellent introduction to central Machine Learning features with understandable pedagogical links to other methods like Neural Networks, Support Vector Machines etc
        • -
      • Analytical expression for the fitting parameters \( \boldsymbol{\beta} \)
        • -
      • Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more
        • -
      • Analytical relation with probabilistic interpretations
        • -
      • Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics
        • -
      • Easy to code! And links well with classification problems and logistic regression and neural networks
        • -
      • Allows for easy hands-on understanding of gradient descent methods
        • -
      • and many more features
        • -
      -

      For more discussions of Ridge and Lasso regression, Wessel van Wieringen's article is highly recommended. -Similarly, Mehta et al's article is also recommended. -

      @@ -394,7 +376,7 @@ Similarly, Mehta et al

    3390. 47
    3391. 48
    3392. ...
      3393. -
    3393. 66
      3394. +
    3394. 62
    3395. »
      3396. diff --git a/doc/pub/week34/html/._week34-bs039.html b/doc/pub/week34/html/._week34-bs039.html index b4a839241..17aa7bf6e 100644 --- a/doc/pub/week34/html/._week34-bs039.html +++ b/doc/pub/week34/html/._week34-bs039.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3396. Installing R, C++, cython or Julia
    3397. Installing R, C++, cython, Numba etc
    3398. Numpy examples and Important Matrix and vector handling packages
      3399. -
    3399. Basic Matrix Features
      3400. -
    3400.    Some famous Matrices
      3401. -
    3401.    More Basic Matrix Features
      3402. -
    3402. Numpy and arrays
      3403. -
    3403. Matrices in Python
      3404. -
    3404. Meet the Pandas
      3405. -
    3405. Friday August 27
      3406. -
    3406.    Simple linear regression model using scikit-learn
      3407. -
    3407.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3408. -
    3408.    Organizing our data
      3409. -
    3409.    Seeing the wood for the trees
      3410. -
    3410.    And what about using neural networks?
      3411. -
    3411. A first summary
      3412. -
    3412. Why Linear Regression (aka Ordinary Least Squares and family)
      3413. -
    3413. Regression analysis, overarching aims
      3414. -
    3414. Regression analysis, overarching aims II
      3415. -
    3415. Examples
      3416. -
    3416. General linear models
      3417. -
    3417. Rewriting the fitting procedure as a linear algebra problem
      3418. -
    3418. Rewriting the fitting procedure as a linear algebra problem, more details
      3419. -
    3419. Generalizing the fitting procedure as a linear algebra problem
      3420. -
    3420. Generalizing the fitting procedure as a linear algebra problem
      3421. -
    3421. Optimizing our parameters
      3422. -
    3422. Our model for the nuclear binding energies
      3423. -
    3423. Optimizing our parameters, more details
      3424. -
    3424. Interpretations and optimizing our parameters
      3425. -
    3425. Interpretations and optimizing our parameters
      3426. -
    3426. Some useful matrix and vector expressions
      3427. -
    3427. Interpretations and optimizing our parameters
      3428. -
    3428. Own code for Ordinary Least Squares
      3429. -
    3429. Adding error analysis and training set up
      3430. -
    3430. The \( \chi^2 \) function
      3431. -
    3431. The \( \chi^2 \) function
      3432. -
    3432. The \( \chi^2 \) function
      3433. -
    3433. The \( \chi^2 \) function
      3434. -
    3434. The \( \chi^2 \) function
      3435. -
    3435. The \( \chi^2 \) function
      3436. -
    3436. Fitting an Equation of State for Dense Nuclear Matter
      3437. -
    3437. The code
      3438. -
    3438. Splitting our Data in Training and Test data
      3439. -
    3439. Exercises
      3440. -
    3440. Exercise 1: Setting up various Python environments
      3441. -
    3441. Exercise 2: making your own data and exploring scikit-learn
      3442. -
    3442. Exercise 3: Normalizing our data
      3443. +
    3443. Numpy and arrays
      3444. +
    3444. Matrices in Python
      3445. +
    3445. Meet the Pandas
      3446. +
    3446.    Simple linear regression model using scikit-learn
      3447. +
    3447.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3448. +
    3448.    Organizing our data
      3449. +
    3449.    And what about using neural networks?
      3450. +
    3450. A first summary
      3451. +
    3451. Why Linear Regression (aka Ordinary Least Squares and family)
      3452. +
    3452. Regression analysis, overarching aims
      3453. +
    3453. Regression analysis, overarching aims II
      3454. +
    3454. Examples
      3455. +
    3455. General linear models
      3456. +
    3456. Rewriting the fitting procedure as a linear algebra problem
      3457. +
    3457. Rewriting the fitting procedure as a linear algebra problem, more details
      3458. +
    3458. Generalizing the fitting procedure as a linear algebra problem
      3459. +
    3459. Generalizing the fitting procedure as a linear algebra problem
      3460. +
    3460. Optimizing our parameters
      3461. +
    3461. Our model for the nuclear binding energies
      3462. +
    3462. Optimizing our parameters, more details
      3463. +
    3463. Interpretations and optimizing our parameters
      3464. +
    3464. Interpretations and optimizing our parameters
      3465. +
    3465. Some useful matrix and vector expressions
      3466. +
    3466. Interpretations and optimizing our parameters
      3467. +
    3467. Own code for Ordinary Least Squares
      3468. +
    3468. Adding error analysis and training set up
      3469. +
    3469. The \( \chi^2 \) function
      3470. +
    3470. The \( \chi^2 \) function
      3471. +
    3471. The \( \chi^2 \) function
      3472. +
    3472. The \( \chi^2 \) function
      3473. +
    3473. The \( \chi^2 \) function
      3474. +
    3474. The \( \chi^2 \) function
      3475. +
    3475. Fitting an Equation of State for Dense Nuclear Matter
      3476. +
    3476. The code
      3477. +
    3477. Splitting our Data in Training and Test data
      3478. +
    3478. Exercises
      3479. +
    3479. Exercise 1: Setting up various Python environments
      3480. +
    3480. Exercise 2: making your own data and exploring scikit-learn
      3481. +
    3481. Exercise 3: Split data in test and training data
      3482. @@ -351,22 +335,20 @@ MathJax.Hub.Config({

       

       

       

      -

      Regression analysis, overarching aims

      +

      Rewriting the fitting procedure as a linear algebra problem

      - -

      Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) and how it varies as function of another variable or a set of such variables \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). -The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) is called the independent variable, or the predictor variable or the explanatory variable. -

      - -

      A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with

      -
        -
      • \( n \) cases \( i = 0, 1, 2, \dots, n-1 \)
        • -
      • Response (target, dependent or outcome) variable \( y_i \) with \( i = 0, 1, 2, \dots, n-1 \)
        • -
      • \( p \) so-called explanatory (independent or predictor) variables \( \boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}] \) with \( i = 0, 1, 2, \dots, n-1 \) and explanatory variables running from \( 0 \) to \( p-1 \). See below for more explicit examples.
        • -
      -

      The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.

      +

      For every set of values \( y_i,x_i \) we have thus the corresponding set of equations

      +$$ +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$
      @@ -396,7 +378,7 @@ The first variable is called the dependent, the outcome or the
    3482. 48
    3483. 49
    3484. ...
      3485. -
    3485. 66
      3486. +
    3486. 62
    3487. »
      3488. diff --git a/doc/pub/week34/html/._week34-bs040.html b/doc/pub/week34/html/._week34-bs040.html index cfbf779dc..ad3664320 100644 --- a/doc/pub/week34/html/._week34-bs040.html +++ b/doc/pub/week34/html/._week34-bs040.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3488. Installing R, C++, cython or Julia
    3489. Installing R, C++, cython, Numba etc
    3490. Numpy examples and Important Matrix and vector handling packages
      3491. -
    3491. Basic Matrix Features
      3492. -
    3492.    Some famous Matrices
      3493. -
    3493.    More Basic Matrix Features
      3494. -
    3494. Numpy and arrays
      3495. -
    3495. Matrices in Python
      3496. -
    3496. Meet the Pandas
      3497. -
    3497. Friday August 27
      3498. -
    3498.    Simple linear regression model using scikit-learn
      3499. -
    3499.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3500. -
    3500.    Organizing our data
      3501. -
    3501.    Seeing the wood for the trees
      3502. -
    3502.    And what about using neural networks?
      3503. -
    3503. A first summary
      3504. -
    3504. Why Linear Regression (aka Ordinary Least Squares and family)
      3505. -
    3505. Regression analysis, overarching aims
      3506. -
    3506. Regression analysis, overarching aims II
      3507. -
    3507. Examples
      3508. -
    3508. General linear models
      3509. -
    3509. Rewriting the fitting procedure as a linear algebra problem
      3510. -
    3510. Rewriting the fitting procedure as a linear algebra problem, more details
      3511. -
    3511. Generalizing the fitting procedure as a linear algebra problem
      3512. -
    3512. Generalizing the fitting procedure as a linear algebra problem
      3513. -
    3513. Optimizing our parameters
      3514. -
    3514. Our model for the nuclear binding energies
      3515. -
    3515. Optimizing our parameters, more details
      3516. -
    3516. Interpretations and optimizing our parameters
      3517. -
    3517. Interpretations and optimizing our parameters
      3518. -
    3518. Some useful matrix and vector expressions
      3519. -
    3519. Interpretations and optimizing our parameters
      3520. -
    3520. Own code for Ordinary Least Squares
      3521. -
    3521. Adding error analysis and training set up
      3522. -
    3522. The \( \chi^2 \) function
      3523. -
    3523. The \( \chi^2 \) function
      3524. -
    3524. The \( \chi^2 \) function
      3525. -
    3525. The \( \chi^2 \) function
      3526. -
    3526. The \( \chi^2 \) function
      3527. -
    3527. The \( \chi^2 \) function
      3528. -
    3528. Fitting an Equation of State for Dense Nuclear Matter
      3529. -
    3529. The code
      3530. -
    3530. Splitting our Data in Training and Test data
      3531. -
    3531. Exercises
      3532. -
    3532. Exercise 1: Setting up various Python environments
      3533. -
    3533. Exercise 2: making your own data and exploring scikit-learn
      3534. -
    3534. Exercise 3: Normalizing our data
      3535. +
    3535. Numpy and arrays
      3536. +
    3536. Matrices in Python
      3537. +
    3537. Meet the Pandas
      3538. +
    3538.    Simple linear regression model using scikit-learn
      3539. +
    3539.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3540. +
    3540.    Organizing our data
      3541. +
    3541.    And what about using neural networks?
      3542. +
    3542. A first summary
      3543. +
    3543. Why Linear Regression (aka Ordinary Least Squares and family)
      3544. +
    3544. Regression analysis, overarching aims
      3545. +
    3545. Regression analysis, overarching aims II
      3546. +
    3546. Examples
      3547. +
    3547. General linear models
      3548. +
    3548. Rewriting the fitting procedure as a linear algebra problem
      3549. +
    3549. Rewriting the fitting procedure as a linear algebra problem, more details
      3550. +
    3550. Generalizing the fitting procedure as a linear algebra problem
      3551. +
    3551. Generalizing the fitting procedure as a linear algebra problem
      3552. +
    3552. Optimizing our parameters
      3553. +
    3553. Our model for the nuclear binding energies
      3554. +
    3554. Optimizing our parameters, more details
      3555. +
    3555. Interpretations and optimizing our parameters
      3556. +
    3556. Interpretations and optimizing our parameters
      3557. +
    3557. Some useful matrix and vector expressions
      3558. +
    3558. Interpretations and optimizing our parameters
      3559. +
    3559. Own code for Ordinary Least Squares
      3560. +
    3560. Adding error analysis and training set up
      3561. +
    3561. The \( \chi^2 \) function
      3562. +
    3562. The \( \chi^2 \) function
      3563. +
    3563. The \( \chi^2 \) function
      3564. +
    3564. The \( \chi^2 \) function
      3565. +
    3565. The \( \chi^2 \) function
      3566. +
    3566. The \( \chi^2 \) function
      3567. +
    3567. Fitting an Equation of State for Dense Nuclear Matter
      3568. +
    3568. The code
      3569. +
    3569. Splitting our Data in Training and Test data
      3570. +
    3570. Exercises
      3571. +
    3571. Exercise 1: Setting up various Python environments
      3572. +
    3572. Exercise 2: making your own data and exploring scikit-learn
      3573. +
    3573. Exercise 3: Split data in test and training data
      3574. @@ -351,30 +335,43 @@ MathJax.Hub.Config({

       

       

       

      -

      Regression analysis, overarching aims II

      +

      Rewriting the fitting procedure as a linear algebra problem, more details

      +

      Defining the vectors

      +$$ +\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, +$$ -

      Consider an experiment in which \( p \) characteristics of \( n \) samples are -measured. The data from this experiment, for various explanatory variables \( p \) are normally represented by a matrix -\( \mathbf{X} \). -

      +

      and

      +$$ +\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, +$$ -

      The matrix \( \mathbf{X} \) is called the design -matrix. Additional information of the samples is available in the -form of \( \boldsymbol{y} \) (also as above). The variable \( \boldsymbol{y} \) is -generally referred to as the response variable. The aim of -regression analysis is to explain \( \boldsymbol{y} \) in terms of -\( \boldsymbol{X} \) through a functional relationship like \( y_i = -f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of -\( f(\cdot) \) is available, it is common to assume a linear relationship -between \( \boldsymbol{X} \) and \( \boldsymbol{y} \). This assumption gives rise to -the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ldots, -\beta_{p-1}]^{T} \) are the regression parameters. -

      +

      and

      +$$ +\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, +$$ -

      Linear regression gives us a set of analytical equations for the parameters \( \beta_j \).

      +

      and the design matrix

      +$$ +\boldsymbol{X}= +\begin{bmatrix} +1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ +1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ +1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ +\end{bmatrix} +$$ + +

      we can rewrite our equations as

      +$$ +\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. +$$ + +

      The above design matrix is called a Vandermonde matrix.

      @@ -404,7 +401,7 @@ the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ld
    3574. 49
    3575. 50
    3576. ...
      3577. -
    3577. 66
      3578. +
    3578. 62
    3579. »
      3580. diff --git a/doc/pub/week34/html/._week34-bs041.html b/doc/pub/week34/html/._week34-bs041.html index b729d273c..873636cbc 100644 --- a/doc/pub/week34/html/._week34-bs041.html +++ b/doc/pub/week34/html/._week34-bs041.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3580. Installing R, C++, cython or Julia
    3581. Installing R, C++, cython, Numba etc
    3582. Numpy examples and Important Matrix and vector handling packages
      3583. -
    3583. Basic Matrix Features
      3584. -
    3584.    Some famous Matrices
      3585. -
    3585.    More Basic Matrix Features
      3586. -
    3586. Numpy and arrays
      3587. -
    3587. Matrices in Python
      3588. -
    3588. Meet the Pandas
      3589. -
    3589. Friday August 27
      3590. -
    3590.    Simple linear regression model using scikit-learn
      3591. -
    3591.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3592. -
    3592.    Organizing our data
      3593. -
    3593.    Seeing the wood for the trees
      3594. -
    3594.    And what about using neural networks?
      3595. -
    3595. A first summary
      3596. -
    3596. Why Linear Regression (aka Ordinary Least Squares and family)
      3597. -
    3597. Regression analysis, overarching aims
      3598. -
    3598. Regression analysis, overarching aims II
      3599. -
    3599. Examples
      3600. -
    3600. General linear models
      3601. -
    3601. Rewriting the fitting procedure as a linear algebra problem
      3602. -
    3602. Rewriting the fitting procedure as a linear algebra problem, more details
      3603. -
    3603. Generalizing the fitting procedure as a linear algebra problem
      3604. -
    3604. Generalizing the fitting procedure as a linear algebra problem
      3605. -
    3605. Optimizing our parameters
      3606. -
    3606. Our model for the nuclear binding energies
      3607. -
    3607. Optimizing our parameters, more details
      3608. -
    3608. Interpretations and optimizing our parameters
      3609. -
    3609. Interpretations and optimizing our parameters
      3610. -
    3610. Some useful matrix and vector expressions
      3611. -
    3611. Interpretations and optimizing our parameters
      3612. -
    3612. Own code for Ordinary Least Squares
      3613. -
    3613. Adding error analysis and training set up
      3614. -
    3614. The \( \chi^2 \) function
      3615. -
    3615. The \( \chi^2 \) function
      3616. -
    3616. The \( \chi^2 \) function
      3617. -
    3617. The \( \chi^2 \) function
      3618. -
    3618. The \( \chi^2 \) function
      3619. -
    3619. The \( \chi^2 \) function
      3620. -
    3620. Fitting an Equation of State for Dense Nuclear Matter
      3621. -
    3621. The code
      3622. -
    3622. Splitting our Data in Training and Test data
      3623. -
    3623. Exercises
      3624. -
    3624. Exercise 1: Setting up various Python environments
      3625. -
    3625. Exercise 2: making your own data and exploring scikit-learn
      3626. -
    3626. Exercise 3: Normalizing our data
      3627. +
    3627. Numpy and arrays
      3628. +
    3628. Matrices in Python
      3629. +
    3629. Meet the Pandas
      3630. +
    3630.    Simple linear regression model using scikit-learn
      3631. +
    3631.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3632. +
    3632.    Organizing our data
      3633. +
    3633.    And what about using neural networks?
      3634. +
    3634. A first summary
      3635. +
    3635. Why Linear Regression (aka Ordinary Least Squares and family)
      3636. +
    3636. Regression analysis, overarching aims
      3637. +
    3637. Regression analysis, overarching aims II
      3638. +
    3638. Examples
      3639. +
    3639. General linear models
      3640. +
    3640. Rewriting the fitting procedure as a linear algebra problem
      3641. +
    3641. Rewriting the fitting procedure as a linear algebra problem, more details
      3642. +
    3642. Generalizing the fitting procedure as a linear algebra problem
      3643. +
    3643. Generalizing the fitting procedure as a linear algebra problem
      3644. +
    3644. Optimizing our parameters
      3645. +
    3645. Our model for the nuclear binding energies
      3646. +
    3646. Optimizing our parameters, more details
      3647. +
    3647. Interpretations and optimizing our parameters
      3648. +
    3648. Interpretations and optimizing our parameters
      3649. +
    3649. Some useful matrix and vector expressions
      3650. +
    3650. Interpretations and optimizing our parameters
      3651. +
    3651. Own code for Ordinary Least Squares
      3652. +
    3652. Adding error analysis and training set up
      3653. +
    3653. The \( \chi^2 \) function
      3654. +
    3654. The \( \chi^2 \) function
      3655. +
    3655. The \( \chi^2 \) function
      3656. +
    3656. The \( \chi^2 \) function
      3657. +
    3657. The \( \chi^2 \) function
      3658. +
    3658. The \( \chi^2 \) function
      3659. +
    3659. Fitting an Equation of State for Dense Nuclear Matter
      3660. +
    3660. The code
      3661. +
    3661. Splitting our Data in Training and Test data
      3662. +
    3662. Exercises
      3663. +
    3663. Exercise 1: Setting up various Python environments
      3664. +
    3664. Exercise 2: making your own data and exploring scikit-learn
      3665. +
    3665. Exercise 3: Split data in test and training data
      3666. @@ -351,29 +335,32 @@ MathJax.Hub.Config({

       

       

       

      -

      Examples

      +

      Generalizing the fitting procedure as a linear algebra problem

      -

      In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), -consider the model we discussed for describing nuclear binding energies. + +

      We are obviously not limited to the above polynomial expansions. We +could replace the various powers of \( x \) with elements of Fourier +series or instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j +x_i)} \), or time series or other orthogonal functions. For every set +of values \( y_i,x_i \) we can then generalize the equations to

      -

      There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. -Assuming -

      $$ -BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} $$ -

      we have five predictors, that is the intercept, the \( A \) dependent term, the \( A^{2/3} \) term and the \( A^{-1/3} \) and \( A^{-1} \) terms. -This gives \( p=0,1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is a -\( p\times n \) matrix \( \boldsymbol{X} \). -

      -

      Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the -so-called credit card default data from Taiwan. The data set contains data on \( n=30000 \) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \( 24 \) such predictors or attributes leading to a design matrix of dimensionality \( 24 \times 30000 \). This is however a classification problem and we will come back to it when we discuss Logistic Regression. -

      +Note that we have \( p=n \) here. The matrix is symmetric. This is generally not the case!
      @@ -403,7 +390,7 @@ so-called 50
    3666. 51
    3667. ...
      3668. -
    3668. 66
      3669. +
    3669. 62
    3670. »
      3671. diff --git a/doc/pub/week34/html/._week34-bs042.html b/doc/pub/week34/html/._week34-bs042.html index 20c977b6d..67ddc608d 100644 --- a/doc/pub/week34/html/._week34-bs042.html +++ b/doc/pub/week34/html/._week34-bs042.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3671. Installing R, C++, cython or Julia
    3672. Installing R, C++, cython, Numba etc
    3673. Numpy examples and Important Matrix and vector handling packages
      3674. -
    3674. Basic Matrix Features
      3675. -
    3675.    Some famous Matrices
      3676. -
    3676.    More Basic Matrix Features
      3677. -
    3677. Numpy and arrays
      3678. -
    3678. Matrices in Python
      3679. -
    3679. Meet the Pandas
      3680. -
    3680. Friday August 27
      3681. -
    3681.    Simple linear regression model using scikit-learn
      3682. -
    3682.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3683. -
    3683.    Organizing our data
      3684. -
    3684.    Seeing the wood for the trees
      3685. -
    3685.    And what about using neural networks?
      3686. -
    3686. A first summary
      3687. -
    3687. Why Linear Regression (aka Ordinary Least Squares and family)
      3688. -
    3688. Regression analysis, overarching aims
      3689. -
    3689. Regression analysis, overarching aims II
      3690. -
    3690. Examples
      3691. -
    3691. General linear models
      3692. -
    3692. Rewriting the fitting procedure as a linear algebra problem
      3693. -
    3693. Rewriting the fitting procedure as a linear algebra problem, more details
      3694. -
    3694. Generalizing the fitting procedure as a linear algebra problem
      3695. -
    3695. Generalizing the fitting procedure as a linear algebra problem
      3696. -
    3696. Optimizing our parameters
      3697. -
    3697. Our model for the nuclear binding energies
      3698. -
    3698. Optimizing our parameters, more details
      3699. -
    3699. Interpretations and optimizing our parameters
      3700. -
    3700. Interpretations and optimizing our parameters
      3701. -
    3701. Some useful matrix and vector expressions
      3702. -
    3702. Interpretations and optimizing our parameters
      3703. -
    3703. Own code for Ordinary Least Squares
      3704. -
    3704. Adding error analysis and training set up
      3705. -
    3705. The \( \chi^2 \) function
      3706. -
    3706. The \( \chi^2 \) function
      3707. -
    3707. The \( \chi^2 \) function
      3708. -
    3708. The \( \chi^2 \) function
      3709. -
    3709. The \( \chi^2 \) function
      3710. -
    3710. The \( \chi^2 \) function
      3711. -
    3711. Fitting an Equation of State for Dense Nuclear Matter
      3712. -
    3712. The code
      3713. -
    3713. Splitting our Data in Training and Test data
      3714. -
    3714. Exercises
      3715. -
    3715. Exercise 1: Setting up various Python environments
      3716. -
    3716. Exercise 2: making your own data and exploring scikit-learn
      3717. -
    3717. Exercise 3: Normalizing our data
      3718. +
    3718. Numpy and arrays
      3719. +
    3719. Matrices in Python
      3720. +
    3720. Meet the Pandas
      3721. +
    3721.    Simple linear regression model using scikit-learn
      3722. +
    3722.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3723. +
    3723.    Organizing our data
      3724. +
    3724.    And what about using neural networks?
      3725. +
    3725. A first summary
      3726. +
    3726. Why Linear Regression (aka Ordinary Least Squares and family)
      3727. +
    3727. Regression analysis, overarching aims
      3728. +
    3728. Regression analysis, overarching aims II
      3729. +
    3729. Examples
      3730. +
    3730. General linear models
      3731. +
    3731. Rewriting the fitting procedure as a linear algebra problem
      3732. +
    3732. Rewriting the fitting procedure as a linear algebra problem, more details
      3733. +
    3733. Generalizing the fitting procedure as a linear algebra problem
      3734. +
    3734. Generalizing the fitting procedure as a linear algebra problem
      3735. +
    3735. Optimizing our parameters
      3736. +
    3736. Our model for the nuclear binding energies
      3737. +
    3737. Optimizing our parameters, more details
      3738. +
    3738. Interpretations and optimizing our parameters
      3739. +
    3739. Interpretations and optimizing our parameters
      3740. +
    3740. Some useful matrix and vector expressions
      3741. +
    3741. Interpretations and optimizing our parameters
      3742. +
    3742. Own code for Ordinary Least Squares
      3743. +
    3743. Adding error analysis and training set up
      3744. +
    3744. The \( \chi^2 \) function
      3745. +
    3745. The \( \chi^2 \) function
      3746. +
    3746. The \( \chi^2 \) function
      3747. +
    3747. The \( \chi^2 \) function
      3748. +
    3748. The \( \chi^2 \) function
      3749. +
    3749. The \( \chi^2 \) function
      3750. +
    3750. Fitting an Equation of State for Dense Nuclear Matter
      3751. +
    3751. The code
      3752. +
    3752. Splitting our Data in Training and Test data
      3753. +
    3753. Exercises
      3754. +
    3754. Exercise 1: Setting up various Python environments
      3755. +
    3755. Exercise 2: making your own data and exploring scikit-learn
      3756. +
    3756. Exercise 3: Split data in test and training data
      3757. @@ -351,18 +335,28 @@ MathJax.Hub.Config({

       

       

       

      -

      General linear models

      +

      Generalizing the fitting procedure as a linear algebra problem

      -

      Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.

      - -

      Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \( y \) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \( n-1 \) with \( n \) points, that is

      +

      We redefine in turn the matrix \( \boldsymbol{X} \) as

      $$ -y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, +\boldsymbol{X}= +\begin{bmatrix} +x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ +x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ +x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ +\end{bmatrix} $$ -

      where \( \epsilon_i \) is the error in our approximation.

      +

      and without loss of generality we rewrite again our equations as

      +$$ +\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. +$$ + +

      The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\epsilon} \) and the parameter vector \( \boldsymbol{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?

      @@ -392,7 +386,7 @@ $$
    3757. 51
    3758. 52
    3759. ...
      3760. -
    3760. 66
      3761. +
    3761. 62
    3762. »
      3763. diff --git a/doc/pub/week34/html/._week34-bs043.html b/doc/pub/week34/html/._week34-bs043.html index 927066a19..f49f21f94 100644 --- a/doc/pub/week34/html/._week34-bs043.html +++ b/doc/pub/week34/html/._week34-bs043.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3763. Installing R, C++, cython or Julia
    3764. Installing R, C++, cython, Numba etc
    3765. Numpy examples and Important Matrix and vector handling packages
      3766. -
    3766. Basic Matrix Features
      3767. -
    3767.    Some famous Matrices
      3768. -
    3768.    More Basic Matrix Features
      3769. -
    3769. Numpy and arrays
      3770. -
    3770. Matrices in Python
      3771. -
    3771. Meet the Pandas
      3772. -
    3772. Friday August 27
      3773. -
    3773.    Simple linear regression model using scikit-learn
      3774. -
    3774.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3775. -
    3775.    Organizing our data
      3776. -
    3776.    Seeing the wood for the trees
      3777. -
    3777.    And what about using neural networks?
      3778. -
    3778. A first summary
      3779. -
    3779. Why Linear Regression (aka Ordinary Least Squares and family)
      3780. -
    3780. Regression analysis, overarching aims
      3781. -
    3781. Regression analysis, overarching aims II
      3782. -
    3782. Examples
      3783. -
    3783. General linear models
      3784. -
    3784. Rewriting the fitting procedure as a linear algebra problem
      3785. -
    3785. Rewriting the fitting procedure as a linear algebra problem, more details
      3786. -
    3786. Generalizing the fitting procedure as a linear algebra problem
      3787. -
    3787. Generalizing the fitting procedure as a linear algebra problem
      3788. -
    3788. Optimizing our parameters
      3789. -
    3789. Our model for the nuclear binding energies
      3790. -
    3790. Optimizing our parameters, more details
      3791. -
    3791. Interpretations and optimizing our parameters
      3792. -
    3792. Interpretations and optimizing our parameters
      3793. -
    3793. Some useful matrix and vector expressions
      3794. -
    3794. Interpretations and optimizing our parameters
      3795. -
    3795. Own code for Ordinary Least Squares
      3796. -
    3796. Adding error analysis and training set up
      3797. -
    3797. The \( \chi^2 \) function
      3798. -
    3798. The \( \chi^2 \) function
      3799. -
    3799. The \( \chi^2 \) function
      3800. -
    3800. The \( \chi^2 \) function
      3801. -
    3801. The \( \chi^2 \) function
      3802. -
    3802. The \( \chi^2 \) function
      3803. -
    3803. Fitting an Equation of State for Dense Nuclear Matter
      3804. -
    3804. The code
      3805. -
    3805. Splitting our Data in Training and Test data
      3806. -
    3806. Exercises
      3807. -
    3807. Exercise 1: Setting up various Python environments
      3808. -
    3808. Exercise 2: making your own data and exploring scikit-learn
      3809. -
    3809. Exercise 3: Normalizing our data
      3810. +
    3810. Numpy and arrays
      3811. +
    3811. Matrices in Python
      3812. +
    3812. Meet the Pandas
      3813. +
    3813.    Simple linear regression model using scikit-learn
      3814. +
    3814.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3815. +
    3815.    Organizing our data
      3816. +
    3816.    And what about using neural networks?
      3817. +
    3817. A first summary
      3818. +
    3818. Why Linear Regression (aka Ordinary Least Squares and family)
      3819. +
    3819. Regression analysis, overarching aims
      3820. +
    3820. Regression analysis, overarching aims II
      3821. +
    3821. Examples
      3822. +
    3822. General linear models
      3823. +
    3823. Rewriting the fitting procedure as a linear algebra problem
      3824. +
    3824. Rewriting the fitting procedure as a linear algebra problem, more details
      3825. +
    3825. Generalizing the fitting procedure as a linear algebra problem
      3826. +
    3826. Generalizing the fitting procedure as a linear algebra problem
      3827. +
    3827. Optimizing our parameters
      3828. +
    3828. Our model for the nuclear binding energies
      3829. +
    3829. Optimizing our parameters, more details
      3830. +
    3830. Interpretations and optimizing our parameters
      3831. +
    3831. Interpretations and optimizing our parameters
      3832. +
    3832. Some useful matrix and vector expressions
      3833. +
    3833. Interpretations and optimizing our parameters
      3834. +
    3834. Own code for Ordinary Least Squares
      3835. +
    3835. Adding error analysis and training set up
      3836. +
    3836. The \( \chi^2 \) function
      3837. +
    3837. The \( \chi^2 \) function
      3838. +
    3838. The \( \chi^2 \) function
      3839. +
    3839. The \( \chi^2 \) function
      3840. +
    3840. The \( \chi^2 \) function
      3841. +
    3841. The \( \chi^2 \) function
      3842. +
    3842. Fitting an Equation of State for Dense Nuclear Matter
      3843. +
    3843. The code
      3844. +
    3844. Splitting our Data in Training and Test data
      3845. +
    3845. Exercises
      3846. +
    3846. Exercise 1: Setting up various Python environments
      3847. +
    3847. Exercise 2: making your own data and exploring scikit-learn
      3848. +
    3848. Exercise 3: Split data in test and training data
      3849. @@ -351,20 +335,27 @@ MathJax.Hub.Config({

       

       

       

      -

      Rewriting the fitting procedure as a linear algebra problem

      +

      Optimizing our parameters

      -

      For every set of values \( y_i,x_i \) we have thus the corresponding set of equations

      +

      We have defined the matrix \( \boldsymbol{X} \) via the equations

      $$ \begin{align*} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ \dots & \dots \\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ \end{align*} $$ + +

      As we noted above, we stayed with a system with the design matrix + \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \), that is we have \( p=n \). For reasons to come later (algorithmic arguments) we will hereafter define +our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors refering to the column numbers and the entries \( n \) being the row elements. +

      @@ -394,7 +385,7 @@ $$
    3849. 52
    3850. 53
    3851. ...
      3852. -
    3852. 66
      3853. +
    3853. 62
    3854. »
      3855. diff --git a/doc/pub/week34/html/._week34-bs044.html b/doc/pub/week34/html/._week34-bs044.html index a687ae449..72fcd2b11 100644 --- a/doc/pub/week34/html/._week34-bs044.html +++ b/doc/pub/week34/html/._week34-bs044.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3855. Installing R, C++, cython or Julia
    3856. Installing R, C++, cython, Numba etc
    3857. Numpy examples and Important Matrix and vector handling packages
      3858. -
    3858. Basic Matrix Features
      3859. -
    3859.    Some famous Matrices
      3860. -
    3860.    More Basic Matrix Features
      3861. -
    3861. Numpy and arrays
      3862. -
    3862. Matrices in Python
      3863. -
    3863. Meet the Pandas
      3864. -
    3864. Friday August 27
      3865. -
    3865.    Simple linear regression model using scikit-learn
      3866. -
    3866.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3867. -
    3867.    Organizing our data
      3868. -
    3868.    Seeing the wood for the trees
      3869. -
    3869.    And what about using neural networks?
      3870. -
    3870. A first summary
      3871. -
    3871. Why Linear Regression (aka Ordinary Least Squares and family)
      3872. -
    3872. Regression analysis, overarching aims
      3873. -
    3873. Regression analysis, overarching aims II
      3874. -
    3874. Examples
      3875. -
    3875. General linear models
      3876. -
    3876. Rewriting the fitting procedure as a linear algebra problem
      3877. -
    3877. Rewriting the fitting procedure as a linear algebra problem, more details
      3878. -
    3878. Generalizing the fitting procedure as a linear algebra problem
      3879. -
    3879. Generalizing the fitting procedure as a linear algebra problem
      3880. -
    3880. Optimizing our parameters
      3881. -
    3881. Our model for the nuclear binding energies
      3882. -
    3882. Optimizing our parameters, more details
      3883. -
    3883. Interpretations and optimizing our parameters
      3884. -
    3884. Interpretations and optimizing our parameters
      3885. -
    3885. Some useful matrix and vector expressions
      3886. -
    3886. Interpretations and optimizing our parameters
      3887. -
    3887. Own code for Ordinary Least Squares
      3888. -
    3888. Adding error analysis and training set up
      3889. -
    3889. The \( \chi^2 \) function
      3890. -
    3890. The \( \chi^2 \) function
      3891. -
    3891. The \( \chi^2 \) function
      3892. -
    3892. The \( \chi^2 \) function
      3893. -
    3893. The \( \chi^2 \) function
      3894. -
    3894. The \( \chi^2 \) function
      3895. -
    3895. Fitting an Equation of State for Dense Nuclear Matter
      3896. -
    3896. The code
      3897. -
    3897. Splitting our Data in Training and Test data
      3898. -
    3898. Exercises
      3899. -
    3899. Exercise 1: Setting up various Python environments
      3900. -
    3900. Exercise 2: making your own data and exploring scikit-learn
      3901. -
    3901. Exercise 3: Normalizing our data
      3902. +
    3902. Numpy and arrays
      3903. +
    3903. Matrices in Python
      3904. +
    3904. Meet the Pandas
      3905. +
    3905.    Simple linear regression model using scikit-learn
      3906. +
    3906.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3907. +
    3907.    Organizing our data
      3908. +
    3908.    And what about using neural networks?
      3909. +
    3909. A first summary
      3910. +
    3910. Why Linear Regression (aka Ordinary Least Squares and family)
      3911. +
    3911. Regression analysis, overarching aims
      3912. +
    3912. Regression analysis, overarching aims II
      3913. +
    3913. Examples
      3914. +
    3914. General linear models
      3915. +
    3915. Rewriting the fitting procedure as a linear algebra problem
      3916. +
    3916. Rewriting the fitting procedure as a linear algebra problem, more details
      3917. +
    3917. Generalizing the fitting procedure as a linear algebra problem
      3918. +
    3918. Generalizing the fitting procedure as a linear algebra problem
      3919. +
    3919. Optimizing our parameters
      3920. +
    3920. Our model for the nuclear binding energies
      3921. +
    3921. Optimizing our parameters, more details
      3922. +
    3922. Interpretations and optimizing our parameters
      3923. +
    3923. Interpretations and optimizing our parameters
      3924. +
    3924. Some useful matrix and vector expressions
      3925. +
    3925. Interpretations and optimizing our parameters
      3926. +
    3926. Own code for Ordinary Least Squares
      3927. +
    3927. Adding error analysis and training set up
      3928. +
    3928. The \( \chi^2 \) function
      3929. +
    3929. The \( \chi^2 \) function
      3930. +
    3930. The \( \chi^2 \) function
      3931. +
    3931. The \( \chi^2 \) function
      3932. +
    3932. The \( \chi^2 \) function
      3933. +
    3933. The \( \chi^2 \) function
      3934. +
    3934. Fitting an Equation of State for Dense Nuclear Matter
      3935. +
    3935. The code
      3936. +
    3936. Splitting our Data in Training and Test data
      3937. +
    3937. Exercises
      3938. +
    3938. Exercise 1: Setting up various Python environments
      3939. +
    3939. Exercise 2: making your own data and exploring scikit-learn
      3940. +
    3940. Exercise 3: Split data in test and training data
      3941. @@ -351,46 +335,108 @@ MathJax.Hub.Config({

       

       

       

      -

      Rewriting the fitting procedure as a linear algebra problem, more details

      -
      -
      - -

      Defining the vectors

      -$$ -\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, -$$ +

      Our model for the nuclear binding energies

      -

      and

      -$$ -\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, -$$ +

      In our introductory notes we looked at the so-called liquid drop model. Let us remind ourselves about what we did by looking at the code.

      -

      and

      -$$ -\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, -$$ +

      We restate the parts of the code we are most interested in.

      -

      and the design matrix

      -$$ -\boldsymbol{X}= -\begin{bmatrix} -1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ -1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ -1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ -\end{bmatrix} -$$ + +
      +
      +
      +
      +
      + 
      # Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from IPython.display import display
+import os
 
-
      we can rewrite our equations as
      
-$$
-\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
-$$
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
 
-
      The above design matrix is called a Vandermonde matrix.
      
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("MassEval2016.dat"),'r')
+
+
+# Read the experimental data with Pandas
+Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
+              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
+              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
+              header=39,
+              index_col=False)
+
+# Extrapolated values are indicated by '#' in place of the decimal place, so
+# the Ebinding column won't be numeric. Coerce to float and drop these entries.
+Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
+Masses = Masses.dropna()
+# Convert from keV to MeV.
+Masses['Ebinding'] /= 1000
+
+# Group the DataFrame by nucleon number, A.
+Masses = Masses.groupby('A')
+# Find the rows of the grouped DataFrame with the maximum binding energy.
+Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
+A = Masses['A']
+Z = Masses['Z']
+N = Masses['N']
+Element = Masses['Element']
+Energies = Masses['Ebinding']
+
+# Now we set up the design matrix X
+X = np.zeros((len(A),5))
+X[:,0] = 1
+X[:,1] = A
+X[:,2] = A**(2.0/3.0)
+X[:,3] = A**(-1.0/3.0)
+X[:,4] = A**(-1.0)
+# Then nice printout using pandas
+DesignMatrix = pd.DataFrame(X)
+DesignMatrix.index = A
+DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
+display(DesignMatrix)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as

      +$$ +\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, +$$ + +

      throughout these lectures.

      @@ -417,7 +463,7 @@ $$

    3941. 53
    3942. 54
    3943. ...
      3944. -
    3944. 66
      3945. +
    3945. 62
    3946. »
      3947. diff --git a/doc/pub/week34/html/._week34-bs045.html b/doc/pub/week34/html/._week34-bs045.html index 89bddfcc3..299eee68f 100644 --- a/doc/pub/week34/html/._week34-bs045.html +++ b/doc/pub/week34/html/._week34-bs045.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    3947. Installing R, C++, cython or Julia
    3948. Installing R, C++, cython, Numba etc
    3949. Numpy examples and Important Matrix and vector handling packages
      3950. -
    3950. Basic Matrix Features
      3951. -
    3951.    Some famous Matrices
      3952. -
    3952.    More Basic Matrix Features
      3953. -
    3953. Numpy and arrays
      3954. -
    3954. Matrices in Python
      3955. -
    3955. Meet the Pandas
      3956. -
    3956. Friday August 27
      3957. -
    3957.    Simple linear regression model using scikit-learn
      3958. -
    3958.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3959. -
    3959.    Organizing our data
      3960. -
    3960.    Seeing the wood for the trees
      3961. -
    3961.    And what about using neural networks?
      3962. -
    3962. A first summary
      3963. -
    3963. Why Linear Regression (aka Ordinary Least Squares and family)
      3964. -
    3964. Regression analysis, overarching aims
      3965. -
    3965. Regression analysis, overarching aims II
      3966. -
    3966. Examples
      3967. -
    3967. General linear models
      3968. -
    3968. Rewriting the fitting procedure as a linear algebra problem
      3969. -
    3969. Rewriting the fitting procedure as a linear algebra problem, more details
      3970. -
    3970. Generalizing the fitting procedure as a linear algebra problem
      3971. -
    3971. Generalizing the fitting procedure as a linear algebra problem
      3972. -
    3972. Optimizing our parameters
      3973. -
    3973. Our model for the nuclear binding energies
      3974. -
    3974. Optimizing our parameters, more details
      3975. -
    3975. Interpretations and optimizing our parameters
      3976. -
    3976. Interpretations and optimizing our parameters
      3977. -
    3977. Some useful matrix and vector expressions
      3978. -
    3978. Interpretations and optimizing our parameters
      3979. -
    3979. Own code for Ordinary Least Squares
      3980. -
    3980. Adding error analysis and training set up
      3981. -
    3981. The \( \chi^2 \) function
      3982. -
    3982. The \( \chi^2 \) function
      3983. -
    3983. The \( \chi^2 \) function
      3984. -
    3984. The \( \chi^2 \) function
      3985. -
    3985. The \( \chi^2 \) function
      3986. -
    3986. The \( \chi^2 \) function
      3987. -
    3987. Fitting an Equation of State for Dense Nuclear Matter
      3988. -
    3988. The code
      3989. -
    3989. Splitting our Data in Training and Test data
      3990. -
    3990. Exercises
      3991. -
    3991. Exercise 1: Setting up various Python environments
      3992. -
    3992. Exercise 2: making your own data and exploring scikit-learn
      3993. -
    3993. Exercise 3: Normalizing our data
      3994. +
    3994. Numpy and arrays
      3995. +
    3995. Matrices in Python
      3996. +
    3996. Meet the Pandas
      3997. +
    3997.    Simple linear regression model using scikit-learn
      3998. +
    3998.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      3999. +
    3999.    Organizing our data
      4000. +
    4000.    And what about using neural networks?
      4001. +
    4001. A first summary
      4002. +
    4002. Why Linear Regression (aka Ordinary Least Squares and family)
      4003. +
    4003. Regression analysis, overarching aims
      4004. +
    4004. Regression analysis, overarching aims II
      4005. +
    4005. Examples
      4006. +
    4006. General linear models
      4007. +
    4007. Rewriting the fitting procedure as a linear algebra problem
      4008. +
    4008. Rewriting the fitting procedure as a linear algebra problem, more details
      4009. +
    4009. Generalizing the fitting procedure as a linear algebra problem
      4010. +
    4010. Generalizing the fitting procedure as a linear algebra problem
      4011. +
    4011. Optimizing our parameters
      4012. +
    4012. Our model for the nuclear binding energies
      4013. +
    4013. Optimizing our parameters, more details
      4014. +
    4014. Interpretations and optimizing our parameters
      4015. +
    4015. Interpretations and optimizing our parameters
      4016. +
    4016. Some useful matrix and vector expressions
      4017. +
    4017. Interpretations and optimizing our parameters
      4018. +
    4018. Own code for Ordinary Least Squares
      4019. +
    4019. Adding error analysis and training set up
      4020. +
    4020. The \( \chi^2 \) function
      4021. +
    4021. The \( \chi^2 \) function
      4022. +
    4022. The \( \chi^2 \) function
      4023. +
    4023. The \( \chi^2 \) function
      4024. +
    4024. The \( \chi^2 \) function
      4025. +
    4025. The \( \chi^2 \) function
      4026. +
    4026. Fitting an Equation of State for Dense Nuclear Matter
      4027. +
    4027. The code
      4028. +
    4028. Splitting our Data in Training and Test data
      4029. +
    4029. Exercises
      4030. +
    4030. Exercise 1: Setting up various Python environments
      4031. +
    4031. Exercise 2: making your own data and exploring scikit-learn
      4032. +
    4032. Exercise 3: Split data in test and training data
      4033. @@ -351,32 +335,36 @@ MathJax.Hub.Config({

       

       

       

      -

      Generalizing the fitting procedure as a linear algebra problem

      +

      Optimizing our parameters, more details

      +

      With the above we use the design matrix to define the approximation \( \boldsymbol{\tilde{y}} \) via the unknown quantity \( \boldsymbol{\beta} \) as

      +$$ +\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, +$$ -

      We are obviously not limited to the above polynomial expansions. We -could replace the various powers of \( x \) with elements of Fourier -series or instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j -x_i)} \), or time series or other orthogonal functions. For every set -of values \( y_i,x_i \) we can then generalize the equations to +

      and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parameterized values \( \tilde{y}_i \), namely

      +$$ +C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, +$$ + +

      or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as

      +$$ +C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

      This function is one possible way to define the so-called cost function.

      + +

      It is also common to define +the function \( C \) as

      $$ -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} +C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, $$ - -Note that we have \( p=n \) here. The matrix is symmetric. This is generally not the case! +

      since when taking the first derivative with respect to the unknown parameters \( \beta \), the factor of \( 2 \) cancels out.

      @@ -406,7 +394,7 @@ $$
    4033. 54
    4034. 55
    4035. ...
      4036. -
    4036. 66
      4037. +
    4037. 62
    4038. »
      4039. diff --git a/doc/pub/week34/html/._week34-bs046.html b/doc/pub/week34/html/._week34-bs046.html index e7ea2dc15..dc96babec 100644 --- a/doc/pub/week34/html/._week34-bs046.html +++ b/doc/pub/week34/html/._week34-bs046.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4039. Installing R, C++, cython or Julia
    4040. Installing R, C++, cython, Numba etc
    4041. Numpy examples and Important Matrix and vector handling packages
      4042. -
    4042. Basic Matrix Features
      4043. -
    4043.    Some famous Matrices
      4044. -
    4044.    More Basic Matrix Features
      4045. -
    4045. Numpy and arrays
      4046. -
    4046. Matrices in Python
      4047. -
    4047. Meet the Pandas
      4048. -
    4048. Friday August 27
      4049. -
    4049.    Simple linear regression model using scikit-learn
      4050. -
    4050.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4051. -
    4051.    Organizing our data
      4052. -
    4052.    Seeing the wood for the trees
      4053. -
    4053.    And what about using neural networks?
      4054. -
    4054. A first summary
      4055. -
    4055. Why Linear Regression (aka Ordinary Least Squares and family)
      4056. -
    4056. Regression analysis, overarching aims
      4057. -
    4057. Regression analysis, overarching aims II
      4058. -
    4058. Examples
      4059. -
    4059. General linear models
      4060. -
    4060. Rewriting the fitting procedure as a linear algebra problem
      4061. -
    4061. Rewriting the fitting procedure as a linear algebra problem, more details
      4062. -
    4062. Generalizing the fitting procedure as a linear algebra problem
      4063. -
    4063. Generalizing the fitting procedure as a linear algebra problem
      4064. -
    4064. Optimizing our parameters
      4065. -
    4065. Our model for the nuclear binding energies
      4066. -
    4066. Optimizing our parameters, more details
      4067. -
    4067. Interpretations and optimizing our parameters
      4068. -
    4068. Interpretations and optimizing our parameters
      4069. -
    4069. Some useful matrix and vector expressions
      4070. -
    4070. Interpretations and optimizing our parameters
      4071. -
    4071. Own code for Ordinary Least Squares
      4072. -
    4072. Adding error analysis and training set up
      4073. -
    4073. The \( \chi^2 \) function
      4074. -
    4074. The \( \chi^2 \) function
      4075. -
    4075. The \( \chi^2 \) function
      4076. -
    4076. The \( \chi^2 \) function
      4077. -
    4077. The \( \chi^2 \) function
      4078. -
    4078. The \( \chi^2 \) function
      4079. -
    4079. Fitting an Equation of State for Dense Nuclear Matter
      4080. -
    4080. The code
      4081. -
    4081. Splitting our Data in Training and Test data
      4082. -
    4082. Exercises
      4083. -
    4083. Exercise 1: Setting up various Python environments
      4084. -
    4084. Exercise 2: making your own data and exploring scikit-learn
      4085. -
    4085. Exercise 3: Normalizing our data
      4086. +
    4086. Numpy and arrays
      4087. +
    4087. Matrices in Python
      4088. +
    4088. Meet the Pandas
      4089. +
    4089.    Simple linear regression model using scikit-learn
      4090. +
    4090.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4091. +
    4091.    Organizing our data
      4092. +
    4092.    And what about using neural networks?
      4093. +
    4093. A first summary
      4094. +
    4094. Why Linear Regression (aka Ordinary Least Squares and family)
      4095. +
    4095. Regression analysis, overarching aims
      4096. +
    4096. Regression analysis, overarching aims II
      4097. +
    4097. Examples
      4098. +
    4098. General linear models
      4099. +
    4099. Rewriting the fitting procedure as a linear algebra problem
      4100. +
    4100. Rewriting the fitting procedure as a linear algebra problem, more details
      4101. +
    4101. Generalizing the fitting procedure as a linear algebra problem
      4102. +
    4102. Generalizing the fitting procedure as a linear algebra problem
      4103. +
    4103. Optimizing our parameters
      4104. +
    4104. Our model for the nuclear binding energies
      4105. +
    4105. Optimizing our parameters, more details
      4106. +
    4106. Interpretations and optimizing our parameters
      4107. +
    4107. Interpretations and optimizing our parameters
      4108. +
    4108. Some useful matrix and vector expressions
      4109. +
    4109. Interpretations and optimizing our parameters
      4110. +
    4110. Own code for Ordinary Least Squares
      4111. +
    4111. Adding error analysis and training set up
      4112. +
    4112. The \( \chi^2 \) function
      4113. +
    4113. The \( \chi^2 \) function
      4114. +
    4114. The \( \chi^2 \) function
      4115. +
    4115. The \( \chi^2 \) function
      4116. +
    4116. The \( \chi^2 \) function
      4117. +
    4117. The \( \chi^2 \) function
      4118. +
    4118. Fitting an Equation of State for Dense Nuclear Matter
      4119. +
    4119. The code
      4120. +
    4120. Splitting our Data in Training and Test data
      4121. +
    4121. Exercises
      4122. +
    4122. Exercise 1: Setting up various Python environments
      4123. +
    4123. Exercise 2: making your own data and exploring scikit-learn
      4124. +
    4124. Exercise 3: Split data in test and training data
      4125. @@ -351,28 +335,53 @@ MathJax.Hub.Config({

       

       

       

      -

      Generalizing the fitting procedure as a linear algebra problem

      +

      Interpretations and optimizing our parameters

      -

      We redefine in turn the matrix \( \boldsymbol{X} \) as

      + +

      The function

      $$ -\boldsymbol{X}= -\begin{bmatrix} -x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ -x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ -x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ -\end{bmatrix} +C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, $$ -

      and without loss of generality we rewrite again our equations as

      +

      can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value. +When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value +

      $$ -\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, $$ -

      The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\epsilon} \) and the parameter vector \( \boldsymbol{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?

      +

      where \( \langle y_i \rangle \) is the mean value. Keep in mind also that +till now we have treated \( y_i \) as the exact value. Normally, the +response (dependent or outcome) variable \( y_i \) the outcome of a +numerical experiment or another type of experiment and is thus only an +approximation to the true value. It is then always accompanied by an +error estimate, often limited to a statistical error estimate given by +the standard deviation discussed earlier. In the discussion here we +will treat \( y_i \) as our exact value for the response variable. +

      + +

      In order to find the parameters \( \beta_i \) we will then minimize the spread of \( C(\boldsymbol{\beta}) \), that is we are going to solve the problem

      +$$ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +$$ + +

      In practical terms it means we will require

      +$$ +\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +$$ + +

      which results in

      +$$ +\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, +$$ + +

      or in a matrix-vector form as

      +$$ +\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). +$$
      @@ -402,7 +411,7 @@ $$
    4125. 55
    4126. 56
    4127. ...
      4128. -
    4128. 66
      4129. +
    4129. 62
    4130. »
      4131. diff --git a/doc/pub/week34/html/._week34-bs047.html b/doc/pub/week34/html/._week34-bs047.html index 21fbe57a9..445497db5 100644 --- a/doc/pub/week34/html/._week34-bs047.html +++ b/doc/pub/week34/html/._week34-bs047.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4131. Installing R, C++, cython or Julia
    4132. Installing R, C++, cython, Numba etc
    4133. Numpy examples and Important Matrix and vector handling packages
      4134. -
    4134. Basic Matrix Features
      4135. -
    4135.    Some famous Matrices
      4136. -
    4136.    More Basic Matrix Features
      4137. -
    4137. Numpy and arrays
      4138. -
    4138. Matrices in Python
      4139. -
    4139. Meet the Pandas
      4140. -
    4140. Friday August 27
      4141. -
    4141.    Simple linear regression model using scikit-learn
      4142. -
    4142.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4143. -
    4143.    Organizing our data
      4144. -
    4144.    Seeing the wood for the trees
      4145. -
    4145.    And what about using neural networks?
      4146. -
    4146. A first summary
      4147. -
    4147. Why Linear Regression (aka Ordinary Least Squares and family)
      4148. -
    4148. Regression analysis, overarching aims
      4149. -
    4149. Regression analysis, overarching aims II
      4150. -
    4150. Examples
      4151. -
    4151. General linear models
      4152. -
    4152. Rewriting the fitting procedure as a linear algebra problem
      4153. -
    4153. Rewriting the fitting procedure as a linear algebra problem, more details
      4154. -
    4154. Generalizing the fitting procedure as a linear algebra problem
      4155. -
    4155. Generalizing the fitting procedure as a linear algebra problem
      4156. -
    4156. Optimizing our parameters
      4157. -
    4157. Our model for the nuclear binding energies
      4158. -
    4158. Optimizing our parameters, more details
      4159. -
    4159. Interpretations and optimizing our parameters
      4160. -
    4160. Interpretations and optimizing our parameters
      4161. -
    4161. Some useful matrix and vector expressions
      4162. -
    4162. Interpretations and optimizing our parameters
      4163. -
    4163. Own code for Ordinary Least Squares
      4164. -
    4164. Adding error analysis and training set up
      4165. -
    4165. The \( \chi^2 \) function
      4166. -
    4166. The \( \chi^2 \) function
      4167. -
    4167. The \( \chi^2 \) function
      4168. -
    4168. The \( \chi^2 \) function
      4169. -
    4169. The \( \chi^2 \) function
      4170. -
    4170. The \( \chi^2 \) function
      4171. -
    4171. Fitting an Equation of State for Dense Nuclear Matter
      4172. -
    4172. The code
      4173. -
    4173. Splitting our Data in Training and Test data
      4174. -
    4174. Exercises
      4175. -
    4175. Exercise 1: Setting up various Python environments
      4176. -
    4176. Exercise 2: making your own data and exploring scikit-learn
      4177. -
    4177. Exercise 3: Normalizing our data
      4178. +
    4178. Numpy and arrays
      4179. +
    4179. Matrices in Python
      4180. +
    4180. Meet the Pandas
      4181. +
    4181.    Simple linear regression model using scikit-learn
      4182. +
    4182.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4183. +
    4183.    Organizing our data
      4184. +
    4184.    And what about using neural networks?
      4185. +
    4185. A first summary
      4186. +
    4186. Why Linear Regression (aka Ordinary Least Squares and family)
      4187. +
    4187. Regression analysis, overarching aims
      4188. +
    4188. Regression analysis, overarching aims II
      4189. +
    4189. Examples
      4190. +
    4190. General linear models
      4191. +
    4191. Rewriting the fitting procedure as a linear algebra problem
      4192. +
    4192. Rewriting the fitting procedure as a linear algebra problem, more details
      4193. +
    4193. Generalizing the fitting procedure as a linear algebra problem
      4194. +
    4194. Generalizing the fitting procedure as a linear algebra problem
      4195. +
    4195. Optimizing our parameters
      4196. +
    4196. Our model for the nuclear binding energies
      4197. +
    4197. Optimizing our parameters, more details
      4198. +
    4198. Interpretations and optimizing our parameters
      4199. +
    4199. Interpretations and optimizing our parameters
      4200. +
    4200. Some useful matrix and vector expressions
      4201. +
    4201. Interpretations and optimizing our parameters
      4202. +
    4202. Own code for Ordinary Least Squares
      4203. +
    4203. Adding error analysis and training set up
      4204. +
    4204. The \( \chi^2 \) function
      4205. +
    4205. The \( \chi^2 \) function
      4206. +
    4206. The \( \chi^2 \) function
      4207. +
    4207. The \( \chi^2 \) function
      4208. +
    4208. The \( \chi^2 \) function
      4209. +
    4209. The \( \chi^2 \) function
      4210. +
    4210. Fitting an Equation of State for Dense Nuclear Matter
      4211. +
    4211. The code
      4212. +
    4212. Splitting our Data in Training and Test data
      4213. +
    4213. Exercises
      4214. +
    4214. Exercise 1: Setting up various Python environments
      4215. +
    4215. Exercise 2: making your own data and exploring scikit-learn
      4216. +
    4216. Exercise 3: Split data in test and training data
      4217. @@ -351,31 +335,48 @@ MathJax.Hub.Config({

       

       

       

      -

      Optimizing our parameters

      +

      Interpretations and optimizing our parameters

      -

      We have defined the matrix \( \boldsymbol{X} \) via the equations

      +

      We can rewrite

      $$ -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} +\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), $$ -

      As we noted above, we stayed with a system with the design matrix - \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \), that is we have \( p=n \). For reasons to come later (algorithmic arguments) we will hereafter define -our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors refering to the column numbers and the entries \( n \) being the row elements. +

      as

      +$$ +\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, +$$ + +

      and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the solution

      +$$ +\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +

      We note also that since our design matrix is defined as \( \boldsymbol{X}\in +{\mathbb{R}}^{n\times p} \), the product \( \boldsymbol{X}^T\boldsymbol{X} \in +{\mathbb{R}}^{p\times p} \). In the above case we have that \( p \ll n \), +in our case \( p=5 \) meaning that we end up with inverting a small +\( 5\times 5 \) matrix. This is a rather common situation, in many cases we end up with low-dimensional +matrices to invert. The methods discussed here and for many other +supervised learning algorithms like classification with logistic +regression or support vector machines, exhibit dimensionalities which +allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix +\( \boldsymbol{X}^T\boldsymbol{X} \).

      +
      +
      + +

      Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)? What kind of problems can we expect?

      +
      +
      + +

        @@ -401,7 +402,7 @@ our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predict
      • 56
      • 57
      • ...
        • -
      • 66
        • +
      • 62
      • »
      diff --git a/doc/pub/week34/html/._week34-bs048.html b/doc/pub/week34/html/._week34-bs048.html index 39e23ceb8..1b95dc910 100644 --- a/doc/pub/week34/html/._week34-bs048.html +++ b/doc/pub/week34/html/._week34-bs048.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4217. Installing R, C++, cython or Julia
    4218. Installing R, C++, cython, Numba etc
    4219. Numpy examples and Important Matrix and vector handling packages
      4220. -
    4220. Basic Matrix Features
      4221. -
    4221.    Some famous Matrices
      4222. -
    4222.    More Basic Matrix Features
      4223. -
    4223. Numpy and arrays
      4224. -
    4224. Matrices in Python
      4225. -
    4225. Meet the Pandas
      4226. -
    4226. Friday August 27
      4227. -
    4227.    Simple linear regression model using scikit-learn
      4228. -
    4228.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4229. -
    4229.    Organizing our data
      4230. -
    4230.    Seeing the wood for the trees
      4231. -
    4231.    And what about using neural networks?
      4232. -
    4232. A first summary
      4233. -
    4233. Why Linear Regression (aka Ordinary Least Squares and family)
      4234. -
    4234. Regression analysis, overarching aims
      4235. -
    4235. Regression analysis, overarching aims II
      4236. -
    4236. Examples
      4237. -
    4237. General linear models
      4238. -
    4238. Rewriting the fitting procedure as a linear algebra problem
      4239. -
    4239. Rewriting the fitting procedure as a linear algebra problem, more details
      4240. -
    4240. Generalizing the fitting procedure as a linear algebra problem
      4241. -
    4241. Generalizing the fitting procedure as a linear algebra problem
      4242. -
    4242. Optimizing our parameters
      4243. -
    4243. Our model for the nuclear binding energies
      4244. -
    4244. Optimizing our parameters, more details
      4245. -
    4245. Interpretations and optimizing our parameters
      4246. -
    4246. Interpretations and optimizing our parameters
      4247. -
    4247. Some useful matrix and vector expressions
      4248. -
    4248. Interpretations and optimizing our parameters
      4249. -
    4249. Own code for Ordinary Least Squares
      4250. -
    4250. Adding error analysis and training set up
      4251. -
    4251. The \( \chi^2 \) function
      4252. -
    4252. The \( \chi^2 \) function
      4253. -
    4253. The \( \chi^2 \) function
      4254. -
    4254. The \( \chi^2 \) function
      4255. -
    4255. The \( \chi^2 \) function
      4256. -
    4256. The \( \chi^2 \) function
      4257. -
    4257. Fitting an Equation of State for Dense Nuclear Matter
      4258. -
    4258. The code
      4259. -
    4259. Splitting our Data in Training and Test data
      4260. -
    4260. Exercises
      4261. -
    4261. Exercise 1: Setting up various Python environments
      4262. -
    4262. Exercise 2: making your own data and exploring scikit-learn
      4263. -
    4263. Exercise 3: Normalizing our data
      4264. +
    4264. Numpy and arrays
      4265. +
    4265. Matrices in Python
      4266. +
    4266. Meet the Pandas
      4267. +
    4267.    Simple linear regression model using scikit-learn
      4268. +
    4268.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4269. +
    4269.    Organizing our data
      4270. +
    4270.    And what about using neural networks?
      4271. +
    4271. A first summary
      4272. +
    4272. Why Linear Regression (aka Ordinary Least Squares and family)
      4273. +
    4273. Regression analysis, overarching aims
      4274. +
    4274. Regression analysis, overarching aims II
      4275. +
    4275. Examples
      4276. +
    4276. General linear models
      4277. +
    4277. Rewriting the fitting procedure as a linear algebra problem
      4278. +
    4278. Rewriting the fitting procedure as a linear algebra problem, more details
      4279. +
    4279. Generalizing the fitting procedure as a linear algebra problem
      4280. +
    4280. Generalizing the fitting procedure as a linear algebra problem
      4281. +
    4281. Optimizing our parameters
      4282. +
    4282. Our model for the nuclear binding energies
      4283. +
    4283. Optimizing our parameters, more details
      4284. +
    4284. Interpretations and optimizing our parameters
      4285. +
    4285. Interpretations and optimizing our parameters
      4286. +
    4286. Some useful matrix and vector expressions
      4287. +
    4287. Interpretations and optimizing our parameters
      4288. +
    4288. Own code for Ordinary Least Squares
      4289. +
    4289. Adding error analysis and training set up
      4290. +
    4290. The \( \chi^2 \) function
      4291. +
    4291. The \( \chi^2 \) function
      4292. +
    4292. The \( \chi^2 \) function
      4293. +
    4293. The \( \chi^2 \) function
      4294. +
    4294. The \( \chi^2 \) function
      4295. +
    4295. The \( \chi^2 \) function
      4296. +
    4296. Fitting an Equation of State for Dense Nuclear Matter
      4297. +
    4297. The code
      4298. +
    4298. Splitting our Data in Training and Test data
      4299. +
    4299. Exercises
      4300. +
    4300. Exercise 1: Setting up various Python environments
      4301. +
    4301. Exercise 2: making your own data and exploring scikit-learn
      4302. +
    4302. Exercise 3: Split data in test and training data
      4303. @@ -351,108 +335,11 @@ MathJax.Hub.Config({

       

       

       

      -

      Our model for the nuclear binding energies

      +

      Some useful matrix and vector expressions

      -

      In our introductory notes we looked at the so-called liquid drop model. Let us remind ourselves about what we did by looking at the code.

      +

      See the handwritten notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesExercise5Week452022.pdf

      -

      We restate the parts of the code we are most interested in.

      - - -
      -
      -
      -
      -
      - 
      # Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-import os
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("MassEval2016.dat"),'r')
-
-
-# Read the experimental data with Pandas
-Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
-              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
-              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
-              header=39,
-              index_col=False)
-
-# Extrapolated values are indicated by '#' in place of the decimal place, so
-# the Ebinding column won't be numeric. Coerce to float and drop these entries.
-Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
-Masses = Masses.dropna()
-# Convert from keV to MeV.
-Masses['Ebinding'] /= 1000
-
-# Group the DataFrame by nucleon number, A.
-Masses = Masses.groupby('A')
-# Find the rows of the grouped DataFrame with the maximum binding energy.
-Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
-A = Masses['A']
-Z = Masses['Z']
-N = Masses['N']
-Element = Masses['Element']
-Energies = Masses['Ebinding']
-
-# Now we set up the design matrix X
-X = np.zeros((len(A),5))
-X[:,0] = 1
-X[:,1] = A
-X[:,2] = A**(2.0/3.0)
-X[:,3] = A**(-1.0/3.0)
-X[:,4] = A**(-1.0)
-# Then nice printout using pandas
-DesignMatrix = pd.DataFrame(X)
-DesignMatrix.index = A
-DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
-display(DesignMatrix)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as

      -$$ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, -$$ - -

      throughout these lectures.

      +

      These notes will be discussed during one of the lectures.

      @@ -479,7 +366,7 @@ $$

    4303. 57
    4304. 58
    4305. ...
      4306. -
    4306. 66
      4307. +
    4307. 62
    4308. »
      4309. diff --git a/doc/pub/week34/html/._week34-bs049.html b/doc/pub/week34/html/._week34-bs049.html index 2133bf74e..ff24777d5 100644 --- a/doc/pub/week34/html/._week34-bs049.html +++ b/doc/pub/week34/html/._week34-bs049.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4309. Installing R, C++, cython or Julia
    4310. Installing R, C++, cython, Numba etc
    4311. Numpy examples and Important Matrix and vector handling packages
      4312. -
    4312. Basic Matrix Features
      4313. -
    4313.    Some famous Matrices
      4314. -
    4314.    More Basic Matrix Features
      4315. -
    4315. Numpy and arrays
      4316. -
    4316. Matrices in Python
      4317. -
    4317. Meet the Pandas
      4318. -
    4318. Friday August 27
      4319. -
    4319.    Simple linear regression model using scikit-learn
      4320. -
    4320.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4321. -
    4321.    Organizing our data
      4322. -
    4322.    Seeing the wood for the trees
      4323. -
    4323.    And what about using neural networks?
      4324. -
    4324. A first summary
      4325. -
    4325. Why Linear Regression (aka Ordinary Least Squares and family)
      4326. -
    4326. Regression analysis, overarching aims
      4327. -
    4327. Regression analysis, overarching aims II
      4328. -
    4328. Examples
      4329. -
    4329. General linear models
      4330. -
    4330. Rewriting the fitting procedure as a linear algebra problem
      4331. -
    4331. Rewriting the fitting procedure as a linear algebra problem, more details
      4332. -
    4332. Generalizing the fitting procedure as a linear algebra problem
      4333. -
    4333. Generalizing the fitting procedure as a linear algebra problem
      4334. -
    4334. Optimizing our parameters
      4335. -
    4335. Our model for the nuclear binding energies
      4336. -
    4336. Optimizing our parameters, more details
      4337. -
    4337. Interpretations and optimizing our parameters
      4338. -
    4338. Interpretations and optimizing our parameters
      4339. -
    4339. Some useful matrix and vector expressions
      4340. -
    4340. Interpretations and optimizing our parameters
      4341. -
    4341. Own code for Ordinary Least Squares
      4342. -
    4342. Adding error analysis and training set up
      4343. -
    4343. The \( \chi^2 \) function
      4344. -
    4344. The \( \chi^2 \) function
      4345. -
    4345. The \( \chi^2 \) function
      4346. -
    4346. The \( \chi^2 \) function
      4347. -
    4347. The \( \chi^2 \) function
      4348. -
    4348. The \( \chi^2 \) function
      4349. -
    4349. Fitting an Equation of State for Dense Nuclear Matter
      4350. -
    4350. The code
      4351. -
    4351. Splitting our Data in Training and Test data
      4352. -
    4352. Exercises
      4353. -
    4353. Exercise 1: Setting up various Python environments
      4354. -
    4354. Exercise 2: making your own data and exploring scikit-learn
      4355. -
    4355. Exercise 3: Normalizing our data
      4356. +
    4356. Numpy and arrays
      4357. +
    4357. Matrices in Python
      4358. +
    4358. Meet the Pandas
      4359. +
    4359.    Simple linear regression model using scikit-learn
      4360. +
    4360.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4361. +
    4361.    Organizing our data
      4362. +
    4362.    And what about using neural networks?
      4363. +
    4363. A first summary
      4364. +
    4364. Why Linear Regression (aka Ordinary Least Squares and family)
      4365. +
    4365. Regression analysis, overarching aims
      4366. +
    4366. Regression analysis, overarching aims II
      4367. +
    4367. Examples
      4368. +
    4368. General linear models
      4369. +
    4369. Rewriting the fitting procedure as a linear algebra problem
      4370. +
    4370. Rewriting the fitting procedure as a linear algebra problem, more details
      4371. +
    4371. Generalizing the fitting procedure as a linear algebra problem
      4372. +
    4372. Generalizing the fitting procedure as a linear algebra problem
      4373. +
    4373. Optimizing our parameters
      4374. +
    4374. Our model for the nuclear binding energies
      4375. +
    4375. Optimizing our parameters, more details
      4376. +
    4376. Interpretations and optimizing our parameters
      4377. +
    4377. Interpretations and optimizing our parameters
      4378. +
    4378. Some useful matrix and vector expressions
      4379. +
    4379. Interpretations and optimizing our parameters
      4380. +
    4380. Own code for Ordinary Least Squares
      4381. +
    4381. Adding error analysis and training set up
      4382. +
    4382. The \( \chi^2 \) function
      4383. +
    4383. The \( \chi^2 \) function
      4384. +
    4384. The \( \chi^2 \) function
      4385. +
    4385. The \( \chi^2 \) function
      4386. +
    4386. The \( \chi^2 \) function
      4387. +
    4387. The \( \chi^2 \) function
      4388. +
    4388. Fitting an Equation of State for Dense Nuclear Matter
      4389. +
    4389. The code
      4390. +
    4390. Splitting our Data in Training and Test data
      4391. +
    4391. Exercises
      4392. +
    4392. Exercise 1: Setting up various Python environments
      4393. +
    4393. Exercise 2: making your own data and exploring scikit-learn
      4394. +
    4394. Exercise 3: Split data in test and training data
      4395. @@ -351,40 +335,32 @@ MathJax.Hub.Config({

       

       

       

      -

      Optimizing our parameters, more details

      +

      Interpretations and optimizing our parameters

      -

      With the above we use the design matrix to define the approximation \( \boldsymbol{\tilde{y}} \) via the unknown quantity \( \boldsymbol{\beta} \) as

      +

      The residuals \( \boldsymbol{\epsilon} \) are in turn given by

      $$ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, +\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, $$ -

      and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parameterized values \( \tilde{y}_i \), namely

      +

      and with

      $$ -C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, +\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, $$ -

      or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as

      +

      we have

      $$ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, $$ -

      This function is one possible way to define the so-called cost function.

      - -

      It is also common to define -the function \( C \) as -

      - -$$ -C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, -$$ - -

      since when taking the first derivative with respect to the unknown parameters \( \beta \), the factor of \( 2 \) cancels out.

      +

      meaning that the solution for \( \boldsymbol{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.

      +

      Let us now return to our nuclear binding energies and simply code the above equations.

      +

      diff --git a/doc/pub/week34/html/._week34-bs050.html b/doc/pub/week34/html/._week34-bs050.html index 953f6aa7f..473b48e58 100644 --- a/doc/pub/week34/html/._week34-bs050.html +++ b/doc/pub/week34/html/._week34-bs050.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4395. Installing R, C++, cython or Julia
    4396. Installing R, C++, cython, Numba etc
    4397. Numpy examples and Important Matrix and vector handling packages
      4398. -
    4398. Basic Matrix Features
      4399. -
    4399.    Some famous Matrices
      4400. -
    4400.    More Basic Matrix Features
      4401. -
    4401. Numpy and arrays
      4402. -
    4402. Matrices in Python
      4403. -
    4403. Meet the Pandas
      4404. -
    4404. Friday August 27
      4405. -
    4405.    Simple linear regression model using scikit-learn
      4406. -
    4406.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4407. -
    4407.    Organizing our data
      4408. -
    4408.    Seeing the wood for the trees
      4409. -
    4409.    And what about using neural networks?
      4410. -
    4410. A first summary
      4411. -
    4411. Why Linear Regression (aka Ordinary Least Squares and family)
      4412. -
    4412. Regression analysis, overarching aims
      4413. -
    4413. Regression analysis, overarching aims II
      4414. -
    4414. Examples
      4415. -
    4415. General linear models
      4416. -
    4416. Rewriting the fitting procedure as a linear algebra problem
      4417. -
    4417. Rewriting the fitting procedure as a linear algebra problem, more details
      4418. -
    4418. Generalizing the fitting procedure as a linear algebra problem
      4419. -
    4419. Generalizing the fitting procedure as a linear algebra problem
      4420. -
    4420. Optimizing our parameters
      4421. -
    4421. Our model for the nuclear binding energies
      4422. -
    4422. Optimizing our parameters, more details
      4423. -
    4423. Interpretations and optimizing our parameters
      4424. -
    4424. Interpretations and optimizing our parameters
      4425. -
    4425. Some useful matrix and vector expressions
      4426. -
    4426. Interpretations and optimizing our parameters
      4427. -
    4427. Own code for Ordinary Least Squares
      4428. -
    4428. Adding error analysis and training set up
      4429. -
    4429. The \( \chi^2 \) function
      4430. -
    4430. The \( \chi^2 \) function
      4431. -
    4431. The \( \chi^2 \) function
      4432. -
    4432. The \( \chi^2 \) function
      4433. -
    4433. The \( \chi^2 \) function
      4434. -
    4434. The \( \chi^2 \) function
      4435. -
    4435. Fitting an Equation of State for Dense Nuclear Matter
      4436. -
    4436. The code
      4437. -
    4437. Splitting our Data in Training and Test data
      4438. -
    4438. Exercises
      4439. -
    4439. Exercise 1: Setting up various Python environments
      4440. -
    4440. Exercise 2: making your own data and exploring scikit-learn
      4441. -
    4441. Exercise 3: Normalizing our data
      4442. +
    4442. Numpy and arrays
      4443. +
    4443. Matrices in Python
      4444. +
    4444. Meet the Pandas
      4445. +
    4445.    Simple linear regression model using scikit-learn
      4446. +
    4446.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4447. +
    4447.    Organizing our data
      4448. +
    4448.    And what about using neural networks?
      4449. +
    4449. A first summary
      4450. +
    4450. Why Linear Regression (aka Ordinary Least Squares and family)
      4451. +
    4451. Regression analysis, overarching aims
      4452. +
    4452. Regression analysis, overarching aims II
      4453. +
    4453. Examples
      4454. +
    4454. General linear models
      4455. +
    4455. Rewriting the fitting procedure as a linear algebra problem
      4456. +
    4456. Rewriting the fitting procedure as a linear algebra problem, more details
      4457. +
    4457. Generalizing the fitting procedure as a linear algebra problem
      4458. +
    4458. Generalizing the fitting procedure as a linear algebra problem
      4459. +
    4459. Optimizing our parameters
      4460. +
    4460. Our model for the nuclear binding energies
      4461. +
    4461. Optimizing our parameters, more details
      4462. +
    4462. Interpretations and optimizing our parameters
      4463. +
    4463. Interpretations and optimizing our parameters
      4464. +
    4464. Some useful matrix and vector expressions
      4465. +
    4465. Interpretations and optimizing our parameters
      4466. +
    4466. Own code for Ordinary Least Squares
      4467. +
    4467. Adding error analysis and training set up
      4468. +
    4468. The \( \chi^2 \) function
      4469. +
    4469. The \( \chi^2 \) function
      4470. +
    4470. The \( \chi^2 \) function
      4471. +
    4471. The \( \chi^2 \) function
      4472. +
    4472. The \( \chi^2 \) function
      4473. +
    4473. The \( \chi^2 \) function
      4474. +
    4474. Fitting an Equation of State for Dense Nuclear Matter
      4475. +
    4475. The code
      4476. +
    4476. Splitting our Data in Training and Test data
      4477. +
    4477. Exercises
      4478. +
    4478. Exercise 1: Setting up various Python environments
      4479. +
    4479. Exercise 2: making your own data and exploring scikit-learn
      4480. +
    4480. Exercise 3: Split data in test and training data
      4481. @@ -351,54 +335,95 @@ MathJax.Hub.Config({

       

       

       

      -

      Interpretations and optimizing our parameters

      -
      -
      - +

      Own code for Ordinary Least Squares

      -

      The function

      -$$ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, -$$ - -

      can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value. -When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value -

      -$$ -y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, -$$ - -

      where \( \langle y_i \rangle \) is the mean value. Keep in mind also that -till now we have treated \( y_i \) as the exact value. Normally, the -response (dependent or outcome) variable \( y_i \) the outcome of a -numerical experiment or another type of experiment and is thus only an -approximation to the true value. It is then always accompanied by an -error estimate, often limited to a statistical error estimate given by -the standard deviation discussed earlier. In the discussion here we -will treat \( y_i \) as our exact value for the response variable. +

      It is rather straightforward to implement the matrix inversion and obtain the parameters \( \boldsymbol{\beta} \). After having defined the matrix \( \boldsymbol{X} \) we simply need to +write

      -

      In order to find the parameters \( \beta_i \) we will then minimize the spread of \( C(\boldsymbol{\beta}) \), that is we are going to solve the problem

      -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -$$ - -

      In practical terms it means we will require

      -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, -$$ - -

      which results in

      -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, -$$ - -

      or in a matrix-vector form as

      -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). -$$ + +
      +
      +
      +
      +
      + 
      # matrix inversion to find beta
+beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)
+# and then make the prediction
+ytilde = X @ beta
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      Alternatively, you can use the least squares functionality in Numpy as

      + + +
      +
      +
      +
      +
      + 
      fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
+ytildenp = np.dot(fit,X.T)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      And finally we plot our fit with and compare with data

      + + +
      +
      +
      +
      +
      + 
      Masses['Eapprox']  = ytilde
+# Generate a plot comparing the experimental with the fitted values values.
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$A = N + Z$')
+ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
+ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
+            label='Ame2016')
+ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
+            label='Fit')
+ax.legend()
+save_fig("Masses2016OLS")
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      @@ -427,7 +452,7 @@ $$
    4481. 59
    4482. 60
    4483. ...
      4484. -
    4484. 66
      4485. +
    4485. 62
    4486. »
      4487. diff --git a/doc/pub/week34/html/._week34-bs051.html b/doc/pub/week34/html/._week34-bs051.html index 0448bc1a8..2a8298132 100644 --- a/doc/pub/week34/html/._week34-bs051.html +++ b/doc/pub/week34/html/._week34-bs051.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4487. Installing R, C++, cython or Julia
    4488. Installing R, C++, cython, Numba etc
    4489. Numpy examples and Important Matrix and vector handling packages
      4490. -
    4490. Basic Matrix Features
      4491. -
    4491.    Some famous Matrices
      4492. -
    4492.    More Basic Matrix Features
      4493. -
    4493. Numpy and arrays
      4494. -
    4494. Matrices in Python
      4495. -
    4495. Meet the Pandas
      4496. -
    4496. Friday August 27
      4497. -
    4497.    Simple linear regression model using scikit-learn
      4498. -
    4498.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4499. -
    4499.    Organizing our data
      4500. -
    4500.    Seeing the wood for the trees
      4501. -
    4501.    And what about using neural networks?
      4502. -
    4502. A first summary
      4503. -
    4503. Why Linear Regression (aka Ordinary Least Squares and family)
      4504. -
    4504. Regression analysis, overarching aims
      4505. -
    4505. Regression analysis, overarching aims II
      4506. -
    4506. Examples
      4507. -
    4507. General linear models
      4508. -
    4508. Rewriting the fitting procedure as a linear algebra problem
      4509. -
    4509. Rewriting the fitting procedure as a linear algebra problem, more details
      4510. -
    4510. Generalizing the fitting procedure as a linear algebra problem
      4511. -
    4511. Generalizing the fitting procedure as a linear algebra problem
      4512. -
    4512. Optimizing our parameters
      4513. -
    4513. Our model for the nuclear binding energies
      4514. -
    4514. Optimizing our parameters, more details
      4515. -
    4515. Interpretations and optimizing our parameters
      4516. -
    4516. Interpretations and optimizing our parameters
      4517. -
    4517. Some useful matrix and vector expressions
      4518. -
    4518. Interpretations and optimizing our parameters
      4519. -
    4519. Own code for Ordinary Least Squares
      4520. -
    4520. Adding error analysis and training set up
      4521. -
    4521. The \( \chi^2 \) function
      4522. -
    4522. The \( \chi^2 \) function
      4523. -
    4523. The \( \chi^2 \) function
      4524. -
    4524. The \( \chi^2 \) function
      4525. -
    4525. The \( \chi^2 \) function
      4526. -
    4526. The \( \chi^2 \) function
      4527. -
    4527. Fitting an Equation of State for Dense Nuclear Matter
      4528. -
    4528. The code
      4529. -
    4529. Splitting our Data in Training and Test data
      4530. -
    4530. Exercises
      4531. -
    4531. Exercise 1: Setting up various Python environments
      4532. -
    4532. Exercise 2: making your own data and exploring scikit-learn
      4533. -
    4533. Exercise 3: Normalizing our data
      4534. +
    4534. Numpy and arrays
      4535. +
    4535. Matrices in Python
      4536. +
    4536. Meet the Pandas
      4537. +
    4537.    Simple linear regression model using scikit-learn
      4538. +
    4538.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4539. +
    4539.    Organizing our data
      4540. +
    4540.    And what about using neural networks?
      4541. +
    4541. A first summary
      4542. +
    4542. Why Linear Regression (aka Ordinary Least Squares and family)
      4543. +
    4543. Regression analysis, overarching aims
      4544. +
    4544. Regression analysis, overarching aims II
      4545. +
    4545. Examples
      4546. +
    4546. General linear models
      4547. +
    4547. Rewriting the fitting procedure as a linear algebra problem
      4548. +
    4548. Rewriting the fitting procedure as a linear algebra problem, more details
      4549. +
    4549. Generalizing the fitting procedure as a linear algebra problem
      4550. +
    4550. Generalizing the fitting procedure as a linear algebra problem
      4551. +
    4551. Optimizing our parameters
      4552. +
    4552. Our model for the nuclear binding energies
      4553. +
    4553. Optimizing our parameters, more details
      4554. +
    4554. Interpretations and optimizing our parameters
      4555. +
    4555. Interpretations and optimizing our parameters
      4556. +
    4556. Some useful matrix and vector expressions
      4557. +
    4557. Interpretations and optimizing our parameters
      4558. +
    4558. Own code for Ordinary Least Squares
      4559. +
    4559. Adding error analysis and training set up
      4560. +
    4560. The \( \chi^2 \) function
      4561. +
    4561. The \( \chi^2 \) function
      4562. +
    4562. The \( \chi^2 \) function
      4563. +
    4563. The \( \chi^2 \) function
      4564. +
    4564. The \( \chi^2 \) function
      4565. +
    4565. The \( \chi^2 \) function
      4566. +
    4566. Fitting an Equation of State for Dense Nuclear Matter
      4567. +
    4567. The code
      4568. +
    4568. Splitting our Data in Training and Test data
      4569. +
    4569. Exercises
      4570. +
    4570. Exercise 1: Setting up various Python environments
      4571. +
    4571. Exercise 2: making your own data and exploring scikit-learn
      4572. +
    4572. Exercise 3: Split data in test and training data
      4573. @@ -351,45 +335,111 @@ MathJax.Hub.Config({

       

       

       

      -

      Interpretations and optimizing our parameters

      -
      -
      - -

      We can rewrite

      -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), -$$ +

      Adding error analysis and training set up

      -

      as

      -$$ -\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, -$$ - -

      and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the solution

      -$$ -\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -$$ - -

      We note also that since our design matrix is defined as \( \boldsymbol{X}\in -{\mathbb{R}}^{n\times p} \), the product \( \boldsymbol{X}^T\boldsymbol{X} \in -{\mathbb{R}}^{p\times p} \). In the above case we have that \( p \ll n \), -in our case \( p=5 \) meaning that we end up with inverting a small -\( 5\times 5 \) matrix. This is a rather common situation, in many cases we end up with low-dimensional -matrices to invert. The methods discussed here and for many other -supervised learning algorithms like classification with logistic -regression or support vector machines, exhibit dimensionalities which -allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix -\( \boldsymbol{X}^T\boldsymbol{X} \). +

      We can easily test our fit by computing the \( R2 \) score that we discussed in connection with the functionality of Scikit-Learn in the introductory slides. +Since we are not using Scikit-Learn here we can define our own \( R2 \) function as

      + + +
      +
      +
      +
      +
      + 
      def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      and we would be using it as

      -
      -
      - -

      Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)? What kind of problems can we expect?

      + +
      +
      +
      +
      +
      + 
      print(R2(Energies,ytilde))
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      We can easily add our MSE score as

      + + +
      +
      +
      +
      +
      + 
      def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+print(MSE(Energies,ytilde))
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      and finally the relative error as

      + + +
      +
      +
      +
      +
      + 
      def RelativeError(y_data,y_model):
+    return abs((y_data-y_model)/y_data)
+print(RelativeError(Energies, ytilde))
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      @@ -418,7 +468,7 @@ allow for the usage of direct linear algebra methods such as LU decomposi
    4573. 60
    4574. 61
    4575. ...
      4576. -
    4576. 66
      4577. +
    4577. 62
    4578. »
      4579. diff --git a/doc/pub/week34/html/._week34-bs052.html b/doc/pub/week34/html/._week34-bs052.html index 079491290..9df498381 100644 --- a/doc/pub/week34/html/._week34-bs052.html +++ b/doc/pub/week34/html/._week34-bs052.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4579. Installing R, C++, cython or Julia
    4580. Installing R, C++, cython, Numba etc
    4581. Numpy examples and Important Matrix and vector handling packages
      4582. -
    4582. Basic Matrix Features
      4583. -
    4583.    Some famous Matrices
      4584. -
    4584.    More Basic Matrix Features
      4585. -
    4585. Numpy and arrays
      4586. -
    4586. Matrices in Python
      4587. -
    4587. Meet the Pandas
      4588. -
    4588. Friday August 27
      4589. -
    4589.    Simple linear regression model using scikit-learn
      4590. -
    4590.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4591. -
    4591.    Organizing our data
      4592. -
    4592.    Seeing the wood for the trees
      4593. -
    4593.    And what about using neural networks?
      4594. -
    4594. A first summary
      4595. -
    4595. Why Linear Regression (aka Ordinary Least Squares and family)
      4596. -
    4596. Regression analysis, overarching aims
      4597. -
    4597. Regression analysis, overarching aims II
      4598. -
    4598. Examples
      4599. -
    4599. General linear models
      4600. -
    4600. Rewriting the fitting procedure as a linear algebra problem
      4601. -
    4601. Rewriting the fitting procedure as a linear algebra problem, more details
      4602. -
    4602. Generalizing the fitting procedure as a linear algebra problem
      4603. -
    4603. Generalizing the fitting procedure as a linear algebra problem
      4604. -
    4604. Optimizing our parameters
      4605. -
    4605. Our model for the nuclear binding energies
      4606. -
    4606. Optimizing our parameters, more details
      4607. -
    4607. Interpretations and optimizing our parameters
      4608. -
    4608. Interpretations and optimizing our parameters
      4609. -
    4609. Some useful matrix and vector expressions
      4610. -
    4610. Interpretations and optimizing our parameters
      4611. -
    4611. Own code for Ordinary Least Squares
      4612. -
    4612. Adding error analysis and training set up
      4613. -
    4613. The \( \chi^2 \) function
      4614. -
    4614. The \( \chi^2 \) function
      4615. -
    4615. The \( \chi^2 \) function
      4616. -
    4616. The \( \chi^2 \) function
      4617. -
    4617. The \( \chi^2 \) function
      4618. -
    4618. The \( \chi^2 \) function
      4619. -
    4619. Fitting an Equation of State for Dense Nuclear Matter
      4620. -
    4620. The code
      4621. -
    4621. Splitting our Data in Training and Test data
      4622. -
    4622. Exercises
      4623. -
    4623. Exercise 1: Setting up various Python environments
      4624. -
    4624. Exercise 2: making your own data and exploring scikit-learn
      4625. -
    4625. Exercise 3: Normalizing our data
      4626. +
    4626. Numpy and arrays
      4627. +
    4627. Matrices in Python
      4628. +
    4628. Meet the Pandas
      4629. +
    4629.    Simple linear regression model using scikit-learn
      4630. +
    4630.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4631. +
    4631.    Organizing our data
      4632. +
    4632.    And what about using neural networks?
      4633. +
    4633. A first summary
      4634. +
    4634. Why Linear Regression (aka Ordinary Least Squares and family)
      4635. +
    4635. Regression analysis, overarching aims
      4636. +
    4636. Regression analysis, overarching aims II
      4637. +
    4637. Examples
      4638. +
    4638. General linear models
      4639. +
    4639. Rewriting the fitting procedure as a linear algebra problem
      4640. +
    4640. Rewriting the fitting procedure as a linear algebra problem, more details
      4641. +
    4641. Generalizing the fitting procedure as a linear algebra problem
      4642. +
    4642. Generalizing the fitting procedure as a linear algebra problem
      4643. +
    4643. Optimizing our parameters
      4644. +
    4644. Our model for the nuclear binding energies
      4645. +
    4645. Optimizing our parameters, more details
      4646. +
    4646. Interpretations and optimizing our parameters
      4647. +
    4647. Interpretations and optimizing our parameters
      4648. +
    4648. Some useful matrix and vector expressions
      4649. +
    4649. Interpretations and optimizing our parameters
      4650. +
    4650. Own code for Ordinary Least Squares
      4651. +
    4651. Adding error analysis and training set up
      4652. +
    4652. The \( \chi^2 \) function
      4653. +
    4653. The \( \chi^2 \) function
      4654. +
    4654. The \( \chi^2 \) function
      4655. +
    4655. The \( \chi^2 \) function
      4656. +
    4656. The \( \chi^2 \) function
      4657. +
    4657. The \( \chi^2 \) function
      4658. +
    4658. Fitting an Equation of State for Dense Nuclear Matter
      4659. +
    4659. The code
      4660. +
    4660. Splitting our Data in Training and Test data
      4661. +
    4661. Exercises
      4662. +
    4662. Exercise 1: Setting up various Python environments
      4663. +
    4663. Exercise 2: making your own data and exploring scikit-learn
      4664. +
    4664. Exercise 3: Split data in test and training data
      4665. @@ -351,27 +335,33 @@ MathJax.Hub.Config({

       

       

       

      -

      Some useful matrix and vector expressions

      +

      The \( \chi^2 \) function

      +
      +
      + -

      The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters. +

      Normally, the response (dependent or outcome) variable \( y_i \) is the +outcome of a numerical experiment or another type of experiment and is +thus only an approximation to the true value. It is then always +accompanied by an error estimate, often limited to a statistical error +estimate given by the standard deviation discussed earlier. In the +discussion here we will treat \( y_i \) as our exact value for the +response variable. +

      + +

      Introducing the standard deviation \( \sigma_i \) for each measurement +\( y_i \), we define now the \( \chi^2 \) function (omitting the \( 1/n \) term) +as

      $$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, +\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, $$ -$$ -\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a}, -$$ +

      where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.

      +
      +
      -$$ -\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, -$$ - -$$ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -$$

      @@ -397,8 +387,6 @@ $$

    4665. 60
    4666. 61
    4667. 62
      4668. -
    4668. ...
      4669. -
    4669. 66
    4670. »
      4671. diff --git a/doc/pub/week34/html/._week34-bs053.html b/doc/pub/week34/html/._week34-bs053.html index a3703eeef..ff0386dc0 100644 --- a/doc/pub/week34/html/._week34-bs053.html +++ b/doc/pub/week34/html/._week34-bs053.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4671. Installing R, C++, cython or Julia
    4672. Installing R, C++, cython, Numba etc
    4673. Numpy examples and Important Matrix and vector handling packages
      4674. -
    4674. Basic Matrix Features
      4675. -
    4675.    Some famous Matrices
      4676. -
    4676.    More Basic Matrix Features
      4677. -
    4677. Numpy and arrays
      4678. -
    4678. Matrices in Python
      4679. -
    4679. Meet the Pandas
      4680. -
    4680. Friday August 27
      4681. -
    4681.    Simple linear regression model using scikit-learn
      4682. -
    4682.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4683. -
    4683.    Organizing our data
      4684. -
    4684.    Seeing the wood for the trees
      4685. -
    4685.    And what about using neural networks?
      4686. -
    4686. A first summary
      4687. -
    4687. Why Linear Regression (aka Ordinary Least Squares and family)
      4688. -
    4688. Regression analysis, overarching aims
      4689. -
    4689. Regression analysis, overarching aims II
      4690. -
    4690. Examples
      4691. -
    4691. General linear models
      4692. -
    4692. Rewriting the fitting procedure as a linear algebra problem
      4693. -
    4693. Rewriting the fitting procedure as a linear algebra problem, more details
      4694. -
    4694. Generalizing the fitting procedure as a linear algebra problem
      4695. -
    4695. Generalizing the fitting procedure as a linear algebra problem
      4696. -
    4696. Optimizing our parameters
      4697. -
    4697. Our model for the nuclear binding energies
      4698. -
    4698. Optimizing our parameters, more details
      4699. -
    4699. Interpretations and optimizing our parameters
      4700. -
    4700. Interpretations and optimizing our parameters
      4701. -
    4701. Some useful matrix and vector expressions
      4702. -
    4702. Interpretations and optimizing our parameters
      4703. -
    4703. Own code for Ordinary Least Squares
      4704. -
    4704. Adding error analysis and training set up
      4705. -
    4705. The \( \chi^2 \) function
      4706. -
    4706. The \( \chi^2 \) function
      4707. -
    4707. The \( \chi^2 \) function
      4708. -
    4708. The \( \chi^2 \) function
      4709. -
    4709. The \( \chi^2 \) function
      4710. -
    4710. The \( \chi^2 \) function
      4711. -
    4711. Fitting an Equation of State for Dense Nuclear Matter
      4712. -
    4712. The code
      4713. -
    4713. Splitting our Data in Training and Test data
      4714. -
    4714. Exercises
      4715. -
    4715. Exercise 1: Setting up various Python environments
      4716. -
    4716. Exercise 2: making your own data and exploring scikit-learn
      4717. -
    4717. Exercise 3: Normalizing our data
      4718. +
    4718. Numpy and arrays
      4719. +
    4719. Matrices in Python
      4720. +
    4720. Meet the Pandas
      4721. +
    4721.    Simple linear regression model using scikit-learn
      4722. +
    4722.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4723. +
    4723.    Organizing our data
      4724. +
    4724.    And what about using neural networks?
      4725. +
    4725. A first summary
      4726. +
    4726. Why Linear Regression (aka Ordinary Least Squares and family)
      4727. +
    4727. Regression analysis, overarching aims
      4728. +
    4728. Regression analysis, overarching aims II
      4729. +
    4729. Examples
      4730. +
    4730. General linear models
      4731. +
    4731. Rewriting the fitting procedure as a linear algebra problem
      4732. +
    4732. Rewriting the fitting procedure as a linear algebra problem, more details
      4733. +
    4733. Generalizing the fitting procedure as a linear algebra problem
      4734. +
    4734. Generalizing the fitting procedure as a linear algebra problem
      4735. +
    4735. Optimizing our parameters
      4736. +
    4736. Our model for the nuclear binding energies
      4737. +
    4737. Optimizing our parameters, more details
      4738. +
    4738. Interpretations and optimizing our parameters
      4739. +
    4739. Interpretations and optimizing our parameters
      4740. +
    4740. Some useful matrix and vector expressions
      4741. +
    4741. Interpretations and optimizing our parameters
      4742. +
    4742. Own code for Ordinary Least Squares
      4743. +
    4743. Adding error analysis and training set up
      4744. +
    4744. The \( \chi^2 \) function
      4745. +
    4745. The \( \chi^2 \) function
      4746. +
    4746. The \( \chi^2 \) function
      4747. +
    4747. The \( \chi^2 \) function
      4748. +
    4748. The \( \chi^2 \) function
      4749. +
    4749. The \( \chi^2 \) function
      4750. +
    4750. Fitting an Equation of State for Dense Nuclear Matter
      4751. +
    4751. The code
      4752. +
    4752. Splitting our Data in Training and Test data
      4753. +
    4753. Exercises
      4754. +
    4754. Exercise 1: Setting up various Python environments
      4755. +
    4755. Exercise 2: making your own data and exploring scikit-learn
      4756. +
    4756. Exercise 3: Split data in test and training data
      4757. @@ -351,32 +335,31 @@ MathJax.Hub.Config({

       

       

       

      -

      Interpretations and optimizing our parameters

      +

      The \( \chi^2 \) function

      -

      The residuals \( \boldsymbol{\epsilon} \) are in turn given by

      + +

      In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\boldsymbol{\beta}) \) by requiring

      $$ -\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, $$ -

      and with

      +

      which results in

      $$ -\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, $$ -

      we have

      +

      or in a matrix-vector form as

      $$ -\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). $$ -

      meaning that the solution for \( \boldsymbol{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.

      +

      where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \boldsymbol{b} \) with elements \( b_i = y_i/\sigma_i \).

      -

      Let us now return to our nuclear binding energies and simply code the above equations.

      -

      diff --git a/doc/pub/week34/html/._week34-bs054.html b/doc/pub/week34/html/._week34-bs054.html index 9e1a65afe..18b48cda2 100644 --- a/doc/pub/week34/html/._week34-bs054.html +++ b/doc/pub/week34/html/._week34-bs054.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4757. Installing R, C++, cython or Julia
    4758. Installing R, C++, cython, Numba etc
    4759. Numpy examples and Important Matrix and vector handling packages
      4760. -
    4760. Basic Matrix Features
      4761. -
    4761.    Some famous Matrices
      4762. -
    4762.    More Basic Matrix Features
      4763. -
    4763. Numpy and arrays
      4764. -
    4764. Matrices in Python
      4765. -
    4765. Meet the Pandas
      4766. -
    4766. Friday August 27
      4767. -
    4767.    Simple linear regression model using scikit-learn
      4768. -
    4768.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4769. -
    4769.    Organizing our data
      4770. -
    4770.    Seeing the wood for the trees
      4771. -
    4771.    And what about using neural networks?
      4772. -
    4772. A first summary
      4773. -
    4773. Why Linear Regression (aka Ordinary Least Squares and family)
      4774. -
    4774. Regression analysis, overarching aims
      4775. -
    4775. Regression analysis, overarching aims II
      4776. -
    4776. Examples
      4777. -
    4777. General linear models
      4778. -
    4778. Rewriting the fitting procedure as a linear algebra problem
      4779. -
    4779. Rewriting the fitting procedure as a linear algebra problem, more details
      4780. -
    4780. Generalizing the fitting procedure as a linear algebra problem
      4781. -
    4781. Generalizing the fitting procedure as a linear algebra problem
      4782. -
    4782. Optimizing our parameters
      4783. -
    4783. Our model for the nuclear binding energies
      4784. -
    4784. Optimizing our parameters, more details
      4785. -
    4785. Interpretations and optimizing our parameters
      4786. -
    4786. Interpretations and optimizing our parameters
      4787. -
    4787. Some useful matrix and vector expressions
      4788. -
    4788. Interpretations and optimizing our parameters
      4789. -
    4789. Own code for Ordinary Least Squares
      4790. -
    4790. Adding error analysis and training set up
      4791. -
    4791. The \( \chi^2 \) function
      4792. -
    4792. The \( \chi^2 \) function
      4793. -
    4793. The \( \chi^2 \) function
      4794. -
    4794. The \( \chi^2 \) function
      4795. -
    4795. The \( \chi^2 \) function
      4796. -
    4796. The \( \chi^2 \) function
      4797. -
    4797. Fitting an Equation of State for Dense Nuclear Matter
      4798. -
    4798. The code
      4799. -
    4799. Splitting our Data in Training and Test data
      4800. -
    4800. Exercises
      4801. -
    4801. Exercise 1: Setting up various Python environments
      4802. -
    4802. Exercise 2: making your own data and exploring scikit-learn
      4803. -
    4803. Exercise 3: Normalizing our data
      4804. +
    4804. Numpy and arrays
      4805. +
    4805. Matrices in Python
      4806. +
    4806. Meet the Pandas
      4807. +
    4807.    Simple linear regression model using scikit-learn
      4808. +
    4808.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4809. +
    4809.    Organizing our data
      4810. +
    4810.    And what about using neural networks?
      4811. +
    4811. A first summary
      4812. +
    4812. Why Linear Regression (aka Ordinary Least Squares and family)
      4813. +
    4813. Regression analysis, overarching aims
      4814. +
    4814. Regression analysis, overarching aims II
      4815. +
    4815. Examples
      4816. +
    4816. General linear models
      4817. +
    4817. Rewriting the fitting procedure as a linear algebra problem
      4818. +
    4818. Rewriting the fitting procedure as a linear algebra problem, more details
      4819. +
    4819. Generalizing the fitting procedure as a linear algebra problem
      4820. +
    4820. Generalizing the fitting procedure as a linear algebra problem
      4821. +
    4821. Optimizing our parameters
      4822. +
    4822. Our model for the nuclear binding energies
      4823. +
    4823. Optimizing our parameters, more details
      4824. +
    4824. Interpretations and optimizing our parameters
      4825. +
    4825. Interpretations and optimizing our parameters
      4826. +
    4826. Some useful matrix and vector expressions
      4827. +
    4827. Interpretations and optimizing our parameters
      4828. +
    4828. Own code for Ordinary Least Squares
      4829. +
    4829. Adding error analysis and training set up
      4830. +
    4830. The \( \chi^2 \) function
      4831. +
    4831. The \( \chi^2 \) function
      4832. +
    4832. The \( \chi^2 \) function
      4833. +
    4833. The \( \chi^2 \) function
      4834. +
    4834. The \( \chi^2 \) function
      4835. +
    4835. The \( \chi^2 \) function
      4836. +
    4836. Fitting an Equation of State for Dense Nuclear Matter
      4837. +
    4837. The code
      4838. +
    4838. Splitting our Data in Training and Test data
      4839. +
    4839. Exercises
      4840. +
    4840. Exercise 1: Setting up various Python environments
      4841. +
    4841. Exercise 2: making your own data and exploring scikit-learn
      4842. +
    4842. Exercise 3: Split data in test and training data
      4843. @@ -351,95 +335,26 @@ MathJax.Hub.Config({

       

       

       

      -

      Own code for Ordinary Least Squares

      +

      The \( \chi^2 \) function

      +
      +
      + -

      It is rather straightforward to implement the matrix inversion and obtain the parameters \( \boldsymbol{\beta} \). After having defined the matrix \( \boldsymbol{X} \) we simply need to -write -

      +

      We can rewrite

      +$$ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), +$$ - -
      -
      -
      -
      -
      - 
      # matrix inversion to find beta
-beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)
-# and then make the prediction
-ytilde = X @ beta
-
      +

      as

      +$$ +\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, +$$ + +

      and if the matrix \( \boldsymbol{A}^T\boldsymbol{A} \) is invertible we have the solution

      +$$ +\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. +$$
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Alternatively, you can use the least squares functionality in Numpy as

      - - -
      -
      -
      -
      -
      - 
      fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
-ytildenp = np.dot(fit,X.T)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      And finally we plot our fit with and compare with data

      - - -
      -
      -
      -
      -
      - 
      Masses['Eapprox']  = ytilde
-# Generate a plot comparing the experimental with the fitted values values.
-fig, ax = plt.subplots()
-ax.set_xlabel(r'$A = N + Z$')
-ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
-ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
-            label='Ame2016')
-ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
-            label='Fit')
-ax.legend()
-save_fig("Masses2016OLS")
-plt.show()
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      @@ -465,10 +380,6 @@ plt.show()
    4843. 60
    4844. 61
    4845. 62
      4846. -
    4846. 63
      4847. -
    4847. 64
      4848. -
    4848. ...
      4849. -
    4849. 66
    4850. »
      4851. diff --git a/doc/pub/week34/html/._week34-bs055.html b/doc/pub/week34/html/._week34-bs055.html index b20679e85..723ad5e02 100644 --- a/doc/pub/week34/html/._week34-bs055.html +++ b/doc/pub/week34/html/._week34-bs055.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4851. Installing R, C++, cython or Julia
    4852. Installing R, C++, cython, Numba etc
    4853. Numpy examples and Important Matrix and vector handling packages
      4854. -
    4854. Basic Matrix Features
      4855. -
    4855.    Some famous Matrices
      4856. -
    4856.    More Basic Matrix Features
      4857. -
    4857. Numpy and arrays
      4858. -
    4858. Matrices in Python
      4859. -
    4859. Meet the Pandas
      4860. -
    4860. Friday August 27
      4861. -
    4861.    Simple linear regression model using scikit-learn
      4862. -
    4862.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4863. -
    4863.    Organizing our data
      4864. -
    4864.    Seeing the wood for the trees
      4865. -
    4865.    And what about using neural networks?
      4866. -
    4866. A first summary
      4867. -
    4867. Why Linear Regression (aka Ordinary Least Squares and family)
      4868. -
    4868. Regression analysis, overarching aims
      4869. -
    4869. Regression analysis, overarching aims II
      4870. -
    4870. Examples
      4871. -
    4871. General linear models
      4872. -
    4872. Rewriting the fitting procedure as a linear algebra problem
      4873. -
    4873. Rewriting the fitting procedure as a linear algebra problem, more details
      4874. -
    4874. Generalizing the fitting procedure as a linear algebra problem
      4875. -
    4875. Generalizing the fitting procedure as a linear algebra problem
      4876. -
    4876. Optimizing our parameters
      4877. -
    4877. Our model for the nuclear binding energies
      4878. -
    4878. Optimizing our parameters, more details
      4879. -
    4879. Interpretations and optimizing our parameters
      4880. -
    4880. Interpretations and optimizing our parameters
      4881. -
    4881. Some useful matrix and vector expressions
      4882. -
    4882. Interpretations and optimizing our parameters
      4883. -
    4883. Own code for Ordinary Least Squares
      4884. -
    4884. Adding error analysis and training set up
      4885. -
    4885. The \( \chi^2 \) function
      4886. -
    4886. The \( \chi^2 \) function
      4887. -
    4887. The \( \chi^2 \) function
      4888. -
    4888. The \( \chi^2 \) function
      4889. -
    4889. The \( \chi^2 \) function
      4890. -
    4890. The \( \chi^2 \) function
      4891. -
    4891. Fitting an Equation of State for Dense Nuclear Matter
      4892. -
    4892. The code
      4893. -
    4893. Splitting our Data in Training and Test data
      4894. -
    4894. Exercises
      4895. -
    4895. Exercise 1: Setting up various Python environments
      4896. -
    4896. Exercise 2: making your own data and exploring scikit-learn
      4897. -
    4897. Exercise 3: Normalizing our data
      4898. +
    4898. Numpy and arrays
      4899. +
    4899. Matrices in Python
      4900. +
    4900. Meet the Pandas
      4901. +
    4901.    Simple linear regression model using scikit-learn
      4902. +
    4902.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4903. +
    4903.    Organizing our data
      4904. +
    4904.    And what about using neural networks?
      4905. +
    4905. A first summary
      4906. +
    4906. Why Linear Regression (aka Ordinary Least Squares and family)
      4907. +
    4907. Regression analysis, overarching aims
      4908. +
    4908. Regression analysis, overarching aims II
      4909. +
    4909. Examples
      4910. +
    4910. General linear models
      4911. +
    4911. Rewriting the fitting procedure as a linear algebra problem
      4912. +
    4912. Rewriting the fitting procedure as a linear algebra problem, more details
      4913. +
    4913. Generalizing the fitting procedure as a linear algebra problem
      4914. +
    4914. Generalizing the fitting procedure as a linear algebra problem
      4915. +
    4915. Optimizing our parameters
      4916. +
    4916. Our model for the nuclear binding energies
      4917. +
    4917. Optimizing our parameters, more details
      4918. +
    4918. Interpretations and optimizing our parameters
      4919. +
    4919. Interpretations and optimizing our parameters
      4920. +
    4920. Some useful matrix and vector expressions
      4921. +
    4921. Interpretations and optimizing our parameters
      4922. +
    4922. Own code for Ordinary Least Squares
      4923. +
    4923. Adding error analysis and training set up
      4924. +
    4924. The \( \chi^2 \) function
      4925. +
    4925. The \( \chi^2 \) function
      4926. +
    4926. The \( \chi^2 \) function
      4927. +
    4927. The \( \chi^2 \) function
      4928. +
    4928. The \( \chi^2 \) function
      4929. +
    4929. The \( \chi^2 \) function
      4930. +
    4930. Fitting an Equation of State for Dense Nuclear Matter
      4931. +
    4931. The code
      4932. +
    4932. Splitting our Data in Training and Test data
      4933. +
    4933. Exercises
      4934. +
    4934. Exercise 1: Setting up various Python environments
      4935. +
    4935. Exercise 2: making your own data and exploring scikit-learn
      4936. +
    4936. Exercise 3: Split data in test and training data
      4937. @@ -351,111 +335,31 @@ MathJax.Hub.Config({

       

       

       

      -

      Adding error analysis and training set up

      +

      The \( \chi^2 \) function

      +
      +
      + -

      We can easily test our fit by computing the \( R2 \) score that we discussed in connection with the functionality of Scikit-Learn in the introductory slides. -Since we are not using Scikit-Learn here we can define our own \( R2 \) function as -

      +

      If we then introduce the matrix

      +$$ +\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, +$$ - -
      -
      -
      -
      -
      - 
      def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-
      +

      we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \))

      +$$ +\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} +$$ + +

      We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as (we leave this as an exercise)

      +$$ +\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, +$$ + +

      resulting in

      +$$ +\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! +$$
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      and we would be using it as

      - - -
      -
      -
      -
      -
      - 
      print(R2(Energies,ytilde))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      We can easily add our MSE score as

      - - -
      -
      -
      -
      -
      - 
      def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-print(MSE(Energies,ytilde))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      and finally the relative error as

      - - -
      -
      -
      -
      -
      - 
      def RelativeError(y_data,y_model):
-    return abs((y_data-y_model)/y_data)
-print(RelativeError(Energies, ytilde))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      @@ -480,11 +384,6 @@ Since we are not using Scikit-Learn here we can define our own \( R2 \) f
    4937. 60
    4938. 61
    4939. 62
      4940. -
    4940. 63
      4941. -
    4941. 64
      4942. -
    4942. 65
      4943. -
    4943. ...
      4944. -
    4944. 66
    4945. »
      4946. diff --git a/doc/pub/week34/html/._week34-bs056.html b/doc/pub/week34/html/._week34-bs056.html index 97af0e1c6..42efdd668 100644 --- a/doc/pub/week34/html/._week34-bs056.html +++ b/doc/pub/week34/html/._week34-bs056.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    4946. Installing R, C++, cython or Julia
    4947. Installing R, C++, cython, Numba etc
    4948. Numpy examples and Important Matrix and vector handling packages
      4949. -
    4949. Basic Matrix Features
      4950. -
    4950.    Some famous Matrices
      4951. -
    4951.    More Basic Matrix Features
      4952. -
    4952. Numpy and arrays
      4953. -
    4953. Matrices in Python
      4954. -
    4954. Meet the Pandas
      4955. -
    4955. Friday August 27
      4956. -
    4956.    Simple linear regression model using scikit-learn
      4957. -
    4957.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4958. -
    4958.    Organizing our data
      4959. -
    4959.    Seeing the wood for the trees
      4960. -
    4960.    And what about using neural networks?
      4961. -
    4961. A first summary
      4962. -
    4962. Why Linear Regression (aka Ordinary Least Squares and family)
      4963. -
    4963. Regression analysis, overarching aims
      4964. -
    4964. Regression analysis, overarching aims II
      4965. -
    4965. Examples
      4966. -
    4966. General linear models
      4967. -
    4967. Rewriting the fitting procedure as a linear algebra problem
      4968. -
    4968. Rewriting the fitting procedure as a linear algebra problem, more details
      4969. -
    4969. Generalizing the fitting procedure as a linear algebra problem
      4970. -
    4970. Generalizing the fitting procedure as a linear algebra problem
      4971. -
    4971. Optimizing our parameters
      4972. -
    4972. Our model for the nuclear binding energies
      4973. -
    4973. Optimizing our parameters, more details
      4974. -
    4974. Interpretations and optimizing our parameters
      4975. -
    4975. Interpretations and optimizing our parameters
      4976. -
    4976. Some useful matrix and vector expressions
      4977. -
    4977. Interpretations and optimizing our parameters
      4978. -
    4978. Own code for Ordinary Least Squares
      4979. -
    4979. Adding error analysis and training set up
      4980. -
    4980. The \( \chi^2 \) function
      4981. -
    4981. The \( \chi^2 \) function
      4982. -
    4982. The \( \chi^2 \) function
      4983. -
    4983. The \( \chi^2 \) function
      4984. -
    4984. The \( \chi^2 \) function
      4985. -
    4985. The \( \chi^2 \) function
      4986. -
    4986. Fitting an Equation of State for Dense Nuclear Matter
      4987. -
    4987. The code
      4988. -
    4988. Splitting our Data in Training and Test data
      4989. -
    4989. Exercises
      4990. -
    4990. Exercise 1: Setting up various Python environments
      4991. -
    4991. Exercise 2: making your own data and exploring scikit-learn
      4992. -
    4992. Exercise 3: Normalizing our data
      4993. +
    4993. Numpy and arrays
      4994. +
    4994. Matrices in Python
      4995. +
    4995. Meet the Pandas
      4996. +
    4996.    Simple linear regression model using scikit-learn
      4997. +
    4997.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      4998. +
    4998.    Organizing our data
      4999. +
    4999.    And what about using neural networks?
      5000. +
    5000. A first summary
      5001. +
    5001. Why Linear Regression (aka Ordinary Least Squares and family)
      5002. +
    5002. Regression analysis, overarching aims
      5003. +
    5003. Regression analysis, overarching aims II
      5004. +
    5004. Examples
      5005. +
    5005. General linear models
      5006. +
    5006. Rewriting the fitting procedure as a linear algebra problem
      5007. +
    5007. Rewriting the fitting procedure as a linear algebra problem, more details
      5008. +
    5008. Generalizing the fitting procedure as a linear algebra problem
      5009. +
    5009. Generalizing the fitting procedure as a linear algebra problem
      5010. +
    5010. Optimizing our parameters
      5011. +
    5011. Our model for the nuclear binding energies
      5012. +
    5012. Optimizing our parameters, more details
      5013. +
    5013. Interpretations and optimizing our parameters
      5014. +
    5014. Interpretations and optimizing our parameters
      5015. +
    5015. Some useful matrix and vector expressions
      5016. +
    5016. Interpretations and optimizing our parameters
      5017. +
    5017. Own code for Ordinary Least Squares
      5018. +
    5018. Adding error analysis and training set up
      5019. +
    5019. The \( \chi^2 \) function
      5020. +
    5020. The \( \chi^2 \) function
      5021. +
    5021. The \( \chi^2 \) function
      5022. +
    5022. The \( \chi^2 \) function
      5023. +
    5023. The \( \chi^2 \) function
      5024. +
    5024. The \( \chi^2 \) function
      5025. +
    5025. Fitting an Equation of State for Dense Nuclear Matter
      5026. +
    5026. The code
      5027. +
    5027. Splitting our Data in Training and Test data
      5028. +
    5028. Exercises
      5029. +
    5029. Exercise 1: Setting up various Python environments
      5030. +
    5030. Exercise 2: making your own data and exploring scikit-learn
      5031. +
    5031. Exercise 3: Split data in test and training data
      5032. @@ -355,26 +339,20 @@ MathJax.Hub.Config({
      - -

      Normally, the response (dependent or outcome) variable \( y_i \) is the -outcome of a numerical experiment or another type of experiment and is -thus only an approximation to the true value. It is then always -accompanied by an error estimate, often limited to a statistical error -estimate given by the standard deviation discussed earlier. In the -discussion here we will treat \( y_i \) as our exact value for the -response variable. -

      - -

      Introducing the standard deviation \( \sigma_i \) for each measurement -\( y_i \), we define now the \( \chi^2 \) function (omitting the \( 1/n \) term) -as -

      - +

      The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write

      $$ -\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, +y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. $$ -

      where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.

      +

      By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by

      +$$ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, +$$ + +

      and

      +$$ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. +$$
      @@ -399,10 +377,6 @@ $$
    5032. 60
    5033. 61
    5034. 62
      5035. -
    5035. 63
      5036. -
    5036. 64
      5037. -
    5037. 65
      5038. -
    5038. 66
    5039. »
      5040. diff --git a/doc/pub/week34/html/._week34-bs057.html b/doc/pub/week34/html/._week34-bs057.html index 53cf16b1b..4b7b322f7 100644 --- a/doc/pub/week34/html/._week34-bs057.html +++ b/doc/pub/week34/html/._week34-bs057.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5040. Installing R, C++, cython or Julia
    5041. Installing R, C++, cython, Numba etc
    5042. Numpy examples and Important Matrix and vector handling packages
      5043. -
    5043. Basic Matrix Features
      5044. -
    5044.    Some famous Matrices
      5045. -
    5045.    More Basic Matrix Features
      5046. -
    5046. Numpy and arrays
      5047. -
    5047. Matrices in Python
      5048. -
    5048. Meet the Pandas
      5049. -
    5049. Friday August 27
      5050. -
    5050.    Simple linear regression model using scikit-learn
      5051. -
    5051.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5052. -
    5052.    Organizing our data
      5053. -
    5053.    Seeing the wood for the trees
      5054. -
    5054.    And what about using neural networks?
      5055. -
    5055. A first summary
      5056. -
    5056. Why Linear Regression (aka Ordinary Least Squares and family)
      5057. -
    5057. Regression analysis, overarching aims
      5058. -
    5058. Regression analysis, overarching aims II
      5059. -
    5059. Examples
      5060. -
    5060. General linear models
      5061. -
    5061. Rewriting the fitting procedure as a linear algebra problem
      5062. -
    5062. Rewriting the fitting procedure as a linear algebra problem, more details
      5063. -
    5063. Generalizing the fitting procedure as a linear algebra problem
      5064. -
    5064. Generalizing the fitting procedure as a linear algebra problem
      5065. -
    5065. Optimizing our parameters
      5066. -
    5066. Our model for the nuclear binding energies
      5067. -
    5067. Optimizing our parameters, more details
      5068. -
    5068. Interpretations and optimizing our parameters
      5069. -
    5069. Interpretations and optimizing our parameters
      5070. -
    5070. Some useful matrix and vector expressions
      5071. -
    5071. Interpretations and optimizing our parameters
      5072. -
    5072. Own code for Ordinary Least Squares
      5073. -
    5073. Adding error analysis and training set up
      5074. -
    5074. The \( \chi^2 \) function
      5075. -
    5075. The \( \chi^2 \) function
      5076. -
    5076. The \( \chi^2 \) function
      5077. -
    5077. The \( \chi^2 \) function
      5078. -
    5078. The \( \chi^2 \) function
      5079. -
    5079. The \( \chi^2 \) function
      5080. -
    5080. Fitting an Equation of State for Dense Nuclear Matter
      5081. -
    5081. The code
      5082. -
    5082. Splitting our Data in Training and Test data
      5083. -
    5083. Exercises
      5084. -
    5084. Exercise 1: Setting up various Python environments
      5085. -
    5085. Exercise 2: making your own data and exploring scikit-learn
      5086. -
    5086. Exercise 3: Normalizing our data
      5087. +
    5087. Numpy and arrays
      5088. +
    5088. Matrices in Python
      5089. +
    5089. Meet the Pandas
      5090. +
    5090.    Simple linear regression model using scikit-learn
      5091. +
    5091.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5092. +
    5092.    Organizing our data
      5093. +
    5093.    And what about using neural networks?
      5094. +
    5094. A first summary
      5095. +
    5095. Why Linear Regression (aka Ordinary Least Squares and family)
      5096. +
    5096. Regression analysis, overarching aims
      5097. +
    5097. Regression analysis, overarching aims II
      5098. +
    5098. Examples
      5099. +
    5099. General linear models
      5100. +
    5100. Rewriting the fitting procedure as a linear algebra problem
      5101. +
    5101. Rewriting the fitting procedure as a linear algebra problem, more details
      5102. +
    5102. Generalizing the fitting procedure as a linear algebra problem
      5103. +
    5103. Generalizing the fitting procedure as a linear algebra problem
      5104. +
    5104. Optimizing our parameters
      5105. +
    5105. Our model for the nuclear binding energies
      5106. +
    5106. Optimizing our parameters, more details
      5107. +
    5107. Interpretations and optimizing our parameters
      5108. +
    5108. Interpretations and optimizing our parameters
      5109. +
    5109. Some useful matrix and vector expressions
      5110. +
    5110. Interpretations and optimizing our parameters
      5111. +
    5111. Own code for Ordinary Least Squares
      5112. +
    5112. Adding error analysis and training set up
      5113. +
    5113. The \( \chi^2 \) function
      5114. +
    5114. The \( \chi^2 \) function
      5115. +
    5115. The \( \chi^2 \) function
      5116. +
    5116. The \( \chi^2 \) function
      5117. +
    5117. The \( \chi^2 \) function
      5118. +
    5118. The \( \chi^2 \) function
      5119. +
    5119. Fitting an Equation of State for Dense Nuclear Matter
      5120. +
    5120. The code
      5121. +
    5121. Splitting our Data in Training and Test data
      5122. +
    5122. Exercises
      5123. +
    5123. Exercise 1: Setting up various Python environments
      5124. +
    5124. Exercise 2: making your own data and exploring scikit-learn
      5125. +
    5125. Exercise 3: Split data in test and training data
      5126. @@ -356,22 +340,49 @@ MathJax.Hub.Config({
      -

      In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\boldsymbol{\beta}) \) by requiring

      +

      For a linear fit (a first-order polynomial) we don't need to invert a matrix!! +Defining +

      $$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, +\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, $$ -

      which results in

      + $$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, +\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, $$ -

      or in a matrix-vector form as

      + $$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). +\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), $$ -

      where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \boldsymbol{b} \) with elements \( b_i = y_i/\sigma_i \).

      + +$$ +\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, +$$ + + +$$ +\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, +$$ + +

      we obtain

      + +$$ +\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, +$$ + + +$$ +\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. +$$ + +

      This approach (different linear and non-linear regression) suffers +often from both being underdetermined and overdetermined in the +unknown coefficients \( \beta_i \). A better approach is to use the +Singular Value Decomposition (SVD) method discussed next week. +

      @@ -395,10 +406,6 @@ $$
    5126. 60
    5127. 61
    5128. 62
      5129. -
    5129. 63
      5130. -
    5130. 64
      5131. -
    5131. 65
      5132. -
    5132. 66
    5133. »
      5134. diff --git a/doc/pub/week34/html/._week34-bs058.html b/doc/pub/week34/html/._week34-bs058.html index 447ed05cf..b07af483e 100644 --- a/doc/pub/week34/html/._week34-bs058.html +++ b/doc/pub/week34/html/._week34-bs058.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5134. Installing R, C++, cython or Julia
    5135. Installing R, C++, cython, Numba etc
    5136. Numpy examples and Important Matrix and vector handling packages
      5137. -
    5137. Basic Matrix Features
      5138. -
    5138.    Some famous Matrices
      5139. -
    5139.    More Basic Matrix Features
      5140. -
    5140. Numpy and arrays
      5141. -
    5141. Matrices in Python
      5142. -
    5142. Meet the Pandas
      5143. -
    5143. Friday August 27
      5144. -
    5144.    Simple linear regression model using scikit-learn
      5145. -
    5145.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5146. -
    5146.    Organizing our data
      5147. -
    5147.    Seeing the wood for the trees
      5148. -
    5148.    And what about using neural networks?
      5149. -
    5149. A first summary
      5150. -
    5150. Why Linear Regression (aka Ordinary Least Squares and family)
      5151. -
    5151. Regression analysis, overarching aims
      5152. -
    5152. Regression analysis, overarching aims II
      5153. -
    5153. Examples
      5154. -
    5154. General linear models
      5155. -
    5155. Rewriting the fitting procedure as a linear algebra problem
      5156. -
    5156. Rewriting the fitting procedure as a linear algebra problem, more details
      5157. -
    5157. Generalizing the fitting procedure as a linear algebra problem
      5158. -
    5158. Generalizing the fitting procedure as a linear algebra problem
      5159. -
    5159. Optimizing our parameters
      5160. -
    5160. Our model for the nuclear binding energies
      5161. -
    5161. Optimizing our parameters, more details
      5162. -
    5162. Interpretations and optimizing our parameters
      5163. -
    5163. Interpretations and optimizing our parameters
      5164. -
    5164. Some useful matrix and vector expressions
      5165. -
    5165. Interpretations and optimizing our parameters
      5166. -
    5166. Own code for Ordinary Least Squares
      5167. -
    5167. Adding error analysis and training set up
      5168. -
    5168. The \( \chi^2 \) function
      5169. -
    5169. The \( \chi^2 \) function
      5170. -
    5170. The \( \chi^2 \) function
      5171. -
    5171. The \( \chi^2 \) function
      5172. -
    5172. The \( \chi^2 \) function
      5173. -
    5173. The \( \chi^2 \) function
      5174. -
    5174. Fitting an Equation of State for Dense Nuclear Matter
      5175. -
    5175. The code
      5176. -
    5176. Splitting our Data in Training and Test data
      5177. -
    5177. Exercises
      5178. -
    5178. Exercise 1: Setting up various Python environments
      5179. -
    5179. Exercise 2: making your own data and exploring scikit-learn
      5180. -
    5180. Exercise 3: Normalizing our data
      5181. +
    5181. Numpy and arrays
      5182. +
    5182. Matrices in Python
      5183. +
    5183. Meet the Pandas
      5184. +
    5184.    Simple linear regression model using scikit-learn
      5185. +
    5185.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5186. +
    5186.    Organizing our data
      5187. +
    5187.    And what about using neural networks?
      5188. +
    5188. A first summary
      5189. +
    5189. Why Linear Regression (aka Ordinary Least Squares and family)
      5190. +
    5190. Regression analysis, overarching aims
      5191. +
    5191. Regression analysis, overarching aims II
      5192. +
    5192. Examples
      5193. +
    5193. General linear models
      5194. +
    5194. Rewriting the fitting procedure as a linear algebra problem
      5195. +
    5195. Rewriting the fitting procedure as a linear algebra problem, more details
      5196. +
    5196. Generalizing the fitting procedure as a linear algebra problem
      5197. +
    5197. Generalizing the fitting procedure as a linear algebra problem
      5198. +
    5198. Optimizing our parameters
      5199. +
    5199. Our model for the nuclear binding energies
      5200. +
    5200. Optimizing our parameters, more details
      5201. +
    5201. Interpretations and optimizing our parameters
      5202. +
    5202. Interpretations and optimizing our parameters
      5203. +
    5203. Some useful matrix and vector expressions
      5204. +
    5204. Interpretations and optimizing our parameters
      5205. +
    5205. Own code for Ordinary Least Squares
      5206. +
    5206. Adding error analysis and training set up
      5207. +
    5207. The \( \chi^2 \) function
      5208. +
    5208. The \( \chi^2 \) function
      5209. +
    5209. The \( \chi^2 \) function
      5210. +
    5210. The \( \chi^2 \) function
      5211. +
    5211. The \( \chi^2 \) function
      5212. +
    5212. The \( \chi^2 \) function
      5213. +
    5213. Fitting an Equation of State for Dense Nuclear Matter
      5214. +
    5214. The code
      5215. +
    5215. Splitting our Data in Training and Test data
      5216. +
    5216. Exercises
      5217. +
    5217. Exercise 1: Setting up various Python environments
      5218. +
    5218. Exercise 2: making your own data and exploring scikit-learn
      5219. +
    5219. Exercise 3: Split data in test and training data
      5220. @@ -351,28 +335,27 @@ MathJax.Hub.Config({

       

       

       

      -

      The \( \chi^2 \) function

      -
      -
      - +

      Fitting an Equation of State for Dense Nuclear Matter

      -

      We can rewrite

      -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), -$$ +

      Before we continue, let us introduce yet another example. We are going to fit the +nuclear equation of state using results from many-body calculations. +The equation of state we have made available here, as function of +density, has been derived using modern nucleon-nucleon potentials with +the addition of three-body +forces. This +time the file is presented as a standard csv file. +

      -

      as

      -$$ -\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, -$$ - -

      and if the matrix \( \boldsymbol{A}^T\boldsymbol{A} \) is invertible we have the solution

      -$$ -\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. -$$ -
      -
      +

      The beginning of the Python code here is similar to what you have seen +before, with the same initializations and declarations. We use also +pandas again, rather extensively in order to organize our data. +

      +

      The difference now is that we use Scikit-Learn's regression tools +instead of our own matrix inversion implementation. Furthermore, we +sneak in Ridge regression (to be discussed below) which includes a +hyperparameter \( \lambda \), also to be explained below. +

      @@ -392,10 +375,6 @@ $$

    5220. 60
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    5222. 62
      5223. -
    5223. 63
      5224. -
    5224. 64
      5225. -
    5225. 65
      5226. -
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    5227. »
      5228. diff --git a/doc/pub/week34/html/._week34-bs059.html b/doc/pub/week34/html/._week34-bs059.html index 0eabb58c4..0dd2d071f 100644 --- a/doc/pub/week34/html/._week34-bs059.html +++ b/doc/pub/week34/html/._week34-bs059.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5228. Installing R, C++, cython or Julia
    5229. Installing R, C++, cython, Numba etc
    5230. Numpy examples and Important Matrix and vector handling packages
      5231. -
    5231. Basic Matrix Features
      5232. -
    5232.    Some famous Matrices
      5233. -
    5233.    More Basic Matrix Features
      5234. -
    5234. Numpy and arrays
      5235. -
    5235. Matrices in Python
      5236. -
    5236. Meet the Pandas
      5237. -
    5237. Friday August 27
      5238. -
    5238.    Simple linear regression model using scikit-learn
      5239. -
    5239.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5240. -
    5240.    Organizing our data
      5241. -
    5241.    Seeing the wood for the trees
      5242. -
    5242.    And what about using neural networks?
      5243. -
    5243. A first summary
      5244. -
    5244. Why Linear Regression (aka Ordinary Least Squares and family)
      5245. -
    5245. Regression analysis, overarching aims
      5246. -
    5246. Regression analysis, overarching aims II
      5247. -
    5247. Examples
      5248. -
    5248. General linear models
      5249. -
    5249. Rewriting the fitting procedure as a linear algebra problem
      5250. -
    5250. Rewriting the fitting procedure as a linear algebra problem, more details
      5251. -
    5251. Generalizing the fitting procedure as a linear algebra problem
      5252. -
    5252. Generalizing the fitting procedure as a linear algebra problem
      5253. -
    5253. Optimizing our parameters
      5254. -
    5254. Our model for the nuclear binding energies
      5255. -
    5255. Optimizing our parameters, more details
      5256. -
    5256. Interpretations and optimizing our parameters
      5257. -
    5257. Interpretations and optimizing our parameters
      5258. -
    5258. Some useful matrix and vector expressions
      5259. -
    5259. Interpretations and optimizing our parameters
      5260. -
    5260. Own code for Ordinary Least Squares
      5261. -
    5261. Adding error analysis and training set up
      5262. -
    5262. The \( \chi^2 \) function
      5263. -
    5263. The \( \chi^2 \) function
      5264. -
    5264. The \( \chi^2 \) function
      5265. -
    5265. The \( \chi^2 \) function
      5266. -
    5266. The \( \chi^2 \) function
      5267. -
    5267. The \( \chi^2 \) function
      5268. -
    5268. Fitting an Equation of State for Dense Nuclear Matter
      5269. -
    5269. The code
      5270. -
    5270. Splitting our Data in Training and Test data
      5271. -
    5271. Exercises
      5272. -
    5272. Exercise 1: Setting up various Python environments
      5273. -
    5273. Exercise 2: making your own data and exploring scikit-learn
      5274. -
    5274. Exercise 3: Normalizing our data
      5275. +
    5275. Numpy and arrays
      5276. +
    5276. Matrices in Python
      5277. +
    5277. Meet the Pandas
      5278. +
    5278.    Simple linear regression model using scikit-learn
      5279. +
    5279.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5280. +
    5280.    Organizing our data
      5281. +
    5281.    And what about using neural networks?
      5282. +
    5282. A first summary
      5283. +
    5283. Why Linear Regression (aka Ordinary Least Squares and family)
      5284. +
    5284. Regression analysis, overarching aims
      5285. +
    5285. Regression analysis, overarching aims II
      5286. +
    5286. Examples
      5287. +
    5287. General linear models
      5288. +
    5288. Rewriting the fitting procedure as a linear algebra problem
      5289. +
    5289. Rewriting the fitting procedure as a linear algebra problem, more details
      5290. +
    5290. Generalizing the fitting procedure as a linear algebra problem
      5291. +
    5291. Generalizing the fitting procedure as a linear algebra problem
      5292. +
    5292. Optimizing our parameters
      5293. +
    5293. Our model for the nuclear binding energies
      5294. +
    5294. Optimizing our parameters, more details
      5295. +
    5295. Interpretations and optimizing our parameters
      5296. +
    5296. Interpretations and optimizing our parameters
      5297. +
    5297. Some useful matrix and vector expressions
      5298. +
    5298. Interpretations and optimizing our parameters
      5299. +
    5299. Own code for Ordinary Least Squares
      5300. +
    5300. Adding error analysis and training set up
      5301. +
    5301. The \( \chi^2 \) function
      5302. +
    5302. The \( \chi^2 \) function
      5303. +
    5303. The \( \chi^2 \) function
      5304. +
    5304. The \( \chi^2 \) function
      5305. +
    5305. The \( \chi^2 \) function
      5306. +
    5306. The \( \chi^2 \) function
      5307. +
    5307. Fitting an Equation of State for Dense Nuclear Matter
      5308. +
    5308. The code
      5309. +
    5309. Splitting our Data in Training and Test data
      5310. +
    5310. Exercises
      5311. +
    5311. Exercise 1: Setting up various Python environments
      5312. +
    5312. Exercise 2: making your own data and exploring scikit-learn
      5313. +
    5313. Exercise 3: Split data in test and training data
      5314. @@ -351,33 +335,123 @@ MathJax.Hub.Config({

       

       

       

      -

      The \( \chi^2 \) function

      -
      -
      - +

      The code

      -

      If we then introduce the matrix

      -$$ -\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, -$$ -

      we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \))

      -$$ -\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} -$$ + +
      +
      +
      +
      +
      + 
      # Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import matplotlib.pyplot as plt
+import sklearn.linear_model as skl
+from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
 
-
      We state without proof the expression for the uncertainty  in the parameters \( \beta_j \) as (we leave this as an exercise)
      
-$$
-\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, 
-$$
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
 
-
      resulting in 
      
-$$
-\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
-$$
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+X = np.zeros((len(Density),4))
+X[:,3] = Density**(4.0/3.0)
+X[:,2] = Density
+X[:,1] = Density**(2.0/3.0)
+X[:,0] = 1
+
+# We use now Scikit-Learn's linear regressor and ridge regressor
+# OLS part
+clf = skl.LinearRegression().fit(X, Energies)
+ytilde = clf.predict(X)
+EoS['Eols']  = ytilde
+# The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(Energies, ytilde))
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))
+print(clf.coef_, clf.intercept_)
+
+# The Ridge regression with a hyperparameter lambda = 0.1
+_lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)
+yridge = clf_ridge.predict(X)
+EoS['Eridge']  = yridge
+# The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(Energies, yridge))
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))
+print(clf_ridge.coef_, clf_ridge.intercept_)
+
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$')
+ax.set_ylabel(r'Energy per particle')
+ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,
+            label='Theoretical data')
+ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',
+            label='OLS')
+ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',
+            label='Ridge $\lambda = 0.1$')
+ax.legend()
+save_fig("EoSfitting")
+plt.show()
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +

      The above simple polynomial in density \( \rho \) gives an excellent fit +to the data. +

      + +

      We note also that there is a small deviation between the +standard OLS and the Ridge regression at higher densities. We discuss this in more detail +below. +

      @@ -396,10 +470,6 @@ $$

    5314. 60
    5315. 61
    5316. 62
      5317. -
    5317. 63
      5318. -
    5318. 64
      5319. -
    5319. 65
      5320. -
    5320. 66
    5321. »
      5322. diff --git a/doc/pub/week34/html/._week34-bs060.html b/doc/pub/week34/html/._week34-bs060.html index aba97711b..60e7d6219 100644 --- a/doc/pub/week34/html/._week34-bs060.html +++ b/doc/pub/week34/html/._week34-bs060.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5322. Installing R, C++, cython or Julia
    5323. Installing R, C++, cython, Numba etc
    5324. Numpy examples and Important Matrix and vector handling packages
      5325. -
    5325. Basic Matrix Features
      5326. -
    5326.    Some famous Matrices
      5327. -
    5327.    More Basic Matrix Features
      5328. -
    5328. Numpy and arrays
      5329. -
    5329. Matrices in Python
      5330. -
    5330. Meet the Pandas
      5331. -
    5331. Friday August 27
      5332. -
    5332.    Simple linear regression model using scikit-learn
      5333. -
    5333.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5334. -
    5334.    Organizing our data
      5335. -
    5335.    Seeing the wood for the trees
      5336. -
    5336.    And what about using neural networks?
      5337. -
    5337. A first summary
      5338. -
    5338. Why Linear Regression (aka Ordinary Least Squares and family)
      5339. -
    5339. Regression analysis, overarching aims
      5340. -
    5340. Regression analysis, overarching aims II
      5341. -
    5341. Examples
      5342. -
    5342. General linear models
      5343. -
    5343. Rewriting the fitting procedure as a linear algebra problem
      5344. -
    5344. Rewriting the fitting procedure as a linear algebra problem, more details
      5345. -
    5345. Generalizing the fitting procedure as a linear algebra problem
      5346. -
    5346. Generalizing the fitting procedure as a linear algebra problem
      5347. -
    5347. Optimizing our parameters
      5348. -
    5348. Our model for the nuclear binding energies
      5349. -
    5349. Optimizing our parameters, more details
      5350. -
    5350. Interpretations and optimizing our parameters
      5351. -
    5351. Interpretations and optimizing our parameters
      5352. -
    5352. Some useful matrix and vector expressions
      5353. -
    5353. Interpretations and optimizing our parameters
      5354. -
    5354. Own code for Ordinary Least Squares
      5355. -
    5355. Adding error analysis and training set up
      5356. -
    5356. The \( \chi^2 \) function
      5357. -
    5357. The \( \chi^2 \) function
      5358. -
    5358. The \( \chi^2 \) function
      5359. -
    5359. The \( \chi^2 \) function
      5360. -
    5360. The \( \chi^2 \) function
      5361. -
    5361. The \( \chi^2 \) function
      5362. -
    5362. Fitting an Equation of State for Dense Nuclear Matter
      5363. -
    5363. The code
      5364. -
    5364. Splitting our Data in Training and Test data
      5365. -
    5365. Exercises
      5366. -
    5366. Exercise 1: Setting up various Python environments
      5367. -
    5367. Exercise 2: making your own data and exploring scikit-learn
      5368. -
    5368. Exercise 3: Normalizing our data
      5369. +
    5369. Numpy and arrays
      5370. +
    5370. Matrices in Python
      5371. +
    5371. Meet the Pandas
      5372. +
    5372.    Simple linear regression model using scikit-learn
      5373. +
    5373.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5374. +
    5374.    Organizing our data
      5375. +
    5375.    And what about using neural networks?
      5376. +
    5376. A first summary
      5377. +
    5377. Why Linear Regression (aka Ordinary Least Squares and family)
      5378. +
    5378. Regression analysis, overarching aims
      5379. +
    5379. Regression analysis, overarching aims II
      5380. +
    5380. Examples
      5381. +
    5381. General linear models
      5382. +
    5382. Rewriting the fitting procedure as a linear algebra problem
      5383. +
    5383. Rewriting the fitting procedure as a linear algebra problem, more details
      5384. +
    5384. Generalizing the fitting procedure as a linear algebra problem
      5385. +
    5385. Generalizing the fitting procedure as a linear algebra problem
      5386. +
    5386. Optimizing our parameters
      5387. +
    5387. Our model for the nuclear binding energies
      5388. +
    5388. Optimizing our parameters, more details
      5389. +
    5389. Interpretations and optimizing our parameters
      5390. +
    5390. Interpretations and optimizing our parameters
      5391. +
    5391. Some useful matrix and vector expressions
      5392. +
    5392. Interpretations and optimizing our parameters
      5393. +
    5393. Own code for Ordinary Least Squares
      5394. +
    5394. Adding error analysis and training set up
      5395. +
    5395. The \( \chi^2 \) function
      5396. +
    5396. The \( \chi^2 \) function
      5397. +
    5397. The \( \chi^2 \) function
      5398. +
    5398. The \( \chi^2 \) function
      5399. +
    5399. The \( \chi^2 \) function
      5400. +
    5400. The \( \chi^2 \) function
      5401. +
    5401. Fitting an Equation of State for Dense Nuclear Matter
      5402. +
    5402. The code
      5403. +
    5403. Splitting our Data in Training and Test data
      5404. +
    5404. Exercises
      5405. +
    5405. Exercise 1: Setting up various Python environments
      5406. +
    5406. Exercise 2: making your own data and exploring scikit-learn
      5407. +
    5407. Exercise 3: Split data in test and training data
      5408. @@ -351,25 +335,104 @@ MathJax.Hub.Config({

       

       

       

      -

      The \( \chi^2 \) function

      -
      -
      - -

      The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write

      -$$ -y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. -$$ +

      Splitting our Data in Training and Test data

      -

      By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by

      -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, -$$ +

      It is normal in essentially all Machine Learning studies to split the +data in a training set and a test set (sometimes also an additional +validation set). Scikit-Learn has an own function for this. There +is no explicit recipe for how much data should be included as training +data and say test data. An accepted rule of thumb is to use +approximately \( 2/3 \) to \( 4/5 \) of the data as training data. We will +postpone a discussion of this splitting to the end of these notes and +our discussion of the so-called bias-variance tradeoff. Here we +limit ourselves to repeat the above equation of state fitting example +but now splitting the data into a training set and a test set. +

      -

      and

      -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. -$$ + + +
      +
      +
      +
      +
      + 
      import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organized into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+X = np.zeros((len(Density),5))
+X[:,0] = 1
+X[:,1] = Density**(2.0/3.0)
+X[:,2] = Density
+X[:,3] = Density**(4.0/3.0)
+X[:,4] = Density**(5.0/3.0)
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+# matrix inversion to find beta
+beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
+# and then make the prediction
+ytilde = X_train @ beta
+print("Training R2")
+print(R2(y_train,ytilde))
+print("Training MSE")
+print(MSE(y_train,ytilde))
+ypredict = X_test @ beta
+print("Test R2")
+print(R2(y_test,ypredict))
+print("Test MSE")
+print(MSE(y_test,ypredict))
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      @@ -389,10 +452,6 @@ $$
    5408. 60
    5409. 61
    5410. 62
      5411. -
    5411. 63
      5412. -
    5412. 64
      5413. -
    5413. 65
      5414. -
    5414. 66
    5415. »
      5416. diff --git a/doc/pub/week34/html/._week34-bs061.html b/doc/pub/week34/html/._week34-bs061.html index 59e886c67..e49663ca8 100644 --- a/doc/pub/week34/html/._week34-bs061.html +++ b/doc/pub/week34/html/._week34-bs061.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5416. Installing R, C++, cython or Julia
    5417. Installing R, C++, cython, Numba etc
    5418. Numpy examples and Important Matrix and vector handling packages
      5419. -
    5419. Basic Matrix Features
      5420. -
    5420.    Some famous Matrices
      5421. -
    5421.    More Basic Matrix Features
      5422. -
    5422. Numpy and arrays
      5423. -
    5423. Matrices in Python
      5424. -
    5424. Meet the Pandas
      5425. -
    5425. Friday August 27
      5426. -
    5426.    Simple linear regression model using scikit-learn
      5427. -
    5427.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5428. -
    5428.    Organizing our data
      5429. -
    5429.    Seeing the wood for the trees
      5430. -
    5430.    And what about using neural networks?
      5431. -
    5431. A first summary
      5432. -
    5432. Why Linear Regression (aka Ordinary Least Squares and family)
      5433. -
    5433. Regression analysis, overarching aims
      5434. -
    5434. Regression analysis, overarching aims II
      5435. -
    5435. Examples
      5436. -
    5436. General linear models
      5437. -
    5437. Rewriting the fitting procedure as a linear algebra problem
      5438. -
    5438. Rewriting the fitting procedure as a linear algebra problem, more details
      5439. -
    5439. Generalizing the fitting procedure as a linear algebra problem
      5440. -
    5440. Generalizing the fitting procedure as a linear algebra problem
      5441. -
    5441. Optimizing our parameters
      5442. -
    5442. Our model for the nuclear binding energies
      5443. -
    5443. Optimizing our parameters, more details
      5444. -
    5444. Interpretations and optimizing our parameters
      5445. -
    5445. Interpretations and optimizing our parameters
      5446. -
    5446. Some useful matrix and vector expressions
      5447. -
    5447. Interpretations and optimizing our parameters
      5448. -
    5448. Own code for Ordinary Least Squares
      5449. -
    5449. Adding error analysis and training set up
      5450. -
    5450. The \( \chi^2 \) function
      5451. -
    5451. The \( \chi^2 \) function
      5452. -
    5452. The \( \chi^2 \) function
      5453. -
    5453. The \( \chi^2 \) function
      5454. -
    5454. The \( \chi^2 \) function
      5455. -
    5455. The \( \chi^2 \) function
      5456. -
    5456. Fitting an Equation of State for Dense Nuclear Matter
      5457. -
    5457. The code
      5458. -
    5458. Splitting our Data in Training and Test data
      5459. -
    5459. Exercises
      5460. -
    5460. Exercise 1: Setting up various Python environments
      5461. -
    5461. Exercise 2: making your own data and exploring scikit-learn
      5462. -
    5462. Exercise 3: Normalizing our data
      5463. +
    5463. Numpy and arrays
      5464. +
    5464. Matrices in Python
      5465. +
    5465. Meet the Pandas
      5466. +
    5466.    Simple linear regression model using scikit-learn
      5467. +
    5467.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5468. +
    5468.    Organizing our data
      5469. +
    5469.    And what about using neural networks?
      5470. +
    5470. A first summary
      5471. +
    5471. Why Linear Regression (aka Ordinary Least Squares and family)
      5472. +
    5472. Regression analysis, overarching aims
      5473. +
    5473. Regression analysis, overarching aims II
      5474. +
    5474. Examples
      5475. +
    5475. General linear models
      5476. +
    5476. Rewriting the fitting procedure as a linear algebra problem
      5477. +
    5477. Rewriting the fitting procedure as a linear algebra problem, more details
      5478. +
    5478. Generalizing the fitting procedure as a linear algebra problem
      5479. +
    5479. Generalizing the fitting procedure as a linear algebra problem
      5480. +
    5480. Optimizing our parameters
      5481. +
    5481. Our model for the nuclear binding energies
      5482. +
    5482. Optimizing our parameters, more details
      5483. +
    5483. Interpretations and optimizing our parameters
      5484. +
    5484. Interpretations and optimizing our parameters
      5485. +
    5485. Some useful matrix and vector expressions
      5486. +
    5486. Interpretations and optimizing our parameters
      5487. +
    5487. Own code for Ordinary Least Squares
      5488. +
    5488. Adding error analysis and training set up
      5489. +
    5489. The \( \chi^2 \) function
      5490. +
    5490. The \( \chi^2 \) function
      5491. +
    5491. The \( \chi^2 \) function
      5492. +
    5492. The \( \chi^2 \) function
      5493. +
    5493. The \( \chi^2 \) function
      5494. +
    5494. The \( \chi^2 \) function
      5495. +
    5495. Fitting an Equation of State for Dense Nuclear Matter
      5496. +
    5496. The code
      5497. +
    5497. Splitting our Data in Training and Test data
      5498. +
    5498. Exercises
      5499. +
    5499. Exercise 1: Setting up various Python environments
      5500. +
    5500. Exercise 2: making your own data and exploring scikit-learn
      5501. +
    5501. Exercise 3: Split data in test and training data
      5502. @@ -351,58 +335,210 @@ MathJax.Hub.Config({

       

       

       

      -

      The \( \chi^2 \) function

      -
      -
      - +

      Exercises

      -

      For a linear fit (a first-order polynomial) we don't need to invert a matrix!! -Defining +

      Here are three possible exercises for week 34

      + + +

      Exercise 1: Setting up various Python environments

      + +

      The first exercise here is of a mere technical art. We want you to have

      +
        +
      • git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo GitHub facilities.
        • +
      • Install various Python packages
        • +
      +

      We will make extensive use of Python as programming language and its +myriad of available libraries. You will find +IPython/Jupyter notebooks invaluable in your work. You can run R +codes in the Jupyter/IPython notebooks, with the immediate benefit of +visualizing your data. You can also use compiled languages like C++, +Rust, Fortran etc if you prefer. The focus in these lectures will be +on Python.

      -$$ -\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, -$$ - -$$ -\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, -$$ - - -$$ -\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), -$$ - - -$$ -\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, -$$ - - -$$ -\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, -$$ - -

      we obtain

      - -$$ -\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, -$$ - - -$$ -\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. -$$ - -

      This approach (different linear and non-linear regression) suffers -often from both being underdetermined and overdetermined in the -unknown coefficients \( \beta_i \). A better approach is to use the -Singular Value Decomposition (SVD) method discussed next week. +

      If you have Python installed (we recommend Python3) and you feel +pretty familiar with installing different packages, we recommend that +you install the following Python packages via pip as

      + +
        +
      1. pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow
        2. +
      +

      For Tensorflow, we recommend following the instructions in the text of +Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly +

      + +

      We will come back to tensorflow later.

      + +

      For Python3, replace pip with pip3.

      + +

      For OSX users we recommend, after having installed Xcode, to +install brew. Brew allows for a seamless installation of additional +software via for example +

      + +
        +
      1. brew install python3
        2. +
      +

      For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, +you can use pip as well and simply install Python as +

      + +
        +
      1. sudo apt-get install python3 (or python for Python2.7)
        2. +
      +

      If you don't want to perform these operations separately and venture +into the hassle of exploring how to set up dependencies and paths, we +recommend two widely used distrubutions which set up all relevant +dependencies for Python, namely +

      + + +

      which is an open source +distribution of the Python and R programming languages for large-scale +data processing, predictive analytics, and scientific computing, that +aims to simplify package management and deployment. Package versions +are managed by the package management system conda. +

      + + +

      is a Python +distribution for scientific and analytic computing distribution and +analysis environment, available for free and under a commercial +license. +

      + +

      We recommend using Anaconda if you are not too familiar with setting paths in a terminal environment.

      + + + + +

      Exercise 2: making your own data and exploring scikit-learn

      + +

      We will generate our own dataset for a function \( y(x) \) where \( x \in [0,1] \) and defined by random numbers computed with the uniform distribution. The function \( y \) is a quadratic polynomial in \( x \) with added stochastic noise according to the normal distribution \( \cal {N}(0,1) \). +The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points). +

      + + +
      +
      +
      +
      +
      + 
      x = np.random.rand(100,1)
+y = 2.0+5*x*x+0.1*np.random.randn(100,1)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
        +
      1. Write your own code (following the examples under the regression notes) for computing the parametrization of the data set fitting a second-order polynomial.
        2. +
      2. Use thereafter scikit-learn (see again the examples in the regression slides) and compare with your own code.
        3. +
      3. Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
        4. +
      +$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +$$ + +

      and the \( R^2 \) score function. +If \( \tilde{\boldsymbol{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as +

      +$$ +R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +$$ + +

      where we have defined the mean value of \( \boldsymbol{y} \) as

      +$$ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +$$ + +

      You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. +Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. +

      + + + + +

      Exercise 3: Split data in test and training data

      + +

      In this exercise we want you to to compute the MSE for the training +data and the test data as function of the complexity of a polynomial, +that is the degree of a given polynomial. +

      + +

      The aim is to reproduce Figure 2.11 of Hastie et al.

      + +

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points. You should try to vary \( n \) in your analysis.

      + + +
      +
      +
      +
      +
      + 
      np.random.seed()
+n = 100
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      +
      + +

      where \( y \) is the function we want to fit with a given polynomial.

      + + +

      +a) +Write a first code which sets up a design matrix \( X \) defined by a fifth-order polynomial and split your data set in training and test data. +

      + + + + +

      +b) +Perform an ordinary least squares fitting and compute the means squared error for the training data and the test data. +

      + + + + +

      +c) +Add now a model which allows you to make polynomials up to degree \( 15 \). Perform a standard OLS fitting of the training data and compute the MSE for the training and test data and plot both test and training data MSE as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)? +

      + + + +

        @@ -418,11 +554,6 @@ Singular Value Decomposition (SVD) method discussed next week.
      • 60
      • 61
      • 62
        • -
      • 63
        • -
      • 64
        • -
      • 65
        • -
      • 66
        • -
      • »
      diff --git a/doc/pub/week34/html/week34-bs.html b/doc/pub/week34/html/week34-bs.html index 9aaab6cf1..19b4a2d84 100644 --- a/doc/pub/week34/html/week34-bs.html +++ b/doc/pub/week34/html/week34-bs.html @@ -109,16 +109,9 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'numpy-examples-and-important-matrix-and-vector-handling-packages'), - ('Basic Matrix Features', 2, None, 'basic-matrix-features'), - ('Some famous Matrices', 3, None, 'some-famous-matrices'), - ('More Basic Matrix Features', - 3, - None, - 'more-basic-matrix-features'), ('Numpy and arrays', 2, None, 'numpy-and-arrays'), ('Matrices in Python', 2, None, 'matrices-in-python'), ('Meet the Pandas', 2, None, 'meet-the-pandas'), - ('Friday August 27', 2, None, 'friday-august-27'), ('Simple linear regression model using _scikit-learn_', 3, None, @@ -129,10 +122,6 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d None, 'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'), ('Organizing our data', 3, None, 'organizing-our-data'), - ('Seeing the wood for the trees', - 3, - None, - 'seeing-the-wood-for-the-trees'), ('And what about using neural networks?', 3, None, @@ -229,10 +218,10 @@ doconce format html week34.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'exercise-2-making-your-own-data-and-exploring-scikit-learn'), - ('Exercise 3: Normalizing our data', + ('Exercise 3: Split data in test and training data', 2, None, - 'exercise-3-normalizing-our-data')]} + 'exercise-3-split-data-in-test-and-training-data')]} end of tocinfo --> @@ -296,50 +285,45 @@ MathJax.Hub.Config({
    5502. Installing R, C++, cython or Julia
    5503. Installing R, C++, cython, Numba etc
    5504. Numpy examples and Important Matrix and vector handling packages
      5505. -
    5505. Basic Matrix Features
      5506. -
    5506.    Some famous Matrices
      5507. -
    5507.    More Basic Matrix Features
      5508. -
    5508. Numpy and arrays
      5509. -
    5509. Matrices in Python
      5510. -
    5510. Meet the Pandas
      5511. -
    5511. Friday August 27
      5512. -
    5512.    Simple linear regression model using scikit-learn
      5513. -
    5513.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5514. -
    5514.    Organizing our data
      5515. -
    5515.    Seeing the wood for the trees
      5516. -
    5516.    And what about using neural networks?
      5517. -
    5517. A first summary
      5518. -
    5518. Why Linear Regression (aka Ordinary Least Squares and family)
      5519. -
    5519. Regression analysis, overarching aims
      5520. -
    5520. Regression analysis, overarching aims II
      5521. -
    5521. Examples
      5522. -
    5522. General linear models
      5523. -
    5523. Rewriting the fitting procedure as a linear algebra problem
      5524. -
    5524. Rewriting the fitting procedure as a linear algebra problem, more details
      5525. -
    5525. Generalizing the fitting procedure as a linear algebra problem
      5526. -
    5526. Generalizing the fitting procedure as a linear algebra problem
      5527. -
    5527. Optimizing our parameters
      5528. -
    5528. Our model for the nuclear binding energies
      5529. -
    5529. Optimizing our parameters, more details
      5530. -
    5530. Interpretations and optimizing our parameters
      5531. -
    5531. Interpretations and optimizing our parameters
      5532. -
    5532. Some useful matrix and vector expressions
      5533. -
    5533. Interpretations and optimizing our parameters
      5534. -
    5534. Own code for Ordinary Least Squares
      5535. -
    5535. Adding error analysis and training set up
      5536. -
    5536. The \( \chi^2 \) function
      5537. -
    5537. The \( \chi^2 \) function
      5538. -
    5538. The \( \chi^2 \) function
      5539. -
    5539. The \( \chi^2 \) function
      5540. -
    5540. The \( \chi^2 \) function
      5541. -
    5541. The \( \chi^2 \) function
      5542. -
    5542. Fitting an Equation of State for Dense Nuclear Matter
      5543. -
    5543. The code
      5544. -
    5544. Splitting our Data in Training and Test data
      5545. -
    5545. Exercises
      5546. -
    5546. Exercise 1: Setting up various Python environments
      5547. -
    5547. Exercise 2: making your own data and exploring scikit-learn
      5548. -
    5548. Exercise 3: Normalizing our data
      5549. +
    5549. Numpy and arrays
      5550. +
    5550. Matrices in Python
      5551. +
    5551. Meet the Pandas
      5552. +
    5552.    Simple linear regression model using scikit-learn
      5553. +
    5553.    To our real data: nuclear binding energies. Brief reminder on masses and binding energies
      5554. +
    5554.    Organizing our data
      5555. +
    5555.    And what about using neural networks?
      5556. +
    5556. A first summary
      5557. +
    5557. Why Linear Regression (aka Ordinary Least Squares and family)
      5558. +
    5558. Regression analysis, overarching aims
      5559. +
    5559. Regression analysis, overarching aims II
      5560. +
    5560. Examples
      5561. +
    5561. General linear models
      5562. +
    5562. Rewriting the fitting procedure as a linear algebra problem
      5563. +
    5563. Rewriting the fitting procedure as a linear algebra problem, more details
      5564. +
    5564. Generalizing the fitting procedure as a linear algebra problem
      5565. +
    5565. Generalizing the fitting procedure as a linear algebra problem
      5566. +
    5566. Optimizing our parameters
      5567. +
    5567. Our model for the nuclear binding energies
      5568. +
    5568. Optimizing our parameters, more details
      5569. +
    5569. Interpretations and optimizing our parameters
      5570. +
    5570. Interpretations and optimizing our parameters
      5571. +
    5571. Some useful matrix and vector expressions
      5572. +
    5572. Interpretations and optimizing our parameters
      5573. +
    5573. Own code for Ordinary Least Squares
      5574. +
    5574. Adding error analysis and training set up
      5575. +
    5575. The \( \chi^2 \) function
      5576. +
    5576. The \( \chi^2 \) function
      5577. +
    5577. The \( \chi^2 \) function
      5578. +
    5578. The \( \chi^2 \) function
      5579. +
    5579. The \( \chi^2 \) function
      5580. +
    5580. The \( \chi^2 \) function
      5581. +
    5581. Fitting an Equation of State for Dense Nuclear Matter
      5582. +
    5582. The code
      5583. +
    5583. Splitting our Data in Training and Test data
      5584. +
    5584. Exercises
      5585. +
    5585. Exercise 1: Setting up various Python environments
      5586. +
    5586. Exercise 2: making your own data and exploring scikit-learn
      5587. +
    5587. Exercise 3: Split data in test and training data
      5588. @@ -394,7 +378,7 @@ MathJax.Hub.Config({
    5588. 9
    5589. 10
    5590. ...
      5591. -
    5591. 66
      5592. +
    5592. 62
    5593. »
      5594. diff --git a/doc/pub/week34/html/week34-reveal.html b/doc/pub/week34/html/week34-reveal.html index fca2cf221..4b6297059 100644 --- a/doc/pub/week34/html/week34-reveal.html +++ b/doc/pub/week34/html/week34-reveal.html @@ -199,7 +199,7 @@ MathJax.Hub.Config({

      1. The sessions on Tuesdays and Wednesdays last four hours for each group (four groups in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects.
        2. -

      2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week.
        3. +

      3. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week. Videos and teaching material will be announced in due time.

      4. There are four groups:
          5. @@ -579,9 +579,9 @@ topics and tools as well as showing the power of various Python libraries for machine learning and statistical data analysis.

          -

          Here, we will mainly focus on two +

          Although we have projects where you write your own codes, we will also focus on two specific Python packages for Machine Learning, Scikit-Learn and -Tensorflow (see below for links etc). Moreover, the examples we +Tensorflow with Keras (see below for links etc). Moreover, the examples we introduce will serve as inputs to many of our discussions later, as well as allowing you to set up models and produce your own data and get started with programming. @@ -901,8 +901,8 @@ use Python during our lectures and in various projects and exercises. Those of you already familiar with R should feel free to continue using R, keeping however an eye on the parallel Python set ups. Similarly, if you are a -Python afecionado, feel free to explore R as well. Jupyter/Ipython -notebook allows you to run R codes interactively in your +Python afecionado, feel free to explore R as well. Jupyter(Julia, Python and R) /Ipython +notebook allows you to run R codes and Julia codes interactively in your browser. The software library R is really tailored for statistical data analysis and allows for an easy usage of the tools and algorithms we will discuss in these lectures. @@ -960,10 +960,7 @@ further processing. For example, convert to latex as

          And to add more versatility, the Python package SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python.

          -

          Finally, if you wish to use the light mark-up language -doconce you can convert a standard ascii text file into various HTML -formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using doconce. -

          +

          Finally, we recommend strongly using Autograd or JAX for automatic differentiation.

          @@ -985,103 +982,6 @@ developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly he
        -
        -

        Basic Matrix Features

        - -
        -Matrix properties reminder -

        -

         
        -$$ - \mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -$$ -

         
        - -

        The inverse of a matrix is defined by

        - -

         
        -$$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -$$ -

         
        - - - - - - - - - - - - - -
        Relations Name matrix elements
        \( A=A^{T} \) symmetric \( a_{ij}=a_{ji} \)
        \( A=\left (A^{T}\right )^{-1} \) real orthogonal \( \sum_k a_{ik}a_{jk}=\sum_k a_{ki} a_{kj}=\delta_{ij} \)
        \( A=A^* \) real matrix \( a_{ij}=a_{ij}^* \)
        \( A=A^{\dagger} \) hermitian \( a_{ij}=a_{ji}^* \)
        \( A=\left(A^{\dagger}\right )^{-1} \) unitary \( \sum_k a_{ik}a_{jk}^*=\sum_k a_{ki}^* a_{kj}=\delta_{ij} \)
        -

        -
        - -
        -

        Some famous Matrices

        - -
          - -

        • Diagonal if \( a_{ij}=0 \) for \( i\ne j \)
          • - -

        • Upper triangular if \( a_{ij}=0 \) for \( i>j \)
          • - -

        • Lower triangular if \( a_{ij}=0 \) for \( i < j \)
          • - -

        • Upper Hessenberg if \( a_{ij}=0 \) for \( i>j+1 \)
          • - -

        • Lower Hessenberg if \( a_{ij}=0 \) for \( i < j+1 \)
          • - -

        • Tridiagonal if \( a_{ij}=0 \) for \( |i -j|>1 \)
          • - -

        • Lower banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i>j+p \)
          • - -

        • Upper banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i < j+p \)
          • - -

        • Banded, block upper triangular, block lower triangular....
          • -
        -
        - -
        -

        More Basic Matrix Features

        - -
        -Some Equivalent Statements -

        -

        For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

        - -
          - -

        • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
          • - -

        • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
          • - -

        • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
          • - -

        • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
          • - -

        • \( \mathbf{A} \) is a product of elementary matrices.
          • - -

        • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
          • -
        -
        -
        -

        Numpy and arrays

        Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

        @@ -1837,14 +1737,6 @@ For multidimensional arrays, we recommend strongly Friday August 27 - -

        "Video of Lecture August 27, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust27.mp4?vrtx=view-as-webpage

        - -

        Video of Lecture from fall 2020 and Handwritten notes

        -
        -

        Simple linear regression model using scikit-learn

        @@ -2197,62 +2089,8 @@ $$

        Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).

        We will discuss in more detail these and other functions in the -various lectures. We conclude this part with another example. Instead -of a linear \( x \)-dependence we study now a cubic polynomial and use the -polynomial regression analysis tools of scikit-learn. +various lectures and lab sessions.

        - - - -
        -
        -
        -
        -
        - 
        import matplotlib.pyplot as plt
-import numpy as np
-import random
-from sklearn.linear_model import Ridge
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.pipeline import make_pipeline
-from sklearn.linear_model import LinearRegression
-
-x=np.linspace(0.02,0.98,200)
-noise = np.asarray(random.sample((range(200)),200))
-y=x**3*noise
-yn=x**3*100
-poly3 = PolynomialFeatures(degree=3)
-X = poly3.fit_transform(x[:,np.newaxis])
-clf3 = LinearRegression()
-clf3.fit(X,y)
-
-Xplot=poly3.fit_transform(x[:,np.newaxis])
-poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
-plt.plot(x,yn, color='red', label="True Cubic")
-plt.scatter(x, y, label='Data', color='orange', s=15)
-plt.legend()
-plt.show()
-
-def error(a):
-    for i in y:
-        err=(y-yn)/yn
-    return abs(np.sum(err))/len(err)
-
-print (error(y))
-
        -
        -
        -
        -
        -
        -
        -
        -
        -
        -
        -
        -
        -

        To our real data: nuclear binding energies. Brief reminder on masses and binding energies

        Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding @@ -2669,60 +2507,6 @@ plt.show()

      -

      Seeing the wood for the trees

      - -

      As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!

      - - - -
      -
      -
      -
      -
      - 
      #Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=5)
-regr_2=DecisionTreeRegressor(max_depth=7)
-regr_3=DecisionTreeRegressor(max_depth=9)
-regr_1.fit(X, Energies)
-regr_2.fit(X, Energies)
-regr_3.fit(X, Energies)
-
-
-y_1 = regr_1.predict(X)
-y_2 = regr_2.predict(X)
-y_3=regr_3.predict(X)
-Masses['Eapprox'] = y_3
-# Plot the results
-plt.figure()
-plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
-plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
-plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
-plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
-
-plt.xlabel("$A$")
-plt.ylabel("$E$[MeV]")
-plt.title("Decision Tree Regression")
-plt.legend()
-save_fig("Masses2016Trees")
-plt.show()
-print(Masses)
-print(np.mean( (Energies-y_1)**2))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      And what about using neural networks?

      The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn's multi-layer perceptron (or feed forward neural network) functionality. @@ -2867,7 +2651,7 @@ the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ld

      -

      In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), +

      In order to understand the relation among the predictors (or features or properties) \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), consider the model we discussed for describing nuclear binding energies.

      @@ -3331,33 +3115,9 @@ allow for the usage of direct linear algebra methods such as LU decomposi

      Some useful matrix and vector expressions

      -

      The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters. -

      +

      See the handwritten notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesExercise5Week452022.pdf

      -

       
      -$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, -$$ -

       
      - -

       
      -$$ -\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a}, -$$ -

       
      - -

       
      -$$ -\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, -$$ -

       
      - -

       
      -$$ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -$$ -

       
      +

      These notes will be discussed during one of the lectures.

      @@ -4057,7 +3817,8 @@ ypredict = X_test @ beta

      Exercises

      -

      Here are three possible exercises for weeks 34 and 35.

      + +

      Here are three possible exercises for week 34

      Exercise 1: Setting up various Python environments

      @@ -4206,181 +3967,19 @@ $$ Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.

      - -

      -Solution. -The code here is an example of where we define our own design matrix and fit parameters \( \beta \). -

      - - -
      -
      -
      -
      -
      - 
      import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),3))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x**2
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(beta)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - - - - -

      Exercise 3: Normalizing our data

      - -

      A much used approach before starting to train the data is to preprocess our -data. Normally the data may need a rescaling and/or may be sensitive -to extreme values. Scaling the data renders our inputs much more -suitable for the algorithms we want to employ. -

      - -

      Scikit-Learn has several functions which allow us to rescale the -data, normally resulting in much better results in terms of various -accuracy scores. The StandardScaler function in Scikit-Learn -ensures that for each feature/predictor we study the mean value is -zero and the variance is one (every column in the design/feature -matrix). This scaling has the drawback that it does not ensure that -we have a particular maximum or minimum in our data set. Another -function included in Scikit-Learn is the MinMaxScaler which -ensures that all features are exactly between \( 0 \) and \( 1 \). The -

      - -

      The Normalizer scales each data -point such that the feature vector has a euclidean length of one. In other words, it -projects a data point on the circle (or sphere in the case of higher dimensions) with a -radius of 1. This means every data point is scaled by a different number (by the -inverse of it’s length). -This normalization is often used when only the direction (or angle) of the data matters, -not the length of the feature vector. -

      - -

      The RobustScaler works similarly to the StandardScaler in that it -ensures statistical properties for each feature that guarantee that -they are on the same scale. However, the RobustScaler uses the median -and quartiles, instead of mean and variance. This makes the -RobustScaler ignore data points that are very different from the rest -(like measurement errors). These odd data points are also called -outliers, and might often lead to trouble for other scaling -techniques. -

      - -

      It also common to split the data in a training set and a testing set. A typical split is to use \( 80\% \) of the data for training and the rest -for testing. This can be done as follows with our design matrix \( \boldsymbol{X} \) and data \( \boldsymbol{y} \) (remember to import scikit-learn) -

      - - -
      -
      -
      -
      -
      - 
      # split in training and test data
-X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Then we can use the standard scaler to scale our data as

      - - -
      -
      -
      -
      -
      - 
      scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      +

      Exercise 3: Split data in test and training data

      In this exercise we want you to to compute the MSE for the training data and the test data as function of the complexity of a polynomial, -that is the degree of a given polynomial. We want you also to compute the \( R2 \) score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling. +that is the degree of a given polynomial.

      -

      One of -the aims is to reproduce Figure 2.11 of Hastie et al. -

      +

      The aim is to reproduce Figure 2.11 of Hastie et al.

      -

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points.

      +

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points. You should try to vary \( n \) in your analysis.

      @@ -4390,7 +3989,6 @@ the aims is to reproduce Figure 2.11 of 
      np.random.seed()
 n = 100
-maxdegree = 14
 # Make data set.
 x = np.linspace(-3, 3, n).reshape(-1, 1)
 y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
@@ -4414,7 +4012,7 @@ y = np.exp(-x**2) + SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS)  and is entirely written in Python. 
      
 
-
      Finally, if you wish to use the light mark-up language 
-doconce you can convert a standard ascii text file into various HTML 
-formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using doconce.
-
      
+
      Finally, we recommend strongly using Autograd or JAX for automatic differentiation.
      
 
 









      
 
      Numpy examples and Important Matrix and vector handling packages 
      
@@ -1011,82 +997,6 @@ developed in the 1970s, namely EISPACK and LINPACK.  We describe them shortly he
   
    5594.  LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.
      5595. 
   
    5595.  BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.
      5596. 
 
-









      
-
      Basic Matrix Features 
      
-
-
      
-Matrix properties reminder
-
      
-$$
- \mathbf{A} =
-      \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
-                                 a_{21} & a_{22} & a_{23} & a_{24} \\
-                                   a_{31} & a_{32} & a_{33} & a_{34} \\
-                                  a_{41} & a_{42} & a_{43} & a_{44}
-             \end{bmatrix}\qquad
-\mathbf{I} =
-      \begin{bmatrix} 1 & 0 & 0 & 0 \\
-                                 0 & 1 & 0 & 0 \\
-                                 0 & 0 & 1 & 0 \\
-                                 0 & 0 & 0 & 1
-             \end{bmatrix}
-$$
-
-
      The inverse of a matrix is defined by
      
-
-$$
-\mathbf{A}^{-1} \cdot \mathbf{A} = I
-$$
-
-
-
-
-
-
-
-
-
-
-
-
-
-
                    Relations                      Name                             matrix elements                         
         \( A=A^{T} \)                                 symmetric              \( a_{ij}=a_{ji} \)                                                
         \( A=\left (A^{T}\right )^{-1} \)             real orthogonal        \( \sum_k a_{ik}a_{jk}=\sum_k a_{ki} a_{kj}=\delta_{ij} \)         
         \( A=A^* \)                                   real matrix            \( a_{ij}=a_{ij}^* \)                                              
         \( A=A^{\dagger} \)                           hermitian              \( a_{ij}=a_{ji}^* \)                                              
         \( A=\left(A^{\dagger}\right )^{-1} \)        unitary                \( \sum_k a_{ik}a_{jk}^*=\sum_k a_{ki}^* a_{kj}=\delta_{ij} \)     
      
-
      
-
-
-









      
-
      Some famous Matrices 
      
-
-
        
-  
      •  Diagonal if \( a_{ij}=0 \) for \( i\ne j \)
        • 
-  
      •  Upper triangular if \( a_{ij}=0 \) for \( i>j \)
        • 
-  
      •  Lower triangular if \( a_{ij}=0 \) for \( i < j \)
        • 
-  
      •  Upper Hessenberg if \( a_{ij}=0 \) for \( i>j+1 \)
        • 
-  
      •  Lower Hessenberg if \( a_{ij}=0 \) for \( i < j+1 \)
        • 
-  
      •  Tridiagonal if \( a_{ij}=0 \) for \( |i -j|>1 \)
        • 
-  
      •  Lower banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i>j+p \)
        • 
-  
      •  Upper banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i < j+p \)
        • 
-  
      •  Banded, block upper triangular, block lower triangular....
        • 
-
      
-









      
-
      More Basic Matrix Features 
      
-
-
      
-Some Equivalent Statements
-
      
-
      For an \( N\times N \) matrix  \( \mathbf{A} \) the following properties are all equivalent
      
-
-
        
-  
      •  If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
        • 
-  
      •  The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
        • 
-  
      •  The rows of \( \mathbf{A} \) form a basis of \( R^N \).
        • 
-  
      •  The columns of \( \mathbf{A} \) form a basis of \( R^N \).
        • 
-  
      •  \( \mathbf{A} \) is a product of elementary matrices.
        • 
-  
      •  \( 0 \) is not eigenvalue of \( \mathbf{A} \).
        • 
-
      
-
      
-
-
 









      
 
      Numpy and arrays 
      
 
      Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as
      
@@ -1835,13 +1745,6 @@ As we will see below it leads also to a very concice code close to the mathemati
 For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray. 
 
      
 
-









      
-
      Friday August 27 
      
-
-
      "Video of Lecture August 27, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust27.mp4?vrtx=view-as-webpage
      
-
-
      Video of Lecture from fall 2020  and Handwritten notes
      
-
 









      
 
      Simple linear regression model using scikit-learn 
      
 
@@ -2174,62 +2077,8 @@ $$
 
      Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).
      
 
 
      We will discuss in more detail these and other functions in the
-various lectures.  We conclude this part with another example. Instead
-of a linear \( x \)-dependence we study now a cubic polynomial and use the
-polynomial regression analysis tools of scikit-learn.
+various lectures and lab sessions.
 
      
-
-
-
-
      
-  
      
-    
      
-      
      
-        
      
-  
      import matplotlib.pyplot as plt
-import numpy as np
-import random
-from sklearn.linear_model import Ridge
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.pipeline import make_pipeline
-from sklearn.linear_model import LinearRegression
-
-x=np.linspace(0.02,0.98,200)
-noise = np.asarray(random.sample((range(200)),200))
-y=x**3*noise
-yn=x**3*100
-poly3 = PolynomialFeatures(degree=3)
-X = poly3.fit_transform(x[:,np.newaxis])
-clf3 = LinearRegression()
-clf3.fit(X,y)
-
-Xplot=poly3.fit_transform(x[:,np.newaxis])
-poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
-plt.plot(x,yn, color='red', label="True Cubic")
-plt.scatter(x, y, label='Data', color='orange', s=15)
-plt.legend()
-plt.show()
-
-def error(a):
-    for i in y:
-        err=(y-yn)/yn
-    return abs(np.sum(err))/len(err)
-
-print (error(y))
-
      
-
      
-      
      
-    
      
-  
      
-  
      
-    
      
-      
      
-        
                
-        
      
-      
      
-    
      
-  
      
-
      
 
      To our real data: nuclear binding energies. Brief reminder on masses and binding energies 
      
 
 
      Let us now dive into  nuclear physics and remind ourselves briefly about some basic features about binding
@@ -2630,60 +2479,6 @@ plt.show()
      -

      Seeing the wood for the trees

      - -

      As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!

      - - - -
      -
      -
      -
      -
      - 
      #Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=5)
-regr_2=DecisionTreeRegressor(max_depth=7)
-regr_3=DecisionTreeRegressor(max_depth=9)
-regr_1.fit(X, Energies)
-regr_2.fit(X, Energies)
-regr_3.fit(X, Energies)
-
-
-y_1 = regr_1.predict(X)
-y_2 = regr_2.predict(X)
-y_3=regr_3.predict(X)
-Masses['Eapprox'] = y_3
-# Plot the results
-plt.figure()
-plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
-plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
-plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
-plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
-
-plt.xlabel("$A$")
-plt.ylabel("$E$[MeV]")
-plt.title("Decision Tree Regression")
-plt.legend()
-save_fig("Masses2016Trees")
-plt.show()
-print(Masses)
-print(np.mean( (Energies-y_1)**2))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      And what about using neural networks?

      The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn's multi-layer perceptron (or feed forward neural network) functionality. @@ -2824,7 +2619,7 @@ the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ld

      -

      In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), +

      In order to understand the relation among the predictors (or features or properties) \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), consider the model we discussed for describing nuclear binding energies.

      @@ -3235,25 +3030,9 @@ allow for the usage of direct linear algebra methods such as LU decomposi









      Some useful matrix and vector expressions

      -

      The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters. -

      +

      See the handwritten notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesExercise5Week452022.pdf

      -$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, -$$ - -$$ -\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a}, -$$ - -$$ -\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, -$$ - -$$ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -$$ +

      These notes will be discussed during one of the lectures.











      Interpretations and optimizing our parameters

      @@ -3907,7 +3686,8 @@ ypredict = X_test @ beta









      Exercises

      -

      Here are three possible exercises for weeks 34 and 35.

      + +

      Here are three possible exercises for week 34

      Exercise 1: Setting up various Python environments

      @@ -4042,181 +3822,19 @@ $$ Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.

      - -

      -Solution. -The code here is an example of where we define our own design matrix and fit parameters \( \beta \). -

      - - -
      -
      -
      -
      -
      - 
      import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),3))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x**2
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(beta)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - - - - -

      Exercise 3: Normalizing our data

      - -

      A much used approach before starting to train the data is to preprocess our -data. Normally the data may need a rescaling and/or may be sensitive -to extreme values. Scaling the data renders our inputs much more -suitable for the algorithms we want to employ. -

      - -

      Scikit-Learn has several functions which allow us to rescale the -data, normally resulting in much better results in terms of various -accuracy scores. The StandardScaler function in Scikit-Learn -ensures that for each feature/predictor we study the mean value is -zero and the variance is one (every column in the design/feature -matrix). This scaling has the drawback that it does not ensure that -we have a particular maximum or minimum in our data set. Another -function included in Scikit-Learn is the MinMaxScaler which -ensures that all features are exactly between \( 0 \) and \( 1 \). The -

      - -

      The Normalizer scales each data -point such that the feature vector has a euclidean length of one. In other words, it -projects a data point on the circle (or sphere in the case of higher dimensions) with a -radius of 1. This means every data point is scaled by a different number (by the -inverse of it’s length). -This normalization is often used when only the direction (or angle) of the data matters, -not the length of the feature vector. -

      - -

      The RobustScaler works similarly to the StandardScaler in that it -ensures statistical properties for each feature that guarantee that -they are on the same scale. However, the RobustScaler uses the median -and quartiles, instead of mean and variance. This makes the -RobustScaler ignore data points that are very different from the rest -(like measurement errors). These odd data points are also called -outliers, and might often lead to trouble for other scaling -techniques. -

      - -

      It also common to split the data in a training set and a testing set. A typical split is to use \( 80\% \) of the data for training and the rest -for testing. This can be done as follows with our design matrix \( \boldsymbol{X} \) and data \( \boldsymbol{y} \) (remember to import scikit-learn) -

      - - -
      -
      -
      -
      -
      - 
      # split in training and test data
-X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Then we can use the standard scaler to scale our data as

      - - -
      -
      -
      -
      -
      - 
      scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      +

      Exercise 3: Split data in test and training data

      In this exercise we want you to to compute the MSE for the training data and the test data as function of the complexity of a polynomial, -that is the degree of a given polynomial. We want you also to compute the \( R2 \) score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling. +that is the degree of a given polynomial.

      -

      One of -the aims is to reproduce Figure 2.11 of Hastie et al. -

      +

      The aim is to reproduce Figure 2.11 of Hastie et al.

      -

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points.

      +

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points. You should try to vary \( n \) in your analysis.

      @@ -4226,7 +3844,6 @@ the aims is to reproduce Figure 2.11 of 
      np.random.seed()
 n = 100
-maxdegree = 14
 # Make data set.
 x = np.linspace(-3, 3, n).reshape(-1, 1)
 y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
@@ -4250,7 +3867,7 @@ y = np.exp(-x**2) + SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS)  and is entirely written in Python. 
      
 
-
      Finally, if you wish to use the light mark-up language 
-doconce you can convert a standard ascii text file into various HTML 
-formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using doconce.
-
      
+
      Finally, we recommend strongly using Autograd or JAX for automatic differentiation.
      
 
 









      
 
      Numpy examples and Important Matrix and vector handling packages 
      
@@ -1088,82 +1074,6 @@ developed in the 1970s, namely EISPACK and LINPACK.  We describe them shortly he
   
    5596.  LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.
      5597. 
   
    5597.  BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.
      5598. 
 
-









      
-
      Basic Matrix Features 
      
-
-
      
-Matrix properties reminder
-
      
-$$
- \mathbf{A} =
-      \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
-                                 a_{21} & a_{22} & a_{23} & a_{24} \\
-                                   a_{31} & a_{32} & a_{33} & a_{34} \\
-                                  a_{41} & a_{42} & a_{43} & a_{44}
-             \end{bmatrix}\qquad
-\mathbf{I} =
-      \begin{bmatrix} 1 & 0 & 0 & 0 \\
-                                 0 & 1 & 0 & 0 \\
-                                 0 & 0 & 1 & 0 \\
-                                 0 & 0 & 0 & 1
-             \end{bmatrix}
-$$
-
-
      The inverse of a matrix is defined by
      
-
-$$
-\mathbf{A}^{-1} \cdot \mathbf{A} = I
-$$
-
-
-
-
-
-
-
-
-
-
-
-
-
-
                    Relations                      Name                             matrix elements                         
         \( A=A^{T} \)                                 symmetric              \( a_{ij}=a_{ji} \)                                                
         \( A=\left (A^{T}\right )^{-1} \)             real orthogonal        \( \sum_k a_{ik}a_{jk}=\sum_k a_{ki} a_{kj}=\delta_{ij} \)         
         \( A=A^* \)                                   real matrix            \( a_{ij}=a_{ij}^* \)                                              
         \( A=A^{\dagger} \)                           hermitian              \( a_{ij}=a_{ji}^* \)                                              
         \( A=\left(A^{\dagger}\right )^{-1} \)        unitary                \( \sum_k a_{ik}a_{jk}^*=\sum_k a_{ki}^* a_{kj}=\delta_{ij} \)     
      
-
      
-
-
-









      
-
      Some famous Matrices 
      
-
-
        
-  
      •  Diagonal if \( a_{ij}=0 \) for \( i\ne j \)
        • 
-  
      •  Upper triangular if \( a_{ij}=0 \) for \( i>j \)
        • 
-  
      •  Lower triangular if \( a_{ij}=0 \) for \( i < j \)
        • 
-  
      •  Upper Hessenberg if \( a_{ij}=0 \) for \( i>j+1 \)
        • 
-  
      •  Lower Hessenberg if \( a_{ij}=0 \) for \( i < j+1 \)
        • 
-  
      •  Tridiagonal if \( a_{ij}=0 \) for \( |i -j|>1 \)
        • 
-  
      •  Lower banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i>j+p \)
        • 
-  
      •  Upper banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i < j+p \)
        • 
-  
      •  Banded, block upper triangular, block lower triangular....
        • 
-
      
-









      
-
      More Basic Matrix Features 
      
-
-
      
-Some Equivalent Statements
-
      
-
      For an \( N\times N \) matrix  \( \mathbf{A} \) the following properties are all equivalent
      
-
-
        
-  
      •  If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
        • 
-  
      •  The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
        • 
-  
      •  The rows of \( \mathbf{A} \) form a basis of \( R^N \).
        • 
-  
      •  The columns of \( \mathbf{A} \) form a basis of \( R^N \).
        • 
-  
      •  \( \mathbf{A} \) is a product of elementary matrices.
        • 
-  
      •  \( 0 \) is not eigenvalue of \( \mathbf{A} \).
        • 
-
      
-
      
-
-
 









      
 
      Numpy and arrays 
      
 
      Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as
      
@@ -1912,13 +1822,6 @@ As we will see below it leads also to a very concice code close to the mathemati
 For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray. 
 
      
 
-









      
-
      Friday August 27 
      
-
-
      "Video of Lecture August 27, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust27.mp4?vrtx=view-as-webpage
      
-
-
      Video of Lecture from fall 2020  and Handwritten notes
      
-
 









      
 
      Simple linear regression model using scikit-learn 
      
 
@@ -2251,62 +2154,8 @@ $$
 
      Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).
      
 
 
      We will discuss in more detail these and other functions in the
-various lectures.  We conclude this part with another example. Instead
-of a linear \( x \)-dependence we study now a cubic polynomial and use the
-polynomial regression analysis tools of scikit-learn.
+various lectures and lab sessions.
 
      
-
-
-
-
      
-  
      
-    
      
-      
      
-        
      
-  
      import matplotlib.pyplot as plt
-import numpy as np
-import random
-from sklearn.linear_model import Ridge
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.pipeline import make_pipeline
-from sklearn.linear_model import LinearRegression
-
-x=np.linspace(0.02,0.98,200)
-noise = np.asarray(random.sample((range(200)),200))
-y=x**3*noise
-yn=x**3*100
-poly3 = PolynomialFeatures(degree=3)
-X = poly3.fit_transform(x[:,np.newaxis])
-clf3 = LinearRegression()
-clf3.fit(X,y)
-
-Xplot=poly3.fit_transform(x[:,np.newaxis])
-poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
-plt.plot(x,yn, color='red', label="True Cubic")
-plt.scatter(x, y, label='Data', color='orange', s=15)
-plt.legend()
-plt.show()
-
-def error(a):
-    for i in y:
-        err=(y-yn)/yn
-    return abs(np.sum(err))/len(err)
-
-print (error(y))
-
      
-
      
-      
      
-    
      
-  
      
-  
      
-    
      
-      
      
-        
                
-        
      
-      
      
-    
      
-  
      
-
      
 
      To our real data: nuclear binding energies. Brief reminder on masses and binding energies 
      
 
 
      Let us now dive into  nuclear physics and remind ourselves briefly about some basic features about binding
@@ -2707,60 +2556,6 @@ plt.show()
      -

      Seeing the wood for the trees

      - -

      As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!

      - - - -
      -
      -
      -
      -
      - 
      #Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=5)
-regr_2=DecisionTreeRegressor(max_depth=7)
-regr_3=DecisionTreeRegressor(max_depth=9)
-regr_1.fit(X, Energies)
-regr_2.fit(X, Energies)
-regr_3.fit(X, Energies)
-
-
-y_1 = regr_1.predict(X)
-y_2 = regr_2.predict(X)
-y_3=regr_3.predict(X)
-Masses['Eapprox'] = y_3
-# Plot the results
-plt.figure()
-plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
-plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
-plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
-plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
-
-plt.xlabel("$A$")
-plt.ylabel("$E$[MeV]")
-plt.title("Decision Tree Regression")
-plt.legend()
-save_fig("Masses2016Trees")
-plt.show()
-print(Masses)
-print(np.mean( (Energies-y_1)**2))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -

      And what about using neural networks?

      The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn's multi-layer perceptron (or feed forward neural network) functionality. @@ -2901,7 +2696,7 @@ the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ld

      -

      In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), +

      In order to understand the relation among the predictors (or features or properties) \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), consider the model we discussed for describing nuclear binding energies.

      @@ -3312,25 +3107,9 @@ allow for the usage of direct linear algebra methods such as LU decomposi









      Some useful matrix and vector expressions

      -

      The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters. -

      +

      See the handwritten notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesExercise5Week452022.pdf

      -$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, -$$ - -$$ -\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a}, -$$ - -$$ -\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, -$$ - -$$ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -$$ +

      These notes will be discussed during one of the lectures.











      Interpretations and optimizing our parameters

      @@ -3984,7 +3763,8 @@ ypredict = X_test Exercises -

      Here are three possible exercises for weeks 34 and 35.

      + +

      Here are three possible exercises for week 34

      Exercise 1: Setting up various Python environments

      @@ -4119,181 +3899,19 @@ $$ Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.

      - -

      -Solution. -The code here is an example of where we define our own design matrix and fit parameters \( \beta \). -

      - - -
      -
      -
      -
      -
      - 
      import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-x = np.random.rand(100)
-y = 2.0+5*x*x+0.1*np.random.randn(100)
-
-
-#  The design matrix now as function of a given polynomial
-X = np.zeros((len(x),3))
-X[:,0] = 1.0
-X[:,1] = x
-X[:,2] = x**2
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
-print(beta)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - - - - -

      Exercise 3: Normalizing our data

      - -

      A much used approach before starting to train the data is to preprocess our -data. Normally the data may need a rescaling and/or may be sensitive -to extreme values. Scaling the data renders our inputs much more -suitable for the algorithms we want to employ. -

      - -

      Scikit-Learn has several functions which allow us to rescale the -data, normally resulting in much better results in terms of various -accuracy scores. The StandardScaler function in Scikit-Learn -ensures that for each feature/predictor we study the mean value is -zero and the variance is one (every column in the design/feature -matrix). This scaling has the drawback that it does not ensure that -we have a particular maximum or minimum in our data set. Another -function included in Scikit-Learn is the MinMaxScaler which -ensures that all features are exactly between \( 0 \) and \( 1 \). The -

      - -

      The Normalizer scales each data -point such that the feature vector has a euclidean length of one. In other words, it -projects a data point on the circle (or sphere in the case of higher dimensions) with a -radius of 1. This means every data point is scaled by a different number (by the -inverse of it’s length). -This normalization is often used when only the direction (or angle) of the data matters, -not the length of the feature vector. -

      - -

      The RobustScaler works similarly to the StandardScaler in that it -ensures statistical properties for each feature that guarantee that -they are on the same scale. However, the RobustScaler uses the median -and quartiles, instead of mean and variance. This makes the -RobustScaler ignore data points that are very different from the rest -(like measurement errors). These odd data points are also called -outliers, and might often lead to trouble for other scaling -techniques. -

      - -

      It also common to split the data in a training set and a testing set. A typical split is to use \( 80\% \) of the data for training and the rest -for testing. This can be done as follows with our design matrix \( \boldsymbol{X} \) and data \( \boldsymbol{y} \) (remember to import scikit-learn) -

      - - -
      -
      -
      -
      -
      - 
      # split in training and test data
-X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      - -

      Then we can use the standard scaler to scale our data as

      - - -
      -
      -
      -
      -
      - 
      scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      -
      +

      Exercise 3: Split data in test and training data

      In this exercise we want you to to compute the MSE for the training data and the test data as function of the complexity of a polynomial, -that is the degree of a given polynomial. We want you also to compute the \( R2 \) score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling. +that is the degree of a given polynomial.

      -

      One of -the aims is to reproduce Figure 2.11 of Hastie et al. -

      +

      The aim is to reproduce Figure 2.11 of Hastie et al.

      -

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points.

      +

      Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points. You should try to vary \( n \) in your analysis.

      @@ -4303,7 +3921,6 @@ the aims is to reproduce Figure 2.11 of 
      np.random.seed()
 n = 100
-maxdegree = 14
 # Make data set.
 x = np.linspace(-3, 3, n).reshape(-1, 1)
 y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
@@ -4327,7 +3944,7 @@ y = np.e
 
 
      
 a)
-Write a first code which sets up a design matrix \( X \) defined by a fifth-order polynomial.  Scale your data and split it in training and test data.
+Write a first code which sets up a design matrix \( X \) defined by a fifth-order polynomial and split your data set in training and test data.
 
      
 
 
@@ -4335,7 +3952,7 @@ Write a first code which sets up a design matrix \( X \) defined by a fifth-orde
 
 
      
 b)
-Perform an ordinary least squares and compute the means squared error and the \( R2 \) factor for the training data and the test data, with and without scaling.
+Perform an ordinary least squares fitting and compute the means squared error for the training data and the test data.
 
      
 
 
@@ -4343,7 +3960,7 @@ Perform an ordinary least squares and compute the means squared error and the \(
 
 
      
 c)
-Add now a model which allows you to make polynomials up to degree \( 15 \).  Perform a standard OLS fitting of the training data and compute the MSE and \( R2 \) for the training and test data and plot both test and training data MSE and \( R2 \) as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
+Add now a model which allows you to make polynomials up to degree \( 15 \).  Perform a standard OLS fitting of the training data and compute the MSE for the training and test data and plot both test and training data MSE as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
 
      
 
 
diff --git a/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz b/doc/pub/week34/ipynb/ipynb-week34-src.tar.gz
index ad652e490a73db69d3f61dad65f45577eac56fdb..8bfb3b048a509ac72044be6e0f5a14db947d111b 100644
GIT binary patch
delta 21
dcmcb!g6+-\n",
@@ -12,8 +14,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fcd73f8e",
-   "metadata": {},
+   "id": "ad682ef2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "# Week 34: Introduction to the course, Logistics and Practicalities\n",
     "**Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, USA\n",
@@ -23,14 +27,16 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e1da964a",
-   "metadata": {},
+   "id": "a3c9b4cb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Overview of first week\n",
     "\n",
     "1. The sessions on Tuesdays and Wednesdays last four hours for each group (four groups in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects.\n",
     "\n",
-    "2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week.\n",
+    "2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week. Videos and teaching material will be announced in due time.\n",
     "\n",
     "3. There are four groups:\n",
     "\n",
@@ -50,8 +56,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cbeb2f85",
-   "metadata": {},
+   "id": "ca50e792",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Schedule first week\n",
     "\n",
@@ -64,8 +72,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4835b20a",
-   "metadata": {},
+   "id": "63fcf456",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Reading Recommendations\n",
     "\n",
@@ -85,8 +95,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3e30978b",
-   "metadata": {},
+   "id": "4d7d32a1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Lectures and ComputerLab\n",
     "\n",
@@ -107,8 +119,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7917611c",
-   "metadata": {},
+   "id": "2421a034",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Communication channels\n",
     "\n",
@@ -119,8 +133,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7ccfe004",
-   "metadata": {},
+   "id": "0675b78c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Course Format\n",
     "\n",
@@ -141,8 +157,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a747992",
-   "metadata": {},
+   "id": "53e77356",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Teachers\n",
     "\n",
@@ -170,8 +188,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4410a68b",
-   "metadata": {},
+   "id": "0aa0c646",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Deadlines for projects (tentative)\n",
     "\n",
@@ -186,8 +206,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "19812ab6",
-   "metadata": {},
+   "id": "c03f73ef",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Recommended textbooks\n",
     "\n",
@@ -208,8 +230,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "66147219",
-   "metadata": {},
+   "id": "83371a9e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Prerequisites\n",
     "\n",
@@ -226,8 +250,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "42a134be",
-   "metadata": {},
+   "id": "1d9b6bd4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Learning outcomes\n",
     "\n",
@@ -268,8 +294,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa87b03b",
-   "metadata": {},
+   "id": "d2b3cdf9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Topics covered in this course: Statistical analysis and optimization of data\n",
     "\n",
@@ -301,8 +329,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8cca3e5b",
-   "metadata": {},
+   "id": "26eea71c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Topics covered in this course\n",
     "\n",
@@ -337,8 +367,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6f1b8ac6",
-   "metadata": {},
+   "id": "850df98b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Extremely useful tools, strongly recommended\n",
     "\n",
@@ -351,8 +383,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2d590bee",
-   "metadata": {},
+   "id": "ccaadbac",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Other courses on Data science and Machine Learning  at UiO\n",
     "\n",
@@ -380,8 +414,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "687b1a20",
-   "metadata": {},
+   "id": "fbb62679",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Introduction\n",
     "\n",
@@ -411,9 +447,9 @@
     "topics and tools as well as showing the power of various Python \n",
     "libraries for machine learning and statistical data analysis.  \n",
     "\n",
-    "Here, we will mainly focus on two\n",
+    "Although we have projects where you write your own codes, we will also focus on two\n",
     "specific Python packages for Machine Learning, Scikit-Learn and\n",
-    "Tensorflow (see below for links etc).  Moreover, the examples we\n",
+    "Tensorflow with Keras (see below for links etc).  Moreover, the examples we\n",
     "introduce will serve as inputs to many of our discussions later, as\n",
     "well as allowing you to set up models and produce your own data and\n",
     "get started with programming."
@@ -421,8 +457,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "09496a03",
-   "metadata": {},
+   "id": "1b51b4ee",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## What is Machine Learning?\n",
     "\n",
@@ -492,8 +530,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "07f9251d",
-   "metadata": {},
+   "id": "777bf300",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Types of Machine Learning\n",
     "\n",
@@ -519,8 +559,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6ab5119",
-   "metadata": {},
+   "id": "d4f5891d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Essential elements of ML\n",
     "\n",
@@ -535,8 +577,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e705506",
-   "metadata": {},
+   "id": "5d8b4a2f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## An optimization/minimization problem\n",
     "\n",
@@ -545,8 +589,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "114abc74",
-   "metadata": {},
+   "id": "4fc2547d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## A Frequentist approach to data analysis\n",
     "\n",
@@ -578,8 +624,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8b2bbf6d",
-   "metadata": {},
+   "id": "b134aa47",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## What is a good model?\n",
     "\n",
@@ -606,8 +654,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e64bf89",
-   "metadata": {},
+   "id": "54c53efd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## What is a good model? Can we define it?\n",
     "\n",
@@ -635,8 +685,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5483611c",
-   "metadata": {},
+   "id": "b25ea1d1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Software and needed installations\n",
     "\n",
@@ -672,8 +724,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9eac6e63",
-   "metadata": {},
+   "id": "37eba906",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Python installers\n",
     "\n",
@@ -703,8 +757,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b4f5215",
-   "metadata": {},
+   "id": "1fd5ce9b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Useful Python libraries\n",
     "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n",
@@ -736,8 +792,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "95ed9491",
-   "metadata": {},
+   "id": "7a563428",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Installing R, C++, cython or Julia\n",
     "\n",
@@ -746,8 +804,8 @@
     "Those of you\n",
     "already familiar with **R** should feel free to continue using **R**, keeping\n",
     "however an eye on the parallel Python set ups. Similarly, if you are a\n",
-    "Python afecionado, feel free to explore **R** as well.  Jupyter/Ipython\n",
-    "notebook allows you to run **R** codes interactively in your\n",
+    "Python afecionado, feel free to explore **R** as well.  Jupyter(Julia, Python and R) /Ipython\n",
+    "notebook allows you to run **R** codes and **Julia** codes interactively in your\n",
     "browser. The software library **R** is really tailored  for statistical data analysis\n",
     "and allows for an easy usage of the tools and algorithms we will discuss in these\n",
     "lectures.\n",
@@ -758,8 +816,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "374079cd",
-   "metadata": {},
+   "id": "67c90d71",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Installing R, C++, cython, Numba etc\n",
     "\n",
@@ -783,28 +843,32 @@
   },
   {
    "cell_type": "markdown",
-   "id": "af22d31e",
-   "metadata": {},
+   "id": "23ec7d03",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "        pycod jupyter nbconvert filename.ipynb --to latex \n"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "6a65b86c",
-   "metadata": {},
+   "id": "7bf3cec8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS)  and is entirely written in Python. \n",
     "\n",
-    "Finally, if you wish to use the light mark-up language \n",
-    "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n",
-    "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**."
+    "Finally, we recommend strongly using Autograd or JAX for automatic differentiation."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "ddd0d581",
-   "metadata": {},
+   "id": "2ffa9ff0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Numpy examples and Important Matrix and vector handling packages\n",
     "\n",
@@ -822,126 +886,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fa618fa",
-   "metadata": {},
-   "source": [
-    "## Basic Matrix Features\n",
-    "\n",
-    "**Matrix properties reminder.**"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d849e19e",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\mathbf{A} =\n",
-    "      \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n",
-    "                                 a_{21} & a_{22} & a_{23} & a_{24} \\\\\n",
-    "                                   a_{31} & a_{32} & a_{33} & a_{34} \\\\\n",
-    "                                  a_{41} & a_{42} & a_{43} & a_{44}\n",
-    "             \\end{bmatrix}\\qquad\n",
-    "\\mathbf{I} =\n",
-    "      \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n",
-    "                                 0 & 1 & 0 & 0 \\\\\n",
-    "                                 0 & 0 & 1 & 0 \\\\\n",
-    "                                 0 & 0 & 0 & 1\n",
-    "             \\end{bmatrix}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "43dd13c9",
-   "metadata": {},
-   "source": [
-    "The inverse of a matrix is defined by"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d5b42841",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "d2756d9d",
-   "metadata": {},
-   "source": [
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "
                  Relations                    Name                           matrix elements                       
         $A=A^{T}$                                 symmetric              $a_{ij}=a_{ji}$                                                
         $A=\\left (A^{T}\\right )^{-1}$             real orthogonal        $\\sum_k a_{ik}a_{jk}=\\sum_k a_{ki} a_{kj}=\\delta_{ij}$         
         $A=A^*$                                   real matrix            $a_{ij}=a_{ij}^*$                                              
         $A=A^{\\dagger}$                           hermitian              $a_{ij}=a_{ji}^*$                                              
         $A=\\left(A^{\\dagger}\\right )^{-1}$        unitary                $\\sum_k a_{ik}a_{jk}^*=\\sum_k a_{ki}^* a_{kj}=\\delta_{ij}$     
      "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f7f5934c",
-   "metadata": {},
-   "source": [
-    "### Some famous Matrices\n",
-    "\n",
-    "  * Diagonal if $a_{ij}=0$ for $i\\ne j$\n",
-    "\n",
-    "  * Upper triangular if $a_{ij}=0$ for $i>j$\n",
-    "\n",
-    "  * Lower triangular if $a_{ij}=0$ for $ij+1$\n",
-    "\n",
-    "  * Lower Hessenberg if $a_{ij}=0$ for $i1$\n",
-    "\n",
-    "  * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i>j+p$\n",
-    "\n",
-    "  * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i\n",
+    "\n",
+    "These notes will be discussed during one of the lectures."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "888346fc",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\frac{\\partial (\\boldsymbol{b}^T\\boldsymbol{a})}{\\partial \\boldsymbol{a}} = \\boldsymbol{b},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0b88b7f5",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\frac{\\partial (\\boldsymbol{a}^T\\boldsymbol{A}\\boldsymbol{a})}{\\partial \\boldsymbol{a}} = (\\boldsymbol{A}+\\boldsymbol{A}^T)\\boldsymbol{a},\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0108eefb",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\frac{\\partial tr(\\boldsymbol{B}\\boldsymbol{A})}{\\partial \\boldsymbol{A}} = \\boldsymbol{B}^T,\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "1c3aeec5",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "0c808963",
-   "metadata": {},
+   "id": "1158b8d6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Interpretations and optimizing our parameters\n",
     "The residuals $\\boldsymbol{\\epsilon}$ are in turn given by"
@@ -3482,8 +3663,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58a7a52b",
-   "metadata": {},
+   "id": "633e1d2c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n",
@@ -3492,16 +3675,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31380692",
-   "metadata": {},
+   "id": "f81d31f8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and with"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "338d57bf",
-   "metadata": {},
+   "id": "dc428bfa",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n",
@@ -3510,16 +3697,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "83acfd23",
-   "metadata": {},
+   "id": "8713d26e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e1096ec3",
-   "metadata": {},
+   "id": "6da49a7a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n",
@@ -3528,8 +3719,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cd841096",
-   "metadata": {},
+   "id": "82436bba",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "meaning that the solution for $\\boldsymbol{\\beta}$ is the one which minimizes the residuals.  Later we will link this with the maximum likelihood approach.\n",
     "\n",
@@ -3538,8 +3731,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a5eb6867",
-   "metadata": {},
+   "id": "3221853e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Own code for Ordinary Least Squares\n",
     "\n",
@@ -3549,9 +3744,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 39,
-   "id": "0c33b4a6",
-   "metadata": {},
+   "execution_count": 37,
+   "id": "32634369",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
@@ -3562,17 +3760,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "30b8c643",
-   "metadata": {},
+   "id": "5fdb2784",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "Alternatively, you can use the least squares functionality in **Numpy** as"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
-   "id": "2f1c46d5",
-   "metadata": {},
+   "execution_count": 38,
+   "id": "0a7052de",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n",
@@ -3581,17 +3784,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "902a28cd",
-   "metadata": {},
+   "id": "0c3fadf2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "And finally we plot our fit with and compare with data"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
-   "id": "5c5d6920",
-   "metadata": {},
+   "execution_count": 39,
+   "id": "49865a8e",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "Masses['Eapprox']  = ytilde\n",
@@ -3610,8 +3818,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c6ab2ba2",
-   "metadata": {},
+   "id": "d0d445d7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Adding error analysis and training set up\n",
     "\n",
@@ -3621,9 +3831,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
-   "id": "cb5a5325",
-   "metadata": {},
+   "execution_count": 40,
+   "id": "d15cdce6",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "def R2(y_data, y_model):\n",
@@ -3632,17 +3845,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "25b4cfd8",
-   "metadata": {},
+   "id": "2275cd6d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and we would be using it as"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
-   "id": "5c2cc37e",
-   "metadata": {},
+   "execution_count": 41,
+   "id": "f07b786c",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "print(R2(Energies,ytilde))"
@@ -3650,17 +3868,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "463a5289",
-   "metadata": {},
+   "id": "0af606bc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We can easily add our **MSE** score as"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
-   "id": "a5f77d4b",
-   "metadata": {},
+   "execution_count": 42,
+   "id": "0df8ce2d",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "def MSE(y_data,y_model):\n",
@@ -3672,17 +3895,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ac24929",
-   "metadata": {},
+   "id": "70e31542",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and finally the relative error as"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
-   "id": "231ad359",
-   "metadata": {},
+   "execution_count": 43,
+   "id": "b22073d9",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "def RelativeError(y_data,y_model):\n",
@@ -3692,8 +3920,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d826bba2",
-   "metadata": {},
+   "id": "36b83437",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "\n",
@@ -3712,8 +3942,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0a66d959",
-   "metadata": {},
+   "id": "a29b9c4c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\chi^2(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\frac{\\left(y_i-\\tilde{y}_i\\right)^2}{\\sigma_i^2}=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\frac{1}{\\boldsymbol{\\Sigma^2}}\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n",
@@ -3722,16 +3954,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f9be426f",
-   "metadata": {},
+   "id": "488d3565",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where the matrix $\\boldsymbol{\\Sigma}$ is a diagonal matrix with $\\sigma_i$ as matrix elements."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "bc82c650",
-   "metadata": {},
+   "id": "7fdcf0e7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "\n",
@@ -3740,8 +3976,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "28087a6d",
-   "metadata": {},
+   "id": "412c2315",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)^2\\right]=0,\n",
@@ -3750,16 +3988,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "edb55e07",
-   "metadata": {},
+   "id": "4fc8aa35",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "which results in"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "aaded591",
-   "metadata": {},
+   "id": "80b19a0f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}\\frac{x_{ij}}{\\sigma_i}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)\\right]=0,\n",
@@ -3768,16 +4010,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5308f0fd",
-   "metadata": {},
+   "id": "58e46ba8",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "or in a matrix-vector form as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "cdbeb539",
-   "metadata": {},
+   "id": "c5f6428e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right).\n",
@@ -3786,16 +4032,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6c3176b9",
-   "metadata": {},
+   "id": "026e3e70",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined the matrix $\\boldsymbol{A} =\\boldsymbol{X}/\\boldsymbol{\\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\\sigma_i$ and the vector $\\boldsymbol{b}$ with elements $b_i = y_i/\\sigma_i$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "c9171b84",
-   "metadata": {},
+   "id": "051702cb",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "\n",
@@ -3804,8 +4054,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "005b23f5",
-   "metadata": {},
+   "id": "1e532239",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right),\n",
@@ -3814,16 +4066,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "011b1cf3",
-   "metadata": {},
+   "id": "5f5358c1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "eecb33b0",
-   "metadata": {},
+   "id": "8aa7f812",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{A}^T\\boldsymbol{b} = \\boldsymbol{A}^T\\boldsymbol{A}\\boldsymbol{\\beta},\n",
@@ -3832,16 +4088,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "477dda35",
-   "metadata": {},
+   "id": "3acb7998",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and if the matrix $\\boldsymbol{A}^T\\boldsymbol{A}$ is invertible we have the solution"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "954eea3f",
-   "metadata": {},
+   "id": "4dd73372",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{\\beta} =\\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1}\\boldsymbol{A}^T\\boldsymbol{b}.\n",
@@ -3850,8 +4110,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3513171",
-   "metadata": {},
+   "id": "0a2d1ab2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "\n",
@@ -3860,8 +4122,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b61882ef",
-   "metadata": {},
+   "id": "5e740c4d",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\boldsymbol{H} =  \\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1},\n",
@@ -3870,16 +4134,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d3b19486",
-   "metadata": {},
+   "id": "f332b423",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we have then the following expression for the parameters $\\beta_j$ (the matrix elements of $\\boldsymbol{H}$ are $h_{ij}$)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "f26a6317",
-   "metadata": {},
+   "id": "cab61b59",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_j = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}\\frac{y_i}{\\sigma_i}\\frac{x_{ik}}{\\sigma_i} = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}b_ia_{ik}\n",
@@ -3888,16 +4156,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ed55e7b5",
-   "metadata": {},
+   "id": "20b9d29f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "We state without proof the expression for the uncertainty  in the parameters $\\beta_j$ as (we leave this as an exercise)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0d4ff691",
-   "metadata": {},
+   "id": "7bfbf5e2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma^2(\\beta_j) = \\sum_{i=0}^{n-1}\\sigma_i^2\\left( \\frac{\\partial \\beta_j}{\\partial y_i}\\right)^2,\n",
@@ -3906,16 +4178,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5da9cdc6",
-   "metadata": {},
+   "id": "71366bb6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "resulting in"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "85883ee8",
-   "metadata": {},
+   "id": "9f0e6e6e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\sigma^2(\\beta_j) = \\left(\\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}a_{ik}\\right)\\left(\\sum_{l=0}^{p-1}h_{jl}\\sum_{m=0}^{n-1}a_{ml}\\right) = h_{jj}!\n",
@@ -3924,8 +4200,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3ac5064",
-   "metadata": {},
+   "id": "c6ecc5fc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "The first step here is to approximate the function $y$ with a first-order polynomial, that is we write"
@@ -3933,8 +4211,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3121513b",
-   "metadata": {},
+   "id": "0eaa78b9",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "y=y(x) \\rightarrow y(x_i) \\approx \\beta_0+\\beta_1 x_i.\n",
@@ -3943,16 +4223,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9fc0e9d",
-   "metadata": {},
+   "id": "89650d96",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "By computing the derivatives of $\\chi^2$ with respect to $\\beta_0$ and $\\beta_1$ show that these are given by"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "50ee610e",
-   "metadata": {},
+   "id": "86b81e9f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -2\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0,\n",
@@ -3961,16 +4245,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94fba98c",
-   "metadata": {},
+   "id": "c3cdd79a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "496da037",
-   "metadata": {},
+   "id": "58bae61c",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_i\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0.\n",
@@ -3979,8 +4267,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9c8f2972",
-   "metadata": {},
+   "id": "2a7c0936",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The $\\chi^2$ function\n",
     "\n",
@@ -3990,8 +4280,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e49831a8",
-   "metadata": {},
+   "id": "b72360a2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\gamma =  \\sum_{i=0}^{n-1}\\frac{1}{\\sigma_i^2},\n",
@@ -4000,8 +4292,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb152733",
-   "metadata": {},
+   "id": "89188b6f",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\gamma_x =  \\sum_{i=0}^{n-1}\\frac{x_{i}}{\\sigma_i^2},\n",
@@ -4010,8 +4304,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6bee7350",
-   "metadata": {},
+   "id": "b2483bb6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\gamma_y = \\sum_{i=0}^{n-1}\\left(\\frac{y_i}{\\sigma_i^2}\\right),\n",
@@ -4020,8 +4316,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b20214ae",
-   "metadata": {},
+   "id": "19b6390a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\gamma_{xx} =  \\sum_{i=0}^{n-1}\\frac{x_ix_{i}}{\\sigma_i^2},\n",
@@ -4030,8 +4328,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5b69a59e",
-   "metadata": {},
+   "id": "54ce44a6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\gamma_{xy} = \\sum_{i=0}^{n-1}\\frac{y_ix_{i}}{\\sigma_i^2},\n",
@@ -4040,16 +4340,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3431f6b",
-   "metadata": {},
+   "id": "eeb4a6a2",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "we obtain"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "b4be888f",
-   "metadata": {},
+   "id": "6b8c03ae",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_0 = \\frac{\\gamma_{xx}\\gamma_y-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2},\n",
@@ -4058,8 +4362,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50edcbf3",
-   "metadata": {},
+   "id": "625f295e",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\beta_1 = \\frac{\\gamma_{xy}\\gamma-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2}.\n",
@@ -4068,8 +4374,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "801af8e7",
-   "metadata": {},
+   "id": "0f0c6fca",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "This approach (different linear and non-linear regression) suffers\n",
     "often from both being underdetermined and overdetermined in the\n",
@@ -4079,8 +4387,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6270d66",
-   "metadata": {},
+   "id": "df57fba0",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Fitting an Equation of State for Dense Nuclear Matter\n",
     "\n",
@@ -4104,17 +4414,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b1011c6",
-   "metadata": {},
+   "id": "14d5df9a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## The code"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 46,
-   "id": "21e0cf86",
-   "metadata": {},
+   "execution_count": 44,
+   "id": "90958307",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "# Common imports\n",
@@ -4206,8 +4521,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b46604cc",
-   "metadata": {},
+   "id": "f42c2ece",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "The above simple polynomial in density $\\rho$ gives an excellent fit\n",
     "to the data. \n",
@@ -4219,8 +4536,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61d3190d",
-   "metadata": {},
+   "id": "305220dd",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Splitting our Data in Training and Test data\n",
     "\n",
@@ -4238,9 +4557,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
-   "id": "74ee63ce",
-   "metadata": {},
+   "execution_count": 45,
+   "id": "70133882",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "import os\n",
@@ -4311,17 +4633,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbed49d1",
-   "metadata": {},
+   "id": "b300dcd4",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Exercises\n",
-    "Here are three possible exercises for weeks 34 and  35."
+    "\n",
+    "Here are three possible exercises for week 34"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "98b47c09",
-   "metadata": {},
+   "id": "abfe4134",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Exercise 1: Setting up various Python environments\n",
     "\n",
@@ -4387,8 +4714,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "53417c28",
-   "metadata": {},
+   "id": "685c34f3",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "## Exercise 2: making your own data and exploring scikit-learn\n",
     "\n",
@@ -4398,9 +4727,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
-   "id": "583b5833",
-   "metadata": {},
+   "execution_count": 46,
+   "id": "4068e855",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "x = np.random.rand(100,1)\n",
@@ -4409,8 +4741,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9e4d6a0",
-   "metadata": {},
+   "id": "902ac5a1",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "1. Write your own code (following the examples under the [regression notes](https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html)) for computing the parametrization of the data set fitting a second-order polynomial. \n",
     "\n",
@@ -4421,8 +4755,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7763507",
-   "metadata": {},
+   "id": "72d229b7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
@@ -4432,8 +4768,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "644c77b3",
-   "metadata": {},
+   "id": "60320f76",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "and the $R^2$ score function.\n",
     "If $\\tilde{\\boldsymbol{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as"
@@ -4441,8 +4779,10 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0e315d8",
-   "metadata": {},
+   "id": "8ba3e83a",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
@@ -4451,16 +4791,20 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a52d1ec",
-   "metadata": {},
+   "id": "aca39d18",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where we have defined the mean value  of $\\boldsymbol{y}$ as"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "387deb50",
-   "metadata": {},
+   "id": "ae5d4ec6",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "$$\n",
     "\\bar{y} =  \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n",
@@ -4469,174 +4813,45 @@
   },
   {
    "cell_type": "markdown",
-   "id": "256391d7",
-   "metadata": {},
+   "id": "1a059b8b",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. \n",
-    "Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.\n",
-    "\n",
-    "\n",
-    "**Solution.**\n",
-    "The code here is an example of where we define our own design matrix and fit parameters $\\beta$."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 49,
-   "id": "d5c01490",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "import os\n",
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import matplotlib.pyplot as plt\n",
-    "from sklearn.model_selection import train_test_split\n",
-    "\n",
-    "def save_fig(fig_id):\n",
-    "    plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
-    "\n",
-    "def R2(y_data, y_model):\n",
-    "    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
-    "def MSE(y_data,y_model):\n",
-    "    n = np.size(y_model)\n",
-    "    return np.sum((y_data-y_model)**2)/n\n",
-    "\n",
-    "x = np.random.rand(100)\n",
-    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
-    "\n",
-    "\n",
-    "#  The design matrix now as function of a given polynomial\n",
-    "X = np.zeros((len(x),3))\n",
-    "X[:,0] = 1.0\n",
-    "X[:,1] = x\n",
-    "X[:,2] = x**2\n",
-    "# We split the data in test and training data\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
-    "# matrix inversion to find beta\n",
-    "beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n",
-    "print(beta)\n",
-    "# and then make the prediction\n",
-    "ytilde = X_train @ beta\n",
-    "print(\"Training R2\")\n",
-    "print(R2(y_train,ytilde))\n",
-    "print(\"Training MSE\")\n",
-    "print(MSE(y_train,ytilde))\n",
-    "ypredict = X_test @ beta\n",
-    "print(\"Test R2\")\n",
-    "print(R2(y_test,ypredict))\n",
-    "print(\"Test MSE\")\n",
-    "print(MSE(y_test,ypredict))"
+    "Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "9e8ca2ba",
-   "metadata": {},
+   "id": "e921f4b5",
+   "metadata": {
+    "editable": true
+   },
    "source": [
-    ""
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "843419d0",
-   "metadata": {},
-   "source": [
-    "## Exercise 3: Normalizing our data\n",
+    "## Exercise 3: Split data in test and training data\n",
     "\n",
-    "A much used approach before starting to train the data is  to preprocess our\n",
-    "data. Normally the data may need a rescaling and/or may be sensitive\n",
-    "to extreme values. Scaling the data renders our inputs much more\n",
-    "suitable for the algorithms we want to employ.\n",
-    "\n",
-    "**Scikit-Learn** has several functions which allow us to rescale the\n",
-    "data, normally resulting in much better results in terms of various\n",
-    "accuracy scores.  The **StandardScaler** function in **Scikit-Learn**\n",
-    "ensures that for each feature/predictor we study the mean value is\n",
-    "zero and the variance is one (every column in the design/feature\n",
-    "matrix).  This scaling has the drawback that it does not ensure that\n",
-    "we have a particular maximum or minimum in our data set. Another\n",
-    "function included in **Scikit-Learn** is the **MinMaxScaler** which\n",
-    "ensures that all features are exactly between $0$ and $1$. The\n",
-    "\n",
-    "The **Normalizer** scales each data\n",
-    "point such that the feature vector has a euclidean length of one. In other words, it\n",
-    "projects a data point on the circle (or sphere in the case of higher dimensions) with a\n",
-    "radius of 1. This means every data point is scaled by a different number (by the\n",
-    "inverse of it’s length).\n",
-    "This normalization is often used when only the direction (or angle) of the data matters,\n",
-    "not the length of the feature vector.\n",
-    "\n",
-    "The **RobustScaler** works similarly to the StandardScaler in that it\n",
-    "ensures statistical properties for each feature that guarantee that\n",
-    "they are on the same scale. However, the RobustScaler uses the median\n",
-    "and quartiles, instead of mean and variance. This makes the\n",
-    "RobustScaler ignore data points that are very different from the rest\n",
-    "(like measurement errors). These odd data points are also called\n",
-    "outliers, and might often lead to trouble for other scaling\n",
-    "techniques.\n",
-    "\n",
-    "It also common to split the data in a **training** set and a **testing** set. A typical split is to use $80\\%$ of the data for training and the rest\n",
-    "for testing. This can be done as follows with our design matrix $\\boldsymbol{X}$ and data $\\boldsymbol{y}$ (remember to import **scikit-learn**)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 50,
-   "id": "0d678956",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# split in training and test data\n",
-    "X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "e735e6b4",
-   "metadata": {},
-   "source": [
-    "Then we can use the standard scaler to scale our data as"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 51,
-   "id": "180127f6",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "scaler = StandardScaler()\n",
-    "scaler.fit(X_train)\n",
-    "X_train_scaled = scaler.transform(X_train)\n",
-    "X_test_scaled = scaler.transform(X_test)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "85ca394f",
-   "metadata": {},
-   "source": [
     "In this exercise we want you to to compute the MSE for the training\n",
     "data and the test data as function of the complexity of a polynomial,\n",
-    "that is the degree of a given polynomial. We want you also to compute the $R2$ score as function of the complexity of the model for both training data and test data.  You should also run the calculation with and without scaling. \n",
+    "that is the degree of a given polynomial.\n",
     "\n",
-    "One of \n",
-    "the aims is to reproduce Figure 2.11 of [Hastie et al](https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf).\n",
+    "The aim is to reproduce Figure 2.11 of [Hastie et al](https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf).\n",
     "\n",
-    "Our data is defined by $x\\in [-3,3]$ with a total of for example $100$ data points."
+    "Our data is defined by $x\\in [-3,3]$ with a total of for example $100$ data points. You should try to vary $n$ in your analysis."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 52,
-   "id": "da7e6454",
-   "metadata": {},
+   "execution_count": 47,
+   "id": "366035cc",
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
    "outputs": [],
    "source": [
     "np.random.seed()\n",
     "n = 100\n",
-    "maxdegree = 14\n",
     "# Make data set.\n",
     "x = np.linspace(-3, 3, n).reshape(-1, 1)\n",
     "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)"
@@ -4644,59 +4859,49 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ded6b167",
-   "metadata": {},
+   "id": "520218c7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "where $y$ is the function we want to fit with a given polynomial."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "980f436a",
-   "metadata": {},
+   "id": "890cb990",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "**a)**\n",
-    "Write a first code which sets up a design matrix $X$ defined by a fifth-order polynomial.  Scale your data and split it in training and test data."
+    "Write a first code which sets up a design matrix $X$ defined by a fifth-order polynomial and split your data set in training and test data."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "e6fa80d2",
-   "metadata": {},
+   "id": "c9d625cc",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "**b)**\n",
-    "Perform an ordinary least squares and compute the means squared error and the $R2$ factor for the training data and the test data, with and without scaling."
+    "Perform an ordinary least squares fitting and compute the means squared error for the training data and the test data."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "d1f9e0b8",
-   "metadata": {},
+   "id": "068e3fc7",
+   "metadata": {
+    "editable": true
+   },
    "source": [
     "**c)**\n",
-    "Add now a model which allows you to make polynomials up to degree $15$.  Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?"
+    "Add now a model which allows you to make polynomials up to degree $15$.  Perform a standard OLS fitting of the training data and compute the MSE for the training and test data and plot both test and training data MSE as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?"
    ]
   }
  ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.9.16"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 5
 }
diff --git a/doc/src/week34/week34.do.txt b/doc/src/week34/week34.do.txt
index 78471bab3..fa8458260 100644
--- a/doc/src/week34/week34.do.txt
+++ b/doc/src/week34/week34.do.txt
@@ -2463,6 +2463,7 @@ print(MSE(y_test,ypredict))
 
 !split
 ===== Exercises =====
+
 Here are three possible exercises for week 34
 
 
@@ -2569,14 +2570,12 @@ In this exercise we want you to to compute the MSE for the training
 data and the test data as function of the complexity of a polynomial,
 that is the degree of a given polynomial.
 
-One of 
-the aims is to reproduce Figure 2.11 of "Hastie et al":"https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf".
+The aim is to reproduce Figure 2.11 of "Hastie et al":"https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf".
 
-Our data is defined by $x\in [-3,3]$ with a total of for example $100$ data points.
+Our data is defined by $x\in [-3,3]$ with a total of for example $100$ data points. You should try to vary $n$ in your analysis.
 !bc pycod
 np.random.seed()
 n = 100
-maxdegree = 14
 # Make data set.
 x = np.linspace(-3, 3, n).reshape(-1, 1)
 y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
@@ -2587,7 +2586,7 @@ Write a first code which sets up a design matrix $X$ defined by a fifth-order po
 !esubex
 
 !bsubex
-Perform an ordinary least squares and compute the means squared error for the training data and the test data.
+Perform an ordinary least squares fitting and compute the means squared error for the training data and the test data.
 !esubex
 
 !bsubex