Morten Hjorth-Jensen
447 lines
28 KiB
HTML
447 lines
28 KiB
HTML
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('Seeing the wood for the trees',
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('And what about using neural networks?',
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('A first summary', 2, None, 'a-first-summary'),
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('Why Linear Regression (aka Ordinary Least Squares and family)',
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2,
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('Regression analysis, overarching aims',
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('Regression analysis, overarching aims II',
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2,
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None,
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('Rewriting the fitting procedure as a linear algebra problem, '
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('Optimizing our parameters',
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('Interpretations and optimizing our parameters',
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('Interpretations and optimizing our parameters',
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('Own code for Ordinary Least Squares',
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('Adding error analysis and training set up',
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('Fitting an Equation of State for Dense Nuclear Matter',
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None,
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'fitting-an-equation-of-state-for-dense-nuclear-matter'),
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('The code', 2, None, 'the-code'),
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('Splitting our Data in Training and Test data',
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('Exercises', 2, None, 'exercises'),
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None,
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'exercise-1-setting-up-various-python-environments'),
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('Exercise 2: making your own data and exploring scikit-learn',
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None,
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'exercise-2-making-your-own-data-and-exploring-scikit-learn'),
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('Exercise 3: Normalizing our data',
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2,
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None,
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'exercise-3-normalizing-our-data')]}
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<a class="navbar-brand" href="week34-bs.html">Week 34: Introduction to the course, Logistics and Practicalities</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs003.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs004.html#communication-channels" style="font-size: 80%;"><b>Communication channels</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs005.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs006.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs007.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs008.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs009.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs010.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs011.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course" style="font-size: 80%;"><b>Topics covered in this course</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs013.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs014.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs015.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs016.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs017.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs018.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs019.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs020.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs021.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs023.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs024.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs025.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs026.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs028.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs029.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs030.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs031.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs033.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs034.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs035.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#simple-linear-regression-model-using-scikit-learn" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs036.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs037.html#why-linear-regression-aka-ordinary-least-squares-and-family" style="font-size: 80%;"><b>Why Linear Regression (aka Ordinary Least Squares and family)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs038.html#regression-analysis-overarching-aims" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs039.html#regression-analysis-overarching-aims-ii" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs040.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs041.html#general-linear-models" style="font-size: 80%;"><b>General linear models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs042.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs043.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs045.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs045.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs046.html#optimizing-our-parameters" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs047.html#our-model-for-the-nuclear-binding-energies" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs048.html#optimizing-our-parameters-more-details" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs052.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs051.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs052.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs053.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs054.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs060.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#fitting-an-equation-of-state-for-dense-nuclear-matter" style="font-size: 80%;"><b>Fitting an Equation of State for Dense Nuclear Matter</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs062.html#the-code" style="font-size: 80%;"><b>The code</b></a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="interpretations-and-optimizing-our-parameters" class="anchor">Interpretations and optimizing our parameters </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>The function </p>
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$$
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C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\},
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$$
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<p>can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value.
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When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value
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</p>
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$$
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y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i,
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$$
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<p>where \( \langle y_i \rangle \) is the mean value. Keep in mind also that
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till now we have treated \( y_i \) as the exact value. Normally, the
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response (dependent or outcome) variable \( y_i \) the outcome of a
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numerical experiment or another type of experiment and is thus only an
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approximation to the true value. It is then always accompanied by an
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error estimate, often limited to a statistical error estimate given by
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the standard deviation discussed earlier. In the discussion here we
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will treat \( y_i \) as our exact value for the response variable.
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</p>
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<p>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</p>
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$$
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{\displaystyle \min_{\boldsymbol{\beta}\in
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{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
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$$
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<p>In practical terms it means we will require</p>
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$$
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\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
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$$
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<p>which results in</p>
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$$
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\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
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$$
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<p>or in a matrix-vector form as</p>
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$$
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\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right).
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$$
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