From a4a41ce5d4670452e398fe3e5db44a87bc10e676 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 11 Oct 2018 05:27:04 +0200 Subject: [PATCH] hints on project 2 --- .../Regression/html/._Regression-bs000.html | 45 +- .../Regression/html/._Regression-bs001.html | 43 +- .../Regression/html/._Regression-bs002.html | 43 +- .../Regression/html/._Regression-bs003.html | 43 +- .../Regression/html/._Regression-bs004.html | 43 +- .../Regression/html/._Regression-bs005.html | 43 +- .../Regression/html/._Regression-bs006.html | 43 +- .../Regression/html/._Regression-bs007.html | 43 +- .../Regression/html/._Regression-bs008.html | 43 +- .../Regression/html/._Regression-bs009.html | 43 +- .../Regression/html/._Regression-bs010.html | 43 +- .../Regression/html/._Regression-bs011.html | 43 +- .../Regression/html/._Regression-bs012.html | 43 +- .../Regression/html/._Regression-bs013.html | 43 +- .../Regression/html/._Regression-bs014.html | 43 +- .../Regression/html/._Regression-bs015.html | 43 +- 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create mode 100644 doc/pub/Regression/html/._Regression-bs109.html create mode 100644 doc/pub/Regression/html/._Regression-bs110.html create mode 100644 doc/pub/Regression/html/._Regression-bs111.html create mode 100644 doc/pub/Regression/html/._Regression-bs112.html create mode 100644 doc/pub/Regression/html/._Regression-bs113.html create mode 100644 doc/pub/Regression/html/._Regression-bs114.html diff --git a/doc/pub/Regression/html/._Regression-bs000.html b/doc/pub/Regression/html/._Regression-bs000.html index 833e9dec8..e6ce300e7 100644 --- a/doc/pub/Regression/html/._Regression-bs000.html +++ b/doc/pub/Regression/html/._Regression-bs000.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -399,7 +438,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 25, 2018

    +

    Oct 11, 2018


    @@ -423,7 +462,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
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    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs001.html b/doc/pub/Regression/html/._Regression-bs001.html index 294497cd4..3ab77c47d 100644 --- a/doc/pub/Regression/html/._Regression-bs001.html +++ b/doc/pub/Regression/html/._Regression-bs001.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -420,7 +459,7 @@ A regression model aims at finding a likelihood function \( p(y\vert \hat{x}) \)
  • 10
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  • ...
    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs002.html b/doc/pub/Regression/html/._Regression-bs002.html index 7551f6711..1a4bca496 100644 --- a/doc/pub/Regression/html/._Regression-bs002.html +++ b/doc/pub/Regression/html/._Regression-bs002.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ response is equal to \( \beta_j \).
  • 11
  • 12
  • ...
    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs003.html b/doc/pub/Regression/html/._Regression-bs003.html index 0dfcdd55f..5b47217af 100644 --- a/doc/pub/Regression/html/._Regression-bs003.html +++ b/doc/pub/Regression/html/._Regression-bs003.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -418,7 +457,7 @@ where \( \epsilon_i \) is the error in our approximation.
  • 12
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  • ...
    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs004.html b/doc/pub/Regression/html/._Regression-bs004.html index 209d2107f..457e18668 100644 --- a/doc/pub/Regression/html/._Regression-bs004.html +++ b/doc/pub/Regression/html/._Regression-bs004.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -418,7 +457,7 @@ $$
  • 13
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  • ...
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs005.html b/doc/pub/Regression/html/._Regression-bs005.html index 216e627f6..49d58a223 100644 --- a/doc/pub/Regression/html/._Regression-bs005.html +++ b/doc/pub/Regression/html/._Regression-bs005.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +479,7 @@ $$
  • 14
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  • ...
    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs006.html b/doc/pub/Regression/html/._Regression-bs006.html index b8ef06a07..7fe5349cf 100644 --- a/doc/pub/Regression/html/._Regression-bs006.html +++ b/doc/pub/Regression/html/._Regression-bs006.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -423,7 +462,7 @@ $$
  • 15
  • 16
  • ...
    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs007.html b/doc/pub/Regression/html/._Regression-bs007.html index 1810e660e..5da5464cc 100644 --- a/doc/pub/Regression/html/._Regression-bs007.html +++ b/doc/pub/Regression/html/._Regression-bs007.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -429,7 +468,7 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs008.html b/doc/pub/Regression/html/._Regression-bs008.html index f1be330c0..ec9818f45 100644 --- a/doc/pub/Regression/html/._Regression-bs008.html +++ b/doc/pub/Regression/html/._Regression-bs008.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,7 +463,7 @@ $$
  • 17
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs009.html b/doc/pub/Regression/html/._Regression-bs009.html index 8f0099a19..03bf9471b 100644 --- a/doc/pub/Regression/html/._Regression-bs009.html +++ b/doc/pub/Regression/html/._Regression-bs009.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ $$
  • 18
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs010.html b/doc/pub/Regression/html/._Regression-bs010.html index cdcdc50a0..80cfe6d3a 100644 --- a/doc/pub/Regression/html/._Regression-bs010.html +++ b/doc/pub/Regression/html/._Regression-bs010.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +482,7 @@ $$
  • 19
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  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs011.html b/doc/pub/Regression/html/._Regression-bs011.html index c4ae55c01..40049b000 100644 --- a/doc/pub/Regression/html/._Regression-bs011.html +++ b/doc/pub/Regression/html/._Regression-bs011.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ $$
  • 20
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  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs012.html b/doc/pub/Regression/html/._Regression-bs012.html index 4b2724b1f..12e3facfd 100644 --- a/doc/pub/Regression/html/._Regression-bs012.html +++ b/doc/pub/Regression/html/._Regression-bs012.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -432,7 +471,7 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs013.html b/doc/pub/Regression/html/._Regression-bs013.html index 92e245a8c..6ee471d53 100644 --- a/doc/pub/Regression/html/._Regression-bs013.html +++ b/doc/pub/Regression/html/._Regression-bs013.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as
  • 22
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  • ...
    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs014.html b/doc/pub/Regression/html/._Regression-bs014.html index a888bb56e..1a3347bba 100644 --- a/doc/pub/Regression/html/._Regression-bs014.html +++ b/doc/pub/Regression/html/._Regression-bs014.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -432,7 +471,7 @@ where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix
  • 23
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs015.html b/doc/pub/Regression/html/._Regression-bs015.html index 13b640175..6afcd78f7 100644 --- a/doc/pub/Regression/html/._Regression-bs015.html +++ b/doc/pub/Regression/html/._Regression-bs015.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ $$
  • 24
  • 25
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs016.html b/doc/pub/Regression/html/._Regression-bs016.html index fcaf19cef..5b981d515 100644 --- a/doc/pub/Regression/html/._Regression-bs016.html +++ b/doc/pub/Regression/html/._Regression-bs016.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -435,7 +474,7 @@ $$
  • 25
  • 26
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs017.html b/doc/pub/Regression/html/._Regression-bs017.html index 217936ac6..2011042b4 100644 --- a/doc/pub/Regression/html/._Regression-bs017.html +++ b/doc/pub/Regression/html/._Regression-bs017.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -428,7 +467,7 @@ $$
  • 26
  • 27
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs018.html b/doc/pub/Regression/html/._Regression-bs018.html index 4818576d8..a29f515f5 100644 --- a/doc/pub/Regression/html/._Regression-bs018.html +++ b/doc/pub/Regression/html/._Regression-bs018.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -450,7 +489,7 @@ This approach (different linear and non-linear regression) suffers often from bo
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    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs019.html b/doc/pub/Regression/html/._Regression-bs019.html index 60d0f0472..e254a6d69 100644 --- a/doc/pub/Regression/html/._Regression-bs019.html +++ b/doc/pub/Regression/html/._Regression-bs019.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +482,7 @@ We see that, as expected, a linear fit gives a seemingly (from the graph) good r
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    • diff --git a/doc/pub/Regression/html/._Regression-bs020.html b/doc/pub/Regression/html/._Regression-bs020.html index 73da57f2f..192379b2a 100644 --- a/doc/pub/Regression/html/._Regression-bs020.html +++ b/doc/pub/Regression/html/._Regression-bs020.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +473,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs021.html b/doc/pub/Regression/html/._Regression-bs021.html index 01dde831f..0ab9033e7 100644 --- a/doc/pub/Regression/html/._Regression-bs021.html +++ b/doc/pub/Regression/html/._Regression-bs021.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -471,7 +510,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs022.html b/doc/pub/Regression/html/._Regression-bs022.html index b7fb9cd33..40cf3890d 100644 --- a/doc/pub/Regression/html/._Regression-bs022.html +++ b/doc/pub/Regression/html/._Regression-bs022.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ where \( x \) is defined as before.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs023.html b/doc/pub/Regression/html/._Regression-bs023.html index 53bc306cc..ebfc44f7a 100644 --- a/doc/pub/Regression/html/._Regression-bs023.html +++ b/doc/pub/Regression/html/._Regression-bs023.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -416,7 +455,7 @@ have not discussed a more rigorous approach to the cost function.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs024.html b/doc/pub/Regression/html/._Regression-bs024.html index c3a3a1b8a..0080884ed 100644 --- a/doc/pub/Regression/html/._Regression-bs024.html +++ b/doc/pub/Regression/html/._Regression-bs024.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ dimensionless.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs025.html b/doc/pub/Regression/html/._Regression-bs025.html index 69eb92b27..012175de2 100644 --- a/doc/pub/Regression/html/._Regression-bs025.html +++ b/doc/pub/Regression/html/._Regression-bs025.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,7 +463,7 @@ the \( \chi^2 \) function becomes smaller.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs026.html b/doc/pub/Regression/html/._Regression-bs026.html index 77d8d82ed..bf4942717 100644 --- a/doc/pub/Regression/html/._Regression-bs026.html +++ b/doc/pub/Regression/html/._Regression-bs026.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +482,7 @@ relative error.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs027.html b/doc/pub/Regression/html/._Regression-bs027.html index 1ad262d3d..45e0111a4 100644 --- a/doc/pub/Regression/html/._Regression-bs027.html +++ b/doc/pub/Regression/html/._Regression-bs027.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -448,7 +487,7 @@ plt.show()
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs028.html b/doc/pub/Regression/html/._Regression-bs028.html index 62b5c5d1b..07e29b5c6 100644 --- a/doc/pub/Regression/html/._Regression-bs028.html +++ b/doc/pub/Regression/html/._Regression-bs028.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -420,7 +459,7 @@ this function as being similar to the \( \chi^2 \) function defined above.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs029.html b/doc/pub/Regression/html/._Regression-bs029.html index 9277050b0..05602885b 100644 --- a/doc/pub/Regression/html/._Regression-bs029.html +++ b/doc/pub/Regression/html/._Regression-bs029.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs030.html b/doc/pub/Regression/html/._Regression-bs030.html index cc82ea4d6..3ad0d84af 100644 --- a/doc/pub/Regression/html/._Regression-bs030.html +++ b/doc/pub/Regression/html/._Regression-bs030.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -428,7 +467,7 @@ years etc.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs031.html b/doc/pub/Regression/html/._Regression-bs031.html index d8cbdb022..941b82cc4 100644 --- a/doc/pub/Regression/html/._Regression-bs031.html +++ b/doc/pub/Regression/html/._Regression-bs031.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -451,7 +490,7 @@ Using R, we can perform similar studies.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs032.html b/doc/pub/Regression/html/._Regression-bs032.html index 5be20ef41..21a829bdc 100644 --- a/doc/pub/Regression/html/._Regression-bs032.html +++ b/doc/pub/Regression/html/._Regression-bs032.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +473,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs033.html b/doc/pub/Regression/html/._Regression-bs033.html index 00e4bff7d..9ff6edea4 100644 --- a/doc/pub/Regression/html/._Regression-bs033.html +++ b/doc/pub/Regression/html/._Regression-bs033.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ non-random scalar. To specify the parameters of the distribution of
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    • diff --git a/doc/pub/Regression/html/._Regression-bs034.html b/doc/pub/Regression/html/._Regression-bs034.html index 4efbba98b..03ff03934 100644 --- a/doc/pub/Regression/html/._Regression-bs034.html +++ b/doc/pub/Regression/html/._Regression-bs034.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -437,7 +476,7 @@ Hence, \( Y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \beta, \sigma^2) \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs035.html b/doc/pub/Regression/html/._Regression-bs035.html index 07aedc90b..abc095fdd 100644 --- a/doc/pub/Regression/html/._Regression-bs035.html +++ b/doc/pub/Regression/html/._Regression-bs035.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs036.html b/doc/pub/Regression/html/._Regression-bs036.html index f0422e232..a17b0f5ad 100644 --- a/doc/pub/Regression/html/._Regression-bs036.html +++ b/doc/pub/Regression/html/._Regression-bs036.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -454,7 +493,7 @@ This is equivalent to saying that the matrix \( \hat{X} \) has at least an eigen
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    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs037.html b/doc/pub/Regression/html/._Regression-bs037.html index 3e988d009..8f8d6c881 100644 --- a/doc/pub/Regression/html/._Regression-bs037.html +++ b/doc/pub/Regression/html/._Regression-bs037.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -431,7 +470,7 @@ where \( \hat{I} \) is the identity matrix.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs038.html b/doc/pub/Regression/html/._Regression-bs038.html index 73dc89768..db901b2a8 100644 --- a/doc/pub/Regression/html/._Regression-bs038.html +++ b/doc/pub/Regression/html/._Regression-bs038.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ Summarize what you have learned about the relationship between model complexity
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  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs039.html b/doc/pub/Regression/html/._Regression-bs039.html index 2028fd298..a9fd9ad01 100644 --- a/doc/pub/Regression/html/._Regression-bs039.html +++ b/doc/pub/Regression/html/._Regression-bs039.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -418,7 +457,7 @@ Summarize what you think you learned about the relationship of knowing the true
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  • ...
    • -
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    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs040.html b/doc/pub/Regression/html/._Regression-bs040.html index a7a448086..d9630cc13 100644 --- a/doc/pub/Regression/html/._Regression-bs040.html +++ b/doc/pub/Regression/html/._Regression-bs040.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -493,7 +532,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs041.html b/doc/pub/Regression/html/._Regression-bs041.html index eee89e490..76d514737 100644 --- a/doc/pub/Regression/html/._Regression-bs041.html +++ b/doc/pub/Regression/html/._Regression-bs041.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -449,7 +488,7 @@ plt.show()
  • 50
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs042.html b/doc/pub/Regression/html/._Regression-bs042.html index 8a82fad75..d5805a287 100644 --- a/doc/pub/Regression/html/._Regression-bs042.html +++ b/doc/pub/Regression/html/._Regression-bs042.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -414,7 +453,7 @@ decision on its value. How do we do that? Much of the same considerations apply
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    • diff --git a/doc/pub/Regression/html/._Regression-bs043.html b/doc/pub/Regression/html/._Regression-bs043.html index 097d09f61..c19f51113 100644 --- a/doc/pub/Regression/html/._Regression-bs043.html +++ b/doc/pub/Regression/html/._Regression-bs043.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -484,7 +523,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs044.html b/doc/pub/Regression/html/._Regression-bs044.html index 18644f1b3..74e5b648e 100644 --- a/doc/pub/Regression/html/._Regression-bs044.html +++ b/doc/pub/Regression/html/._Regression-bs044.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -517,7 +556,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs045.html b/doc/pub/Regression/html/._Regression-bs045.html index 629b92b10..99d215822 100644 --- a/doc/pub/Regression/html/._Regression-bs045.html +++ b/doc/pub/Regression/html/._Regression-bs045.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,7 +463,7 @@ once using the original training sample.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs046.html b/doc/pub/Regression/html/._Regression-bs046.html index a10d08bb9..b960c16c3 100644 --- a/doc/pub/Regression/html/._Regression-bs046.html +++ b/doc/pub/Regression/html/._Regression-bs046.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -429,7 +468,7 @@ bootstrap is widely used.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs047.html b/doc/pub/Regression/html/._Regression-bs047.html index 45a7ae665..d76a1ddaf 100644 --- a/doc/pub/Regression/html/._Regression-bs047.html +++ b/doc/pub/Regression/html/._Regression-bs047.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -420,7 +459,7 @@ MathJax.Hub.Config({
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  • ...
    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs048.html b/doc/pub/Regression/html/._Regression-bs048.html index ff55bacef..065fd7937 100644 --- a/doc/pub/Regression/html/._Regression-bs048.html +++ b/doc/pub/Regression/html/._Regression-bs048.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -426,7 +465,7 @@ MathJax.Hub.Config({
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs049.html b/doc/pub/Regression/html/._Regression-bs049.html index 6764d378a..e015ee6e5 100644 --- a/doc/pub/Regression/html/._Regression-bs049.html +++ b/doc/pub/Regression/html/._Regression-bs049.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -435,7 +474,7 @@ selection of a large set of these numbers reproduces this PDF.
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  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs050.html b/doc/pub/Regression/html/._Regression-bs050.html index b9d0e2710..9a62bada9 100644 --- a/doc/pub/Regression/html/._Regression-bs050.html +++ b/doc/pub/Regression/html/._Regression-bs050.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs051.html b/doc/pub/Regression/html/._Regression-bs051.html index b24036b0c..b2090828a 100644 --- a/doc/pub/Regression/html/._Regression-bs051.html +++ b/doc/pub/Regression/html/._Regression-bs051.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -442,7 +481,7 @@ qualitatively as the spread of \( p \) around its mean.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs052.html b/doc/pub/Regression/html/._Regression-bs052.html index f4b46a682..4082be61c 100644 --- a/doc/pub/Regression/html/._Regression-bs052.html +++ b/doc/pub/Regression/html/._Regression-bs052.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -435,7 +474,7 @@ $$
  • 61
  • 62
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs053.html b/doc/pub/Regression/html/._Regression-bs053.html index 09c54b61d..7836c4705 100644 --- a/doc/pub/Regression/html/._Regression-bs053.html +++ b/doc/pub/Regression/html/._Regression-bs053.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +475,7 @@ $$
  • 62
  • 63
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs054.html b/doc/pub/Regression/html/._Regression-bs054.html index 22e2bb3a4..34234fd7b 100644 --- a/doc/pub/Regression/html/._Regression-bs054.html +++ b/doc/pub/Regression/html/._Regression-bs054.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ $$
  • 63
  • 64
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs055.html b/doc/pub/Regression/html/._Regression-bs055.html index ffccf342d..f91de8e81 100644 --- a/doc/pub/Regression/html/._Regression-bs055.html +++ b/doc/pub/Regression/html/._Regression-bs055.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +475,7 @@ value of a set of measurements.
  • 64
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  • ...
    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs056.html b/doc/pub/Regression/html/._Regression-bs056.html index 26813c1d2..a7a9d479b 100644 --- a/doc/pub/Regression/html/._Regression-bs056.html +++ b/doc/pub/Regression/html/._Regression-bs056.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -432,7 +471,7 @@ interested in finding the few lowest moments, like the mean
  • 65
  • 66
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs057.html b/doc/pub/Regression/html/._Regression-bs057.html index f1dbfebcf..c0e2becb5 100644 --- a/doc/pub/Regression/html/._Regression-bs057.html +++ b/doc/pub/Regression/html/._Regression-bs057.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -430,7 +469,7 @@ $$
  • 66
  • 67
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs058.html b/doc/pub/Regression/html/._Regression-bs058.html index eec557144..0fdf133fd 100644 --- a/doc/pub/Regression/html/._Regression-bs058.html +++ b/doc/pub/Regression/html/._Regression-bs058.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ and covariance \( \mathrm{cov}(X,Y) \).
  • 67
  • 68
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs059.html b/doc/pub/Regression/html/._Regression-bs059.html index 7e743a687..ea0763aaf 100644 --- a/doc/pub/Regression/html/._Regression-bs059.html +++ b/doc/pub/Regression/html/._Regression-bs059.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +475,7 @@ true PDFs behind, which we usually do not have.
  • 68
  • 69
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs060.html b/doc/pub/Regression/html/._Regression-bs060.html index 1744242ec..9863000eb 100644 --- a/doc/pub/Regression/html/._Regression-bs060.html +++ b/doc/pub/Regression/html/._Regression-bs060.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -426,7 +465,7 @@ means.
  • 69
  • 70
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs061.html b/doc/pub/Regression/html/._Regression-bs061.html index 855ee6b0a..17ca74a2c 100644 --- a/doc/pub/Regression/html/._Regression-bs061.html +++ b/doc/pub/Regression/html/._Regression-bs061.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ And in particular we are interested in its variance \( \mathrm{var}(\overline X_
  • 70
  • 71
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/._Regression-bs062.html b/doc/pub/Regression/html/._Regression-bs062.html index 709cf9fc8..0ec164d3d 100644 --- a/doc/pub/Regression/html/._Regression-bs062.html +++ b/doc/pub/Regression/html/._Regression-bs062.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -429,7 +468,7 @@ $$
  • 71
  • 72
  • ...
    • -
  • 103
    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs063.html b/doc/pub/Regression/html/._Regression-bs063.html index aa964e1d7..b6a58bbf6 100644 --- a/doc/pub/Regression/html/._Regression-bs063.html +++ b/doc/pub/Regression/html/._Regression-bs063.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -433,7 +472,7 @@ estimate of the PDF of each of the \( X_i \), estimating all properties of
  • 72
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs064.html b/doc/pub/Regression/html/._Regression-bs064.html index 73aa4746e..15d11df4f 100644 --- a/doc/pub/Regression/html/._Regression-bs064.html +++ b/doc/pub/Regression/html/._Regression-bs064.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -431,7 +470,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs065.html b/doc/pub/Regression/html/._Regression-bs065.html index cf03c6dfa..61a64d6e7 100644 --- a/doc/pub/Regression/html/._Regression-bs065.html +++ b/doc/pub/Regression/html/._Regression-bs065.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -443,7 +482,7 @@ measurements in the sample.
  • 74
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs066.html b/doc/pub/Regression/html/._Regression-bs066.html index 682017b2d..b18633ddb 100644 --- a/doc/pub/Regression/html/._Regression-bs066.html +++ b/doc/pub/Regression/html/._Regression-bs066.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +478,7 @@ cannot overlook the always present correlations.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs067.html b/doc/pub/Regression/html/._Regression-bs067.html index ac6022240..ce3cf496e 100644 --- a/doc/pub/Regression/html/._Regression-bs067.html +++ b/doc/pub/Regression/html/._Regression-bs067.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -433,7 +472,7 @@ measurements. For uncorrelated measurements this second term is zero.
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs068.html b/doc/pub/Regression/html/._Regression-bs068.html index 18ae22d07..28acb7f65 100644 --- a/doc/pub/Regression/html/._Regression-bs068.html +++ b/doc/pub/Regression/html/._Regression-bs068.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -426,7 +465,7 @@ have to be stored throughout the experiment.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs069.html b/doc/pub/Regression/html/._Regression-bs069.html index 57280b0d6..24689f594 100644 --- a/doc/pub/Regression/html/._Regression-bs069.html +++ b/doc/pub/Regression/html/._Regression-bs069.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -437,7 +476,7 @@ starting always at \( 1 \) for \( d=0 \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs070.html b/doc/pub/Regression/html/._Regression-bs070.html index efcf83cca..d0a08f223 100644 --- a/doc/pub/Regression/html/._Regression-bs070.html +++ b/doc/pub/Regression/html/._Regression-bs070.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -437,7 +476,7 @@ $$
  • 79
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    • -
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs071.html b/doc/pub/Regression/html/._Regression-bs071.html index f132e0346..a7e68c8f4 100644 --- a/doc/pub/Regression/html/._Regression-bs071.html +++ b/doc/pub/Regression/html/._Regression-bs071.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -429,7 +468,7 @@ measurements is very large.
  • 80
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs072.html b/doc/pub/Regression/html/._Regression-bs072.html index 0ab6f8bcd..6b0c75e54 100644 --- a/doc/pub/Regression/html/._Regression-bs072.html +++ b/doc/pub/Regression/html/._Regression-bs072.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -436,7 +475,7 @@ The value of \( \lambda \) which minimizes \( \mbox{AIC}(\lambda) \) corresponds
  • 81
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    • diff --git a/doc/pub/Regression/html/._Regression-bs073.html b/doc/pub/Regression/html/._Regression-bs073.html index 7bc3fd4e5..6bce83806 100644 --- a/doc/pub/Regression/html/._Regression-bs073.html +++ b/doc/pub/Regression/html/._Regression-bs073.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +473,7 @@ some sense) is then selected.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs074.html b/doc/pub/Regression/html/._Regression-bs074.html index 2120de23e..7ddab5a16 100644 --- a/doc/pub/Regression/html/._Regression-bs074.html +++ b/doc/pub/Regression/html/._Regression-bs074.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -415,7 +454,7 @@ The validation set approach is conceptually simple and is easy to implement. But
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    • diff --git a/doc/pub/Regression/html/._Regression-bs075.html b/doc/pub/Regression/html/._Regression-bs075.html index 0f8eae171..302ee9562 100644 --- a/doc/pub/Regression/html/._Regression-bs075.html +++ b/doc/pub/Regression/html/._Regression-bs075.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,7 +463,7 @@ cross-validation (LOOCV).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs076.html b/doc/pub/Regression/html/._Regression-bs076.html index 5b25680e5..1dcd2c9fd 100644 --- a/doc/pub/Regression/html/._Regression-bs076.html +++ b/doc/pub/Regression/html/._Regression-bs076.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +478,7 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs077.html b/doc/pub/Regression/html/._Regression-bs077.html index 68ab6e94d..c902c861e 100644 --- a/doc/pub/Regression/html/._Regression-bs077.html +++ b/doc/pub/Regression/html/._Regression-bs077.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -428,7 +467,7 @@ the design matrix and the parameters \( \beta \).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs078.html b/doc/pub/Regression/html/._Regression-bs078.html index efd0d82d1..8e5d69406 100644 --- a/doc/pub/Regression/html/._Regression-bs078.html +++ b/doc/pub/Regression/html/._Regression-bs078.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -426,7 +465,7 @@ need for bootstrapping.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs079.html b/doc/pub/Regression/html/._Regression-bs079.html index e200bd8c3..3cd8369e9 100644 --- a/doc/pub/Regression/html/._Regression-bs079.html +++ b/doc/pub/Regression/html/._Regression-bs079.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -422,7 +461,7 @@ number \( i \) is left out. Using this notation, define
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    • diff --git a/doc/pub/Regression/html/._Regression-bs080.html b/doc/pub/Regression/html/._Regression-bs080.html index 3d89f5ba0..51764999a 100644 --- a/doc/pub/Regression/html/._Regression-bs080.html +++ b/doc/pub/Regression/html/._Regression-bs080.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -417,7 +456,7 @@ $$
  • 89
  • 90
  • ...
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    • +
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    • diff --git a/doc/pub/Regression/html/._Regression-bs081.html b/doc/pub/Regression/html/._Regression-bs081.html index 854b0125b..fe8949010 100644 --- a/doc/pub/Regression/html/._Regression-bs081.html +++ b/doc/pub/Regression/html/._Regression-bs081.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -439,7 +478,7 @@ t = jackknife(x, stat)
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    • diff --git a/doc/pub/Regression/html/._Regression-bs082.html b/doc/pub/Regression/html/._Regression-bs082.html index 63d63c594..ea5b87c5e 100644 --- a/doc/pub/Regression/html/._Regression-bs082.html +++ b/doc/pub/Regression/html/._Regression-bs082.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ advantages:
  • 91
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    • -
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    • diff --git a/doc/pub/Regression/html/._Regression-bs083.html b/doc/pub/Regression/html/._Regression-bs083.html index d7c3551c4..834ebb88b 100644 --- a/doc/pub/Regression/html/._Regression-bs083.html +++ b/doc/pub/Regression/html/._Regression-bs083.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -419,7 +458,7 @@ estimators.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs084.html b/doc/pub/Regression/html/._Regression-bs084.html index b3b2e3814..e804d4f02 100644 --- a/doc/pub/Regression/html/._Regression-bs084.html +++ b/doc/pub/Regression/html/._Regression-bs084.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ idea is to use the relative frequency of \( \widehat{\theta}^* \)
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    • diff --git a/doc/pub/Regression/html/._Regression-bs085.html b/doc/pub/Regression/html/._Regression-bs085.html index 1513219f9..f5c492eff 100644 --- a/doc/pub/Regression/html/._Regression-bs085.html +++ b/doc/pub/Regression/html/._Regression-bs085.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,7 +463,7 @@ frequency of the observation \( X_i \), just draw the values
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    • diff --git a/doc/pub/Regression/html/._Regression-bs086.html b/doc/pub/Regression/html/._Regression-bs086.html index 184a4db28..4b88fd8ba 100644 --- a/doc/pub/Regression/html/._Regression-bs086.html +++ b/doc/pub/Regression/html/._Regression-bs086.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -420,7 +459,7 @@ When you are done, you can draw a histogram of the relative frequency of \( \wid
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    • diff --git a/doc/pub/Regression/html/._Regression-bs087.html b/doc/pub/Regression/html/._Regression-bs087.html index 5badfa310..b238190ec 100644 --- a/doc/pub/Regression/html/._Regression-bs087.html +++ b/doc/pub/Regression/html/._Regression-bs087.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -454,7 +493,7 @@ plt.show()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs088.html b/doc/pub/Regression/html/._Regression-bs088.html index f9f418dcc..d27461467 100644 --- a/doc/pub/Regression/html/._Regression-bs088.html +++ b/doc/pub/Regression/html/._Regression-bs088.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -429,7 +468,7 @@ moreover, it becomes more accurate the larger \( n \) is.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs089.html b/doc/pub/Regression/html/._Regression-bs089.html index 7e99c3cad..0195fd72b 100644 --- a/doc/pub/Regression/html/._Regression-bs089.html +++ b/doc/pub/Regression/html/._Regression-bs089.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -440,7 +479,7 @@ elements of \( \vec{X}_i \) and let \( n_i \) be the number of elements of
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    • diff --git a/doc/pub/Regression/html/._Regression-bs090.html b/doc/pub/Regression/html/._Regression-bs090.html index 9e0891aaf..a3746cf29 100644 --- a/doc/pub/Regression/html/._Regression-bs090.html +++ b/doc/pub/Regression/html/._Regression-bs090.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -427,7 +466,7 @@ The quantity \( \hat{X} \) is asymptotic uncorrelated by assumption, \( \hat{X}_
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    • diff --git a/doc/pub/Regression/html/._Regression-bs091.html b/doc/pub/Regression/html/._Regression-bs091.html index 72e1def76..13f1056fd 100644 --- a/doc/pub/Regression/html/._Regression-bs091.html +++ b/doc/pub/Regression/html/._Regression-bs091.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -425,7 +464,7 @@ We can show that \( V(\overline{X}_i) = V(\overline{X}_j) \) for all \( 0 \leq i
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    • diff --git a/doc/pub/Regression/html/._Regression-bs092.html b/doc/pub/Regression/html/._Regression-bs092.html index 9a65ab40c..8258b3005 100644 --- a/doc/pub/Regression/html/._Regression-bs092.html +++ b/doc/pub/Regression/html/._Regression-bs092.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -434,7 +473,7 @@ It means we can apply blocking transformations until
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    • diff --git a/doc/pub/Regression/html/._Regression-bs093.html b/doc/pub/Regression/html/._Regression-bs093.html index e1d99a12f..67c893844 100644 --- a/doc/pub/Regression/html/._Regression-bs093.html +++ b/doc/pub/Regression/html/._Regression-bs093.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -631,6 +670,8 @@ dataAnalysis.printOutput()
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    • diff --git a/doc/pub/Regression/html/._Regression-bs094.html b/doc/pub/Regression/html/._Regression-bs094.html index 06e21545b..62bfeba8d 100644 --- a/doc/pub/Regression/html/._Regression-bs094.html +++ b/doc/pub/Regression/html/._Regression-bs094.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -424,6 +463,9 @@ The
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  • 115
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    • diff --git a/doc/pub/Regression/html/._Regression-bs095.html b/doc/pub/Regression/html/._Regression-bs095.html index 21344805e..4b2e34abf 100644 --- a/doc/pub/Regression/html/._Regression-bs095.html +++ b/doc/pub/Regression/html/._Regression-bs095.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -413,6 +452,10 @@ where \( \epsilon \) is normally distributed with mean zero and standard deviati
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  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -415,6 +454,11 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs097.html b/doc/pub/Regression/html/._Regression-bs097.html index 0ca3f9a9d..70b4a1f5c 100644 --- a/doc/pub/Regression/html/._Regression-bs097.html +++ b/doc/pub/Regression/html/._Regression-bs097.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -415,6 +454,12 @@ functional if we drew an infinite number of datasets \( \{\mathcal{L}_1,
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    • diff --git a/doc/pub/Regression/html/._Regression-bs098.html b/doc/pub/Regression/html/._Regression-bs098.html index 9c7699772..e93202174 100644 --- a/doc/pub/Regression/html/._Regression-bs098.html +++ b/doc/pub/Regression/html/._Regression-bs098.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -423,6 +462,13 @@ terms.
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    • diff --git a/doc/pub/Regression/html/._Regression-bs099.html b/doc/pub/Regression/html/._Regression-bs099.html index e5b68b91a..107bd7107 100644 --- a/doc/pub/Regression/html/._Regression-bs099.html +++ b/doc/pub/Regression/html/._Regression-bs099.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -415,6 +454,14 @@ $$
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    • diff --git a/doc/pub/Regression/html/._Regression-bs100.html b/doc/pub/Regression/html/._Regression-bs100.html index d454ff85b..fce74f58b 100644 --- a/doc/pub/Regression/html/._Regression-bs100.html +++ b/doc/pub/Regression/html/._Regression-bs100.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -408,6 +447,15 @@ and measures the deviation of the expectation value of our estimator (i.e. the a
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    • diff --git a/doc/pub/Regression/html/._Regression-bs101.html b/doc/pub/Regression/html/._Regression-bs101.html index ea0da0cbf..051d57739 100644 --- a/doc/pub/Regression/html/._Regression-bs101.html +++ b/doc/pub/Regression/html/._Regression-bs101.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -419,6 +458,16 @@ finite-sized training dataset (smaller variance).
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    • diff --git a/doc/pub/Regression/html/._Regression-bs102.html b/doc/pub/Regression/html/._Regression-bs102.html index d129f2258..adccff3cc 100644 --- a/doc/pub/Regression/html/._Regression-bs102.html +++ b/doc/pub/Regression/html/._Regression-bs102.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -401,6 +440,7 @@ sets. However, if a method has high variance then small changes in the training data can result in large changes in the model. In general, more flexible statistical methods have higher variance. +

    diff --git a/doc/pub/Regression/html/._Regression-bs103.html b/doc/pub/Regression/html/._Regression-bs103.html index 0aa98b52c..32bf8052f 100644 --- a/doc/pub/Regression/html/._Regression-bs103.html +++ b/doc/pub/Regression/html/._Regression-bs103.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -393,20 +419,37 @@ MathJax.Hub.Config({ -

    Linear classifier

    +

    The one-dimensional Ising model, project 2

    -Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset +The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant \( J \) is given by + $$ -\begin{equation} -s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w}, +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, \tag{30} -\end{equation} +\end{align} $$ -where we use the short-hand notation -\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \). +where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the low temperature limit there is no phase transition. +

    +We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies. + +

    + + +

    L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+

    @@ -428,6 +471,12 @@ where we use the short-hand notation

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    • diff --git a/doc/pub/Regression/html/._Regression-bs104.html b/doc/pub/Regression/html/._Regression-bs104.html index f7dcffe23..7f19c515a 100644 --- a/doc/pub/Regression/html/._Regression-bs104.html +++ b/doc/pub/Regression/html/._Regression-bs104.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -393,30 +419,28 @@ MathJax.Hub.Config({ -

    Some selected properties

    +

    Example: The one-dimensional Ising model

    -This function takes values on the entire real axis. In the case of -logistic regression, however, the labels \( y_i \) are discrete -variables. One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to \( \{0,1\} \), -\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed, -this is commonly known as the "perceptron" in the machine learning -literature. This model is extremely simple, and it is favorable in -many cases (e.g. noisy data) to have a ``soft" classifier that outputs -the probability of a given category. For example, given -\( \mathbf{x}_i \), the classifier outputs the probability of being in -category \( m \). One such function is the logistic (or sigmoid) function: - -$$ -\begin{equation} -f(s) = \frac{1}{1+\mathrm e^{-s}}. -\tag{31} -\end{equation} -$$ - -Note that \( 1-f(s)= f(-s) \), which will be useful shortly. +Here we use linear (ordinary least squares), ridge and LASSO +regression to predict the energy in the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use the linear regression models to fit a value for +the coupling constant to achieve this. +

    + +

    import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+

    @@ -437,6 +461,13 @@ Note that \( 1-f(s)= f(-s) \), which will be useful shortly.

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    • diff --git a/doc/pub/Regression/html/._Regression-bs105.html b/doc/pub/Regression/html/._Regression-bs105.html index 1d815b130..35f9a1406 100644 --- a/doc/pub/Regression/html/._Regression-bs105.html +++ b/doc/pub/Regression/html/._Regression-bs105.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -393,26 +419,59 @@ MathJax.Hub.Config({ -

    The cross-entropy as a cost function for logistic regression

    +

    Reformulating the problem to suit regression

    -The perceptron is an example of a ``hard classification": each datapoint is deterministically assigned to a category (i.e \( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" classifier that outputs the probability of a given category rather than a single value. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \). -Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by -$$ -\begin{eqnarray} -P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\ -P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)}, -\end{eqnarray} -$$ +A more general form for the one-dimensional Ising model is -where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data. +$$ +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\tag{31} +\end{align} +$$

    -Notice that in terms of the logistic function, we can write +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form $$ -P(y_i=1) =f(\mathbf{x}_i^T\mathbf{w})=1-P(y_i=0). +\begin{align} + H = X J, +\tag{32} +\end{align} $$ +

    +where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the +elements \( -J_{jk} \). This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +$$ +\begin{align} + y = X\omega + \epsilon, +\tag{33} +\end{align} +$$ + +

    + + +

    X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+

    @@ -432,6 +491,12 @@ $$

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    • diff --git a/doc/pub/Regression/html/._Regression-bs106.html b/doc/pub/Regression/html/._Regression-bs106.html index 7f367c12c..2b8dd41ea 100644 --- a/doc/pub/Regression/html/._Regression-bs106.html +++ b/doc/pub/Regression/html/._Regression-bs106.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
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    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
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  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
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    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -391,32 +417,38 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Maximum likelihood

    +

    Linear regression

    -We now define the cost function for logistic regression using Maximum -Likelihood Estimation (MLE). Recall, that in MLE we choose parameters -to maximize the probability of seeing the observed data. Consider a -dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels -\( y_i\in\{0,1\} \) where the data points are drawn independently. The -likelihood of the seeing the data under our model is just: +The problem at hand is to try to fit the equation $$ \begin{align} -P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\ -\tag{32} + y = f(x) + \epsilon, +\tag{34} \end{align} $$ -from which we can readily compute the log-likelihood: +

    +where \( f(x) \) is some unknown function of the data \( x \) and \( \epsilon \) +is normally distributed with mean zero noise with standard deviation +\( \sigma_{\epsilon} \). Our job is to try to find a predictor which +estimates the function \( f(x) \). In linear regression we assume that we +can formulate the problem as + $$ -\begin{equation} -l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. -\tag{33} -\end{equation} +\begin{align} + y = X\omega + \epsilon, +\tag{35} +\end{align} $$ +

    +where \( X \) and \( \omega \) are now matrices. Our job at hand is now to +find a cost function \( C \), which we wish to minimize in order to find +the best estimate of \( \omega \). +

    @@ -435,6 +467,12 @@ $$

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    • diff --git a/doc/pub/Regression/html/._Regression-bs107.html b/doc/pub/Regression/html/._Regression-bs107.html index 0ff989da8..9eeb9f894 100644 --- a/doc/pub/Regression/html/._Regression-bs107.html +++ b/doc/pub/Regression/html/._Regression-bs107.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -392,22 +418,50 @@ MathJax.Hub.Config({ -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \theta \) + +

    Ordinary least squares

    + +

    +In the ordinary least squares method we choose the cost function $$ -\hat{\mathbf{w}} = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]. +\begin{align} + C(X, \omega) = ||X\omega - y||^2 + = (X\omega - y)^T(X\omega - y) +\tag{36} +\end{align} $$ -Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that +We then find the extremal point of \( C \) by taking the derivative with respect to \( \omega \) and setting it to zero, i.e., + $$ -\begin{eqnarray} -\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\ -&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber -\end{eqnarray} +\begin{align} + \dfrac{\mathrm{d}C}{\mathrm{d}\omega} + = 0. +\tag{37} +\end{align} $$ -This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression, -in practice we usually supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression. +This yields the expression for \( \omega \) to be +$$ +\begin{align} + \omega = \frac{X^T y}{X^T X}, +\tag{38} +\end{align} +$$ +

    +which immediately imposes some requirements on \( X \) as there must exist +an inverse of \( X^T X \). If the expression we are modelling contains an +intercept, i.e., a constant expression we must make sure that the +first column of \( X \) consists of \( 1 \). + +

    + + +

    def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+omega = get_ols_weights_naive(X_train_own, y_train)
+

    @@ -425,6 +479,12 @@ in practice we usually supplement the cross-entropy with additional regularizati

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    • diff --git a/doc/pub/Regression/html/._Regression-bs108.html b/doc/pub/Regression/html/._Regression-bs108.html index 24f214d02..2e1910830 100644 --- a/doc/pub/Regression/html/._Regression-bs108.html +++ b/doc/pub/Regression/html/._Regression-bs108.html @@ -194,46 +194,65 @@ Automatically generated HTML file from DocOnce source '___sec77'), ('Resampling methods: Jackknife', 2, None, '___sec78'), ('Resampling methods: Jackknife estimator', 2, None, '___sec79'), - ('Resampling methods: Bootstrap', 2, None, '___sec80'), - ('Resampling methods: Bootstrap background', 2, None, '___sec81'), + ('Jackknife code example', 2, None, '___sec80'), + ('Resampling methods: Bootstrap', 2, None, '___sec81'), + ('Resampling methods: Bootstrap background', 2, None, '___sec82'), ('Resampling methods: More Bootstrap background', 2, None, - '___sec82'), - ('Resampling methods: Bootstrap approach', 2, None, '___sec83'), - ('Resampling methods: Bootstrap steps', 2, None, '___sec84'), - ('Resampling methods: Blocking', 2, None, '___sec85'), - ('Blocking Transformations', 2, None, '___sec86'), - ('Blocking Transformations', 2, None, '___sec87'), - ('Blocking Transformations, getting there', 2, None, '___sec88'), + '___sec83'), + ('Resampling methods: Bootstrap approach', 2, None, '___sec84'), + ('Resampling methods: Bootstrap steps', 2, None, '___sec85'), + ('Code example for the Bootstrap method', 2, None, '___sec86'), + ('Resampling methods: Blocking', 2, None, '___sec87'), + ('Blocking Transformations', 2, None, '___sec88'), + ('Blocking Transformations', 2, None, '___sec89'), + ('Blocking Transformations, getting there', 2, None, '___sec90'), ('Blocking Transformations, final expressions', 2, None, - '___sec89'), + '___sec91'), ('"Code examples for Blocking, Jackknife and ' 'bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"', 2, None, - '___sec90'), - ('The bias-variance tradeoff', 2, None, '___sec91'), - ('Training and testing data', 2, None, '___sec92'), - ('Procedure to find a predictor', 2, None, '___sec93'), - ('What we want', 2, None, '___sec94'), - ('The expected generalization error', 2, None, '___sec95'), - ('Elaborating a little bit more', 2, None, '___sec96'), - ('The bias', 2, None, '___sec97'), - ('The variance', 2, None, '___sec98'), - ('Summing up', 2, None, '___sec99'), - ('Logistic Regression', 2, None, '___sec100'), - ('Basics', 2, None, '___sec101'), - ('Linear classifier', 2, None, '___sec102'), - ('Some selected properties', 2, None, '___sec103'), - ('The cross-entropy as a cost function for logistic regression', + '___sec92'), + ('The bias-variance tradeoff', 2, None, '___sec93'), + ('Training and testing data', 2, None, '___sec94'), + ('Procedure to find a predictor', 2, None, '___sec95'), + ('What we want', 2, None, '___sec96'), + ('The expected generalization error', 2, None, '___sec97'), + ('Elaborating a little bit more', 2, None, '___sec98'), + ('The bias', 2, None, '___sec99'), + ('The variance', 2, None, '___sec100'), + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', 2, None, '___sec104'), - ('Maximum likelihood', 2, None, '___sec105'), - ('Minimizing the cross entropy', 2, None, '___sec106')]} + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -351,33 +370,40 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife and Bootstrap
  • Resampling methods: Jackknife
  • Resampling methods: Jackknife estimator
    • -
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Resampling methods: Blocking
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations
    • -
  • Blocking Transformations, getting there
    • -
  • Blocking Transformations, final expressions
    • -
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • -
  • The bias-variance tradeoff
    • -
  • Training and testing data
    • -
  • Procedure to find a predictor
    • -
  • What we want
    • -
  • The expected generalization error
    • -
  • Elaborating a little bit more
    • -
  • The bias
    • -
  • The variance
    • -
  • Summing up
    • -
  • Logistic Regression
    • -
  • Basics
    • -
  • Linear classifier
    • -
  • Some selected properties
    • -
  • The cross-entropy as a cost function for logistic regression
    • -
  • Maximum likelihood
    • -
  • Minimizing the cross entropy
    • +
  • Jackknife code example
    • +
  • Resampling methods: Bootstrap
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Resampling methods: Blocking
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations
    • +
  • Blocking Transformations, getting there
    • +
  • Blocking Transformations, final expressions
    • +
  • "Code examples for Blocking, Jackknife and bootstrap":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/ResamplingAnalysisScripts"
    • +
  • The bias-variance tradeoff
    • +
  • Training and testing data
    • +
  • Procedure to find a predictor
    • +
  • What we want
    • +
  • The expected generalization error
    • +
  • Elaborating a little bit more
    • +
  • The bias
    • +
  • The variance
    • +
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -393,27 +419,62 @@ MathJax.Hub.Config({ -

    Minimizing the cross entropy

    - -

    -The cross entropy is a convex function of the weights \( \mathbf{w} \) and, -therefore, any local minimizer is a global minimizer. Minimizing this -cost function leads to the following equation +

    Singular Value decomposition

    +Doing the inversion directly turns out to be a bad idea as the matrix +\( X^TX \) is singular. An alternative approach is to use the singular +value decomposition. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for \( \omega \) as $$ -\begin{equation} -\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i, -\tag{34} -\end{equation} +\begin{align} + \omega = X^{+}y, +\tag{39} +\end{align} +$$ + +where the pseudoinverse of \( X \) is given by +$$ +\begin{align} + X^{+} = \frac{X^T}{X^T X}. +\tag{40} +\end{align} $$

    -where we made use of the logistic function identity \( \partial_z f(z) = -f(z)[1-f(z)] \). This equation defines a transcendental equation for -\( \mathbf{w} \), the solution of which, unlike linear regression, cannot -be written in a closed form. -Here we need gradient descent methods! +Using singular value decomposition we have that \( X = U\Sigma V^T \), +where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for +\( \omega \) to +$$ +\begin{align} + \omega = V\Sigma^{+} U^T y. +\tag{41} +\end{align} +$$ +

    +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a +general matrix. Instead, doing \( QR \)-factorization and solving the +linear system as an equation would reduce this down to +\( \mathcal{O}(n^2) \) operations. + +

    + + +

    def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
    +

    +Before passing in the data to the function we append a column with ones to the training data. + +

    + + +

    omega = get_ols_weights(X_train_own,y_train)
+
    +

    diff --git a/doc/pub/Regression/html/._Regression-bs109.html b/doc/pub/Regression/html/._Regression-bs109.html new file mode 100644 index 000000000..804d9cf08 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs109.html @@ -0,0 +1,515 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Fitting with scikit-learn

    + +

    +Next we fit a LinearRegression-model from Scikit-learn for comparison. + +

    + + +

    clf = skl.LinearRegression().fit(X_train, y_train)
+
    +

    +Extracting the \( J \)-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept. + +

    + + +

    J_own = omega[1:].reshape(L, L)
+J_sk = clf.coef_.reshape(L, L)
+
    +

    +A way of looking at the coefficients in \( J \) is to plot the matrices as images. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_own, **cmap_args)
+plt.title("Home-made OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
    +

    +We can see that our model for the least squares method performes close +to the benchmark from Scikit-learn. It is interesting to note that OLS +considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as +valid matrix elements for \( J \). + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs110.html b/doc/pub/Regression/html/._Regression-bs110.html new file mode 100644 index 000000000..1994449a9 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs110.html @@ -0,0 +1,528 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Ridge regression

    + +

    +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a regularizer. This +involves a new cost function which leads to a new estimate for the +weights \( \omega \). This results in a penalized regression problem. The +cost function is given by + +$$ +\begin{align} + C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2 + = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega. +\tag{42} +\end{align} +$$ + +Finding the extremum of this function yields the weights + +$$ +\begin{align} + \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda}, +\tag{43} +\end{align} +$$ + +

    +where \( \omega_{\text{LS}} \) is the weights from ordinary least +squares. The last assumption assumes that \( X \) is orthogonal, which it +is not. We will therefore resort to solving the equation as it stands +on the left hand side. + +

    + + +

    def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:
+    return x.T @ y @ scl.inv(
+        x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda
+    )
+lambda = 0.1
+omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_own = omega_ridge[1:].reshape(L, L)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_own, **cmap_args)
+plt.title("Home-made ridge regression", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs111.html b/doc/pub/Regression/html/._Regression-bs111.html new file mode 100644 index 000000000..0e1b20c58 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs111.html @@ -0,0 +1,500 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    LASSO regression

    + +

    +In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function. +$$ +\begin{align} + C(X, \omega; \lambda) = + ||X\omega - y||^2 + \lambda ||\omega|| + = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}. +\tag{44} +\end{align} +$$ + +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn. + +

    + + +

    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +\( J_{j, j + 1} = -1 \). + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs112.html b/doc/pub/Regression/html/._Regression-bs112.html new file mode 100644 index 000000000..4c095c632 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs112.html @@ -0,0 +1,495 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Performance of the different models

    + +

    +In order to judge which model performs best at varying values of \( \lambda \) (for ridge and LASSO) we compute \( R^2 \) which is given by + +$$ +\begin{align} + R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2}, +\tag{45} +\end{align} +$$ + +where \( y \) is a vector with the true values of the energy, \( \hat{y} \) is the predicted values of \( y \) from the models and \( \bar{y} \) is the mean of \( \hat{y} \). + +

    + + +

    def r_squared(y, y_hat):
+    return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2)
+
    +

    +This is the same metric used by Scikit-learn for their regression models when scoring. +

    + + +

    y_hat = clf.predict(X_test)
+r_test = r_squared(y_test, y_hat)
+sk_r_test = clf.score(X_test, y_test)
+
+assert abs(r_test - sk_r_test) < 1e-2
+
    +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs113.html b/doc/pub/Regression/html/._Regression-bs113.html new file mode 100644 index 000000000..f33282cad --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs113.html @@ -0,0 +1,542 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Performance as function of the regularization parameter

    + +

    +We see how the different models perform for a different set of values for \( \lambda \). + +

    + + +

    lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    omega = get_ols_weights(X_train_own, y_train)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ols_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ols_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title("Home made OLS")
+    plot_counter += 1
+
+    omega = get_ridge_weights(X_train_own, y_train, _lambda)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ridge_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ridge_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title(r"Home made ridge, $\lambda = %.4f$" % _lambda)
+    plot_counter += 1
+
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
+
+plt.show()
+
    +

    +We can see that LASSO quite fast reaches a good solution for low +values of \( \lambda \), but will "wither" when we increase \( \lambda \) too +much. Ridge is more stable over a larger range of values for +\( \lambda \), but eventually also fades away. + +

    +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs114.html b/doc/pub/Regression/html/._Regression-bs114.html new file mode 100644 index 000000000..9294abbed --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs114.html @@ -0,0 +1,514 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + + Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis +
    + +
    + +
    +
    + + +
    + +

     

     

     

    + + + + +

    Finding the optimal value of \( \lambda \)

    + +

    +To determine which value of \( \lambda \) is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+
+colors = {
+    "ols_own": "b",
+    "ridge_own": "g",
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+#plt.semilogx(lambdas, train_errors["ols_own"], label="Train (OLS own)")
+#plt.semilogx(lambdas, test_errors["ols_own"], label="Test (OLS own)")
+
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
    +

    +From the above figure we can see that LASSO with \( \lambda = 10^{-2} \) +achieve a very good accuracy on the test set. This by far surpases the +other models for all values of \( \lambda \). + +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/Regression/html/Regression-bs.html b/doc/pub/Regression/html/Regression-bs.html index 833e9dec8..e6ce300e7 100644 --- a/doc/pub/Regression/html/Regression-bs.html +++ b/doc/pub/Regression/html/Regression-bs.html @@ -225,7 +225,34 @@ Automatically generated HTML file from DocOnce source ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -365,6 +392,18 @@ MathJax.Hub.Config({
  • The bias
  • The variance
  • Summing up
    • +
  • The one-dimensional Ising model, project 2
    • +
  • Example: The one-dimensional Ising model
    • +
  • Reformulating the problem to suit regression
    • +
  • Linear regression
    • +
  • Ordinary least squares
    • +
  • Singular Value decomposition
    • +
  • Fitting with scikit-learn
    • +
  • Ridge regression
    • +
  • LASSO regression
    • +
  • Performance of the different models
    • +
  • Performance as function of the regularization parameter
    • +
  • Finding the optimal value of \( \lambda \)
    • @@ -399,7 +438,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 25, 2018

    +

    Oct 11, 2018


    @@ -423,7 +462,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
  • 103
    • +
  • 115
  • »
    • diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html index c5cf28468..1959fd9b9 100644 --- a/doc/pub/Regression/html/Regression-reveal.html +++ b/doc/pub/Regression/html/Regression-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 25, 2018

    +

    Oct 11, 2018


    @@ -3541,6 +3541,624 @@ flexible statistical methods have higher variance. +

    +

    The one-dimensional Ising model, project 2

    + +

    +The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant \( J \) is given by + +

     
    +$$ +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\tag{30} +\end{align} +$$ +

     
    + +where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the low temperature limit there is no phase transition. + +

    +We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies. + +

    + + +

    L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
    +
    + + +
    +

    Example: The one-dimensional Ising model

    + +

    +Here we use linear (ordinary least squares), ridge and LASSO +regression to predict the energy in the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use the linear regression models to fit a value for +the coupling constant to achieve this. +

    + + +

    import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
    +
    + + +
    +

    Reformulating the problem to suit regression

    + +

    +A more general form for the one-dimensional Ising model is + +

     
    +$$ +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\tag{31} +\end{align} +$$ +

     
    + +

    +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form +

     
    +$$ +\begin{align} + H = X J, +\tag{32} +\end{align} +$$ +

     
    + +

    +where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the +elements \( -J_{jk} \). This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +

     
    +$$ +\begin{align} + y = X\omega + \epsilon, +\tag{33} +\end{align} +$$ +

     
    + +

    + + +

    X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
    +
    + + +
    +

    Linear regression

    + +

    +The problem at hand is to try to fit the equation +

     
    +$$ +\begin{align} + y = f(x) + \epsilon, +\tag{34} +\end{align} +$$ +

     
    + +

    +where \( f(x) \) is some unknown function of the data \( x \) and \( \epsilon \) +is normally distributed with mean zero noise with standard deviation +\( \sigma_{\epsilon} \). Our job is to try to find a predictor which +estimates the function \( f(x) \). In linear regression we assume that we +can formulate the problem as + +

     
    +$$ +\begin{align} + y = X\omega + \epsilon, +\tag{35} +\end{align} +$$ +

     
    + +

    +where \( X \) and \( \omega \) are now matrices. Our job at hand is now to +find a cost function \( C \), which we wish to minimize in order to find +the best estimate of \( \omega \). +

    + + +
    +

    Ordinary least squares

    + +

    +In the ordinary least squares method we choose the cost function +

     
    +$$ +\begin{align} + C(X, \omega) = ||X\omega - y||^2 + = (X\omega - y)^T(X\omega - y) +\tag{36} +\end{align} +$$ +

     
    + +We then find the extremal point of \( C \) by taking the derivative with respect to \( \omega \) and setting it to zero, i.e., + +

     
    +$$ +\begin{align} + \dfrac{\mathrm{d}C}{\mathrm{d}\omega} + = 0. +\tag{37} +\end{align} +$$ +

     
    + +This yields the expression for \( \omega \) to be +

     
    +$$ +\begin{align} + \omega = \frac{X^T y}{X^T X}, +\tag{38} +\end{align} +$$ +

     
    + +

    +which immediately imposes some requirements on \( X \) as there must exist +an inverse of \( X^T X \). If the expression we are modelling contains an +intercept, i.e., a constant expression we must make sure that the +first column of \( X \) consists of \( 1 \). + +

    + + +

    def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+omega = get_ols_weights_naive(X_train_own, y_train)
+
    +
    + + +
    +

    Singular Value decomposition

    +Doing the inversion directly turns out to be a bad idea as the matrix +\( X^TX \) is singular. An alternative approach is to use the singular +value decomposition. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for \( \omega \) as + +

     
    +$$ +\begin{align} + \omega = X^{+}y, +\tag{39} +\end{align} +$$ +

     
    + +where the pseudoinverse of \( X \) is given by +

     
    +$$ +\begin{align} + X^{+} = \frac{X^T}{X^T X}. +\tag{40} +\end{align} +$$ +

     
    + +

    +Using singular value decomposition we have that \( X = U\Sigma V^T \), +where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for +\( \omega \) to +

     
    +$$ +\begin{align} + \omega = V\Sigma^{+} U^T y. +\tag{41} +\end{align} +$$ +

     
    + +

    +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a +general matrix. Instead, doing \( QR \)-factorization and solving the +linear system as an equation would reduce this down to +\( \mathcal{O}(n^2) \) operations. + +

    + + +

    def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
    +

    +Before passing in the data to the function we append a column with ones to the training data. + +

    + + +

    omega = get_ols_weights(X_train_own,y_train)
+
    +
    + + +
    +

    Fitting with scikit-learn

    + +

    +Next we fit a LinearRegression-model from Scikit-learn for comparison. + +

    + + +

    clf = skl.LinearRegression().fit(X_train, y_train)
+
    +

    +Extracting the \( J \)-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept. + +

    + + +

    J_own = omega[1:].reshape(L, L)
+J_sk = clf.coef_.reshape(L, L)
+
    +

    +A way of looking at the coefficients in \( J \) is to plot the matrices as images. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_own, **cmap_args)
+plt.title("Home-made OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
    +

    +We can see that our model for the least squares method performes close +to the benchmark from Scikit-learn. It is interesting to note that OLS +considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as +valid matrix elements for \( J \). +

    + + +
    +

    Ridge regression

    + +

    +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a regularizer. This +involves a new cost function which leads to a new estimate for the +weights \( \omega \). This results in a penalized regression problem. The +cost function is given by + +

     
    +$$ +\begin{align} + C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2 + = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega. +\tag{42} +\end{align} +$$ +

     
    + +Finding the extremum of this function yields the weights + +

     
    +$$ +\begin{align} + \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda}, +\tag{43} +\end{align} +$$ +

     
    + +

    +where \( \omega_{\text{LS}} \) is the weights from ordinary least +squares. The last assumption assumes that \( X \) is orthogonal, which it +is not. We will therefore resort to solving the equation as it stands +on the left hand side. + +

    + + +

    def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:
+    return x.T @ y @ scl.inv(
+        x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda
+    )
+lambda = 0.1
+omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_own = omega_ridge[1:].reshape(L, L)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_own, **cmap_args)
+plt.title("Home-made ridge regression", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +
    + + +
    +

    LASSO regression

    + +

    +In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function. +

     
    +$$ +\begin{align} + C(X, \omega; \lambda) = + ||X\omega - y||^2 + \lambda ||\omega|| + = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}. +\tag{44} +\end{align} +$$ +

     
    + +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn. + +

    + + +

    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +\( J_{j, j + 1} = -1 \). +

    + + +
    +

    Performance of the different models

    + +

    +In order to judge which model performs best at varying values of \( \lambda \) (for ridge and LASSO) we compute \( R^2 \) which is given by + +

     
    +$$ +\begin{align} + R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2}, +\tag{45} +\end{align} +$$ +

     
    + +where \( y \) is a vector with the true values of the energy, \( \hat{y} \) is the predicted values of \( y \) from the models and \( \bar{y} \) is the mean of \( \hat{y} \). + +

    + + +

    def r_squared(y, y_hat):
+    return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2)
+
    +

    +This is the same metric used by Scikit-learn for their regression models when scoring. +

    + + +

    y_hat = clf.predict(X_test)
+r_test = r_squared(y_test, y_hat)
+sk_r_test = clf.score(X_test, y_test)
+
+assert abs(r_test - sk_r_test) < 1e-2
+
    +
    + + +
    +

    Performance as function of the regularization parameter

    + +

    +We see how the different models perform for a different set of values for \( \lambda \). + +

    + + +

    lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    omega = get_ols_weights(X_train_own, y_train)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ols_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ols_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title("Home made OLS")
+    plot_counter += 1
+
+    omega = get_ridge_weights(X_train_own, y_train, _lambda)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ridge_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ridge_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title(r"Home made ridge, $\lambda = %.4f$" % _lambda)
+    plot_counter += 1
+
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
+
+plt.show()
+
    +

    +We can see that LASSO quite fast reaches a good solution for low +values of \( \lambda \), but will "wither" when we increase \( \lambda \) too +much. Ridge is more stable over a larger range of values for +\( \lambda \), but eventually also fades away. +

    + + +
    +

    Finding the optimal value of \( \lambda \)

    + +

    +To determine which value of \( \lambda \) is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+
+colors = {
+    "ols_own": "b",
+    "ridge_own": "g",
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+#plt.semilogx(lambdas, train_errors["ols_own"], label="Train (OLS own)")
+#plt.semilogx(lambdas, test_errors["ols_own"], label="Test (OLS own)")
+
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
    +

    +From the above figure we can see that LASSO with \( \lambda = 10^{-2} \) +achieve a very good accuracy on the test set. This by far surpases the +other models for all values of \( \lambda \). +

    + + diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html index df9b24537..95341ec36 100644 --- a/doc/pub/Regression/html/Regression-solarized.html +++ b/doc/pub/Regression/html/Regression-solarized.html @@ -245,7 +245,34 @@ div { text-align: justify; text-justify: inter-word; } ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -287,7 +314,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 25, 2018

    +

    Oct 11, 2018












    @@ -3472,6 +3499,585 @@ sets. However, if a method has high variance then small changes in the training data can result in large changes in the model. In general, more flexible statistical methods have higher variance. +

    +









    + +

    The one-dimensional Ising model, project 2

    + +

    +The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant \( J \) is given by + +$$ +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\label{_auto19} +\end{align} +$$ + +where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the low temperature limit there is no phase transition. + +

    +We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies. + +

    + + +

    L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
    +

    +









    + +

    Example: The one-dimensional Ising model

    + +

    +Here we use linear (ordinary least squares), ridge and LASSO +regression to predict the energy in the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use the linear regression models to fit a value for +the coupling constant to achieve this. +

    + + +

    import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
    +

    +









    + +

    Reformulating the problem to suit regression

    + +

    +A more general form for the one-dimensional Ising model is + +$$ +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\label{_auto20} +\end{align} +$$ + +

    +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form +$$ +\begin{align} + H = X J, +\label{_auto21} +\end{align} +$$ + +

    +where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the +elements \( -J_{jk} \). This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +$$ +\begin{align} + y = X\omega + \epsilon, +\label{_auto22} +\end{align} +$$ + +

    + + +

    X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
    +

    +









    + +

    Linear regression

    + +

    +The problem at hand is to try to fit the equation +$$ +\begin{align} + y = f(x) + \epsilon, +\label{_auto23} +\end{align} +$$ + +

    +where \( f(x) \) is some unknown function of the data \( x \) and \( \epsilon \) +is normally distributed with mean zero noise with standard deviation +\( \sigma_{\epsilon} \). Our job is to try to find a predictor which +estimates the function \( f(x) \). In linear regression we assume that we +can formulate the problem as + +$$ +\begin{align} + y = X\omega + \epsilon, +\label{_auto24} +\end{align} +$$ + +

    +where \( X \) and \( \omega \) are now matrices. Our job at hand is now to +find a cost function \( C \), which we wish to minimize in order to find +the best estimate of \( \omega \). + +

    +









    + +

    Ordinary least squares

    + +

    +In the ordinary least squares method we choose the cost function +$$ +\begin{align} + C(X, \omega) = ||X\omega - y||^2 + = (X\omega - y)^T(X\omega - y) +\label{_auto25} +\end{align} +$$ + +We then find the extremal point of \( C \) by taking the derivative with respect to \( \omega \) and setting it to zero, i.e., + +$$ +\begin{align} + \dfrac{\mathrm{d}C}{\mathrm{d}\omega} + = 0. +\label{_auto26} +\end{align} +$$ + +This yields the expression for \( \omega \) to be +$$ +\begin{align} + \omega = \frac{X^T y}{X^T X}, +\label{_auto27} +\end{align} +$$ + +

    +which immediately imposes some requirements on \( X \) as there must exist +an inverse of \( X^T X \). If the expression we are modelling contains an +intercept, i.e., a constant expression we must make sure that the +first column of \( X \) consists of \( 1 \). + +

    + + +

    def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+omega = get_ols_weights_naive(X_train_own, y_train)
+
    +

    +









    + +

    Singular Value decomposition

    +Doing the inversion directly turns out to be a bad idea as the matrix +\( X^TX \) is singular. An alternative approach is to use the singular +value decomposition. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for \( \omega \) as + +$$ +\begin{align} + \omega = X^{+}y, +\label{_auto28} +\end{align} +$$ + +where the pseudoinverse of \( X \) is given by +$$ +\begin{align} + X^{+} = \frac{X^T}{X^T X}. +\label{_auto29} +\end{align} +$$ + +

    +Using singular value decomposition we have that \( X = U\Sigma V^T \), +where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for +\( \omega \) to +$$ +\begin{align} + \omega = V\Sigma^{+} U^T y. +\label{_auto30} +\end{align} +$$ + +

    +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a +general matrix. Instead, doing \( QR \)-factorization and solving the +linear system as an equation would reduce this down to +\( \mathcal{O}(n^2) \) operations. + +

    + + +

    def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
    +

    +Before passing in the data to the function we append a column with ones to the training data. + +

    + + +

    omega = get_ols_weights(X_train_own,y_train)
+
    +

    +









    + +

    Fitting with scikit-learn

    + +

    +Next we fit a LinearRegression-model from Scikit-learn for comparison. + +

    + + +

    clf = skl.LinearRegression().fit(X_train, y_train)
+
    +

    +Extracting the \( J \)-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept. + +

    + + +

    J_own = omega[1:].reshape(L, L)
+J_sk = clf.coef_.reshape(L, L)
+
    +

    +A way of looking at the coefficients in \( J \) is to plot the matrices as images. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_own, **cmap_args)
+plt.title("Home-made OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
    +

    +We can see that our model for the least squares method performes close +to the benchmark from Scikit-learn. It is interesting to note that OLS +considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as +valid matrix elements for \( J \). + +

    +









    + +

    Ridge regression

    + +

    +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a regularizer. This +involves a new cost function which leads to a new estimate for the +weights \( \omega \). This results in a penalized regression problem. The +cost function is given by + +$$ +\begin{align} + C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2 + = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega. +\label{_auto31} +\end{align} +$$ + +Finding the extremum of this function yields the weights + +$$ +\begin{align} + \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda}, +\label{_auto32} +\end{align} +$$ + +

    +where \( \omega_{\text{LS}} \) is the weights from ordinary least +squares. The last assumption assumes that \( X \) is orthogonal, which it +is not. We will therefore resort to solving the equation as it stands +on the left hand side. + +

    + + +

    def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:
+    return x.T @ y @ scl.inv(
+        x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda
+    )
+lambda = 0.1
+omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_own = omega_ridge[1:].reshape(L, L)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_own, **cmap_args)
+plt.title("Home-made ridge regression", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +









    + +

    LASSO regression

    + +

    +In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function. +$$ +\begin{align} + C(X, \omega; \lambda) = + ||X\omega - y||^2 + \lambda ||\omega|| + = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}. +\label{_auto33} +\end{align} +$$ + +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn. + +

    + + +

    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +\( J_{j, j + 1} = -1 \). + +

    +









    + +

    Performance of the different models

    + +

    +In order to judge which model performs best at varying values of \( \lambda \) (for ridge and LASSO) we compute \( R^2 \) which is given by + +$$ +\begin{align} + R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2}, +\label{_auto34} +\end{align} +$$ + +where \( y \) is a vector with the true values of the energy, \( \hat{y} \) is the predicted values of \( y \) from the models and \( \bar{y} \) is the mean of \( \hat{y} \). + +

    + + +

    def r_squared(y, y_hat):
+    return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2)
+
    +

    +This is the same metric used by Scikit-learn for their regression models when scoring. +

    + + +

    y_hat = clf.predict(X_test)
+r_test = r_squared(y_test, y_hat)
+sk_r_test = clf.score(X_test, y_test)
+
+assert abs(r_test - sk_r_test) < 1e-2
+
    +

    +









    + +

    Performance as function of the regularization parameter

    + +

    +We see how the different models perform for a different set of values for \( \lambda \). + +

    + + +

    lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    omega = get_ols_weights(X_train_own, y_train)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ols_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ols_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title("Home made OLS")
+    plot_counter += 1
+
+    omega = get_ridge_weights(X_train_own, y_train, _lambda)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ridge_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ridge_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title(r"Home made ridge, $\lambda = %.4f$" % _lambda)
+    plot_counter += 1
+
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
+
+plt.show()
+
    +

    +We can see that LASSO quite fast reaches a good solution for low +values of \( \lambda \), but will "wither" when we increase \( \lambda \) too +much. Ridge is more stable over a larger range of values for +\( \lambda \), but eventually also fades away. + +

    +









    + +

    Finding the optimal value of \( \lambda \)

    + +

    +To determine which value of \( \lambda \) is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+
+colors = {
+    "ols_own": "b",
+    "ridge_own": "g",
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+#plt.semilogx(lambdas, train_errors["ols_own"], label="Train (OLS own)")
+#plt.semilogx(lambdas, test_errors["ols_own"], label="Test (OLS own)")
+
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
    +

    +From the above figure we can see that LASSO with \( \lambda = 10^{-2} \) +achieve a very good accuracy on the test set. This by far surpases the +other models for all values of \( \lambda \). + diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index 3681d2511..af11e2365 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -250,7 +250,34 @@ div { text-align: justify; text-justify: inter-word; } ('Elaborating a little bit more', 2, None, '___sec98'), ('The bias', 2, None, '___sec99'), ('The variance', 2, None, '___sec100'), - ('Summing up', 2, None, '___sec101')]} + ('Summing up', 2, None, '___sec101'), + ('The one-dimensional Ising model, project 2', + 2, + None, + '___sec102'), + ('Example: The one-dimensional Ising model', + 2, + None, + '___sec103'), + ('Reformulating the problem to suit regression', + 2, + None, + '___sec104'), + ('Linear regression', 2, None, '___sec105'), + ('Ordinary least squares', 2, None, '___sec106'), + ('Singular Value decomposition', 2, None, '___sec107'), + ('Fitting with scikit-learn', 2, None, '___sec108'), + ('Ridge regression', 2, None, '___sec109'), + ('LASSO regression', 2, None, '___sec110'), + ('Performance of the different models', 2, None, '___sec111'), + ('Performance as function of the regularization parameter', + 2, + None, + '___sec112'), + ('Finding the optimal value of $\\lambda$', + 2, + None, + '___sec113')]} end of tocinfo --> @@ -292,7 +319,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 25, 2018

    +

    Oct 11, 2018












    @@ -3477,6 +3504,585 @@ sets. However, if a method has high variance then small changes in the training data can result in large changes in the model. In general, more flexible statistical methods have higher variance. +

    +









    + +

    The one-dimensional Ising model, project 2

    + +

    +The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant \( J \) is given by + +$$ +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\label{_auto19} +\end{align} +$$ + +where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the low temperature limit there is no phase transition. + +

    +We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies. + +

    + + +

    L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
    +

    +









    + +

    Example: The one-dimensional Ising model

    + +

    +Here we use linear (ordinary least squares), ridge and LASSO +regression to predict the energy in the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use the linear regression models to fit a value for +the coupling constant to achieve this. +

    + + +

    import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
    +

    +









    + +

    Reformulating the problem to suit regression

    + +

    +A more general form for the one-dimensional Ising model is + +$$ +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\label{_auto20} +\end{align} +$$ + +

    +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form +$$ +\begin{align} + H = X J, +\label{_auto21} +\end{align} +$$ + +

    +where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the +elements \( -J_{jk} \). This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +$$ +\begin{align} + y = X\omega + \epsilon, +\label{_auto22} +\end{align} +$$ + +

    + + +

    X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
    +

    +









    + +

    Linear regression

    + +

    +The problem at hand is to try to fit the equation +$$ +\begin{align} + y = f(x) + \epsilon, +\label{_auto23} +\end{align} +$$ + +

    +where \( f(x) \) is some unknown function of the data \( x \) and \( \epsilon \) +is normally distributed with mean zero noise with standard deviation +\( \sigma_{\epsilon} \). Our job is to try to find a predictor which +estimates the function \( f(x) \). In linear regression we assume that we +can formulate the problem as + +$$ +\begin{align} + y = X\omega + \epsilon, +\label{_auto24} +\end{align} +$$ + +

    +where \( X \) and \( \omega \) are now matrices. Our job at hand is now to +find a cost function \( C \), which we wish to minimize in order to find +the best estimate of \( \omega \). + +

    +









    + +

    Ordinary least squares

    + +

    +In the ordinary least squares method we choose the cost function +$$ +\begin{align} + C(X, \omega) = ||X\omega - y||^2 + = (X\omega - y)^T(X\omega - y) +\label{_auto25} +\end{align} +$$ + +We then find the extremal point of \( C \) by taking the derivative with respect to \( \omega \) and setting it to zero, i.e., + +$$ +\begin{align} + \dfrac{\mathrm{d}C}{\mathrm{d}\omega} + = 0. +\label{_auto26} +\end{align} +$$ + +This yields the expression for \( \omega \) to be +$$ +\begin{align} + \omega = \frac{X^T y}{X^T X}, +\label{_auto27} +\end{align} +$$ + +

    +which immediately imposes some requirements on \( X \) as there must exist +an inverse of \( X^T X \). If the expression we are modelling contains an +intercept, i.e., a constant expression we must make sure that the +first column of \( X \) consists of \( 1 \). + +

    + + +

    def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+omega = get_ols_weights_naive(X_train_own, y_train)
+
    +

    +









    + +

    Singular Value decomposition

    +Doing the inversion directly turns out to be a bad idea as the matrix +\( X^TX \) is singular. An alternative approach is to use the singular +value decomposition. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for \( \omega \) as + +$$ +\begin{align} + \omega = X^{+}y, +\label{_auto28} +\end{align} +$$ + +where the pseudoinverse of \( X \) is given by +$$ +\begin{align} + X^{+} = \frac{X^T}{X^T X}. +\label{_auto29} +\end{align} +$$ + +

    +Using singular value decomposition we have that \( X = U\Sigma V^T \), +where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for +\( \omega \) to +$$ +\begin{align} + \omega = V\Sigma^{+} U^T y. +\label{_auto30} +\end{align} +$$ + +

    +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a +general matrix. Instead, doing \( QR \)-factorization and solving the +linear system as an equation would reduce this down to +\( \mathcal{O}(n^2) \) operations. + +

    + + +

    def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
    +

    +Before passing in the data to the function we append a column with ones to the training data. + +

    + + +

    omega = get_ols_weights(X_train_own,y_train)
+
    +

    +









    + +

    Fitting with scikit-learn

    + +

    +Next we fit a LinearRegression-model from Scikit-learn for comparison. + +

    + + +

    clf = skl.LinearRegression().fit(X_train, y_train)
+
    +

    +Extracting the \( J \)-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept. + +

    + + +

    J_own = omega[1:].reshape(L, L)
+J_sk = clf.coef_.reshape(L, L)
+
    +

    +A way of looking at the coefficients in \( J \) is to plot the matrices as images. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_own, **cmap_args)
+plt.title("Home-made OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
    +

    +We can see that our model for the least squares method performes close +to the benchmark from Scikit-learn. It is interesting to note that OLS +considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as +valid matrix elements for \( J \). + +

    +









    + +

    Ridge regression

    + +

    +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a regularizer. This +involves a new cost function which leads to a new estimate for the +weights \( \omega \). This results in a penalized regression problem. The +cost function is given by + +$$ +\begin{align} + C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2 + = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega. +\label{_auto31} +\end{align} +$$ + +Finding the extremum of this function yields the weights + +$$ +\begin{align} + \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda}, +\label{_auto32} +\end{align} +$$ + +

    +where \( \omega_{\text{LS}} \) is the weights from ordinary least +squares. The last assumption assumes that \( X \) is orthogonal, which it +is not. We will therefore resort to solving the equation as it stands +on the left hand side. + +

    + + +

    def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:
+    return x.T @ y @ scl.inv(
+        x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda
+    )
+lambda = 0.1
+omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_own = omega_ridge[1:].reshape(L, L)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_own, **cmap_args)
+plt.title("Home-made ridge regression", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +









    + +

    LASSO regression

    + +

    +In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function. +$$ +\begin{align} + C(X, \omega; \lambda) = + ||X\omega - y||^2 + \lambda ||\omega|| + = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}. +\label{_auto33} +\end{align} +$$ + +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn. + +

    + + +

    clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
    +

    +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +\( J_{j, j + 1} = -1 \). + +

    +









    + +

    Performance of the different models

    + +

    +In order to judge which model performs best at varying values of \( \lambda \) (for ridge and LASSO) we compute \( R^2 \) which is given by + +$$ +\begin{align} + R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2}, +\label{_auto34} +\end{align} +$$ + +where \( y \) is a vector with the true values of the energy, \( \hat{y} \) is the predicted values of \( y \) from the models and \( \bar{y} \) is the mean of \( \hat{y} \). + +

    + + +

    def r_squared(y, y_hat):
+    return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2)
+
    +

    +This is the same metric used by Scikit-learn for their regression models when scoring. +

    + + +

    y_hat = clf.predict(X_test)
+r_test = r_squared(y_test, y_hat)
+sk_r_test = clf.score(X_test, y_test)
+
+assert abs(r_test - sk_r_test) < 1e-2
+
    +

    +









    + +

    Performance as function of the regularization parameter

    + +

    +We see how the different models perform for a different set of values for \( \lambda \). + +

    + + +

    lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_own": np.zeros(lambdas.size),
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_own": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    omega = get_ols_weights(X_train_own, y_train)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ols_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ols_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title("Home made OLS")
+    plot_counter += 1
+
+    omega = get_ridge_weights(X_train_own, y_train, _lambda)
+    y_hat_train = X_train_own @ omega
+    y_hat_test = X_test_own @ omega
+
+    train_errors["ridge_own"][i] = r_squared(y_train, y_hat_train)
+    test_errors["ridge_own"][i] = r_squared(y_test, y_hat_test)
+
+    plt.subplot(10, 5, plot_counter)
+    plt.imshow(omega[1:].reshape(L, L), **cmap_args)
+    plt.title(r"Home made ridge, $\lambda = %.4f$" % _lambda)
+    plot_counter += 1
+
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
+
+plt.show()
+
    +

    +We can see that LASSO quite fast reaches a good solution for low +values of \( \lambda \), but will "wither" when we increase \( \lambda \) too +much. Ridge is more stable over a larger range of values for +\( \lambda \), but eventually also fades away. + +

    +









    + +

    Finding the optimal value of \( \lambda \)

    + +

    +To determine which value of \( \lambda \) is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + +

    + + +

    fig = plt.figure(figsize=(20, 14))
+
+colors = {
+    "ols_own": "b",
+    "ridge_own": "g",
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+#plt.semilogx(lambdas, train_errors["ols_own"], label="Train (OLS own)")
+#plt.semilogx(lambdas, test_errors["ols_own"], label="Test (OLS own)")
+
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
    +

    +From the above figure we can see that LASSO with \( \lambda = 10^{-2} \) +achieve a very good accuracy on the test set. This by far surpases the +other models for all values of \( \lambda \). + diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb index c3f5a08af..907fa601a 100644 --- a/doc/pub/Regression/ipynb/Regression.ipynb +++ b/doc/pub/Regression/ipynb/Regression.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Sep 25, 2018**\n", + "Date: **Oct 11, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -4505,7 +4505,903 @@ "estimate for our model should not vary too much between training\n", "sets. However, if a method has high variance then small changes in\n", "the training data can result in large changes in the model. In general, more\n", - "flexible statistical methods have higher variance." + "flexible statistical methods have higher variance.\n", + "\n", + "\n", + "## The one-dimensional Ising model, project 2\n", + "\n", + "The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant $J$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "

    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = -J \\sum_{k}^L s_k s_{k + 1},\n", + "\\label{_auto19} \\tag{30}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the low temperature limit there is no phase transition.\n", + "\n", + "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies." + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "L = 40\n", + "n = int(1e4)\n", + "\n", + "spins = np.random.choice([-1, 1], size=(n, L))\n", + "J = 1.0\n", + "\n", + "energies = np.zeros(n)\n", + "\n", + "for i in range(n):\n", + " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: The one-dimensional Ising model\n", + "\n", + "\n", + "Here we use linear (ordinary least squares), ridge and LASSO\n", + "regression to predict the energy in the nearest neighbor\n", + "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n", + "around. We will use the linear regression models to fit a value for\n", + "the coupling constant to achieve this." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", + "import seaborn as sns\n", + "import scipy.linalg as scl\n", + "from sklearn.model_selection import train_test_split\n", + "import sklearn.linear_model as skl\n", + "import tqdm\n", + "sns.set(color_codes=True)\n", + "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Reformulating the problem to suit regression\n", + "\n", + "A more general form for the one-dimensional Ising model is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n", + "\\label{_auto20} \\tag{31}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we allow for interactions beyond the nearest neighbors and a more\n", + "adaptive coupling matrix. This latter expression can be formulated as\n", + "a matrix-product on the form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = X J,\n", + "\\label{_auto21} \\tag{32}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n", + "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", + "with the form utilized in linear regression, viz." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " y = X\\omega + \\epsilon,\n", + "\\label{_auto22} \\tag{33}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "X = np.zeros((n, L ** 2))\n", + "for i in range(n):\n", + " X[i] = np.outer(spins[i], spins[i]).ravel()\n", + "y = energies\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n", + "\n", + "X_train_own = np.concatenate(\n", + " (np.ones(len(X_train))[:, np.newaxis], X_train),\n", + " axis=1\n", + ")\n", + "\n", + "X_test_own = np.concatenate(\n", + " (np.ones(len(X_test))[:, np.newaxis], X_test),\n", + " axis=1\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear regression\n", + "\n", + "The problem at hand is to try to fit the equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " y = f(x) + \\epsilon,\n", + "\\label{_auto23} \\tag{34}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $f(x)$ is some unknown function of the data $x$ and $\\epsilon$\n", + "is normally distributed with mean zero noise with standard deviation\n", + "$\\sigma_{\\epsilon}$. Our job is to try to find a predictor which\n", + "estimates the function $f(x)$. In linear regression we assume that we\n", + "can formulate the problem as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " y = X\\omega + \\epsilon,\n", + "\\label{_auto24} \\tag{35}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X$ and $\\omega$ are now matrices. Our job at hand is now to\n", + "find a **cost function** $C$, which we wish to minimize in order to find\n", + "the best estimate of $\\omega$.\n", + "\n", + "## Ordinary least squares\n", + "\n", + "In the ordinary least squares method we choose the cost function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(X, \\omega) = ||X\\omega - y||^2\n", + " = (X\\omega - y)^T(X\\omega - y)\n", + "\\label{_auto25} \\tag{36}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We then find the extremal point of $C$ by taking the derivative with respect to $\\omega$ and setting it to zero, i.e.," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\dfrac{\\mathrm{d}C}{\\mathrm{d}\\omega}\n", + " = 0.\n", + "\\label{_auto26} \\tag{37}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This yields the expression for $\\omega$ to be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\omega = \\frac{X^T y}{X^T X},\n", + "\\label{_auto27} \\tag{38}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which immediately imposes some requirements on $X$ as there must exist\n", + "an inverse of $X^T X$. If the expression we are modelling contains an\n", + "intercept, i.e., a constant expression we must make sure that the\n", + "first column of $X$ consists of $1$." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " return scl.inv(x.T @ x) @ (x.T @ y)\n", + "omega = get_ols_weights_naive(X_train_own, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Singular Value decomposition\n", + "Doing the inversion directly turns out to be a bad idea as the matrix\n", + "$X^TX$ is singular. An alternative approach is to use the **singular\n", + "value decomposition**. Using the definition of the Moore-Penrose\n", + "pseudoinverse we can write the equation for $\\omega$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\omega = X^{+}y,\n", + "\\label{_auto28} \\tag{39}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the pseudoinverse of $X$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " X^{+} = \\frac{X^T}{X^T X}.\n", + "\\label{_auto29} \\tag{40}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using singular value decomposition we have that $X = U\\Sigma V^T$,\n", + "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n", + "$\\omega$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\omega = V\\Sigma^{+} U^T y.\n", + "\\label{_auto30} \\tag{41}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that solving this equation by actually doing the pseudoinverse\n", + "(which is what we will do) is not a good idea as this operation scales\n", + "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n", + "general matrix. Instead, doing $QR$-factorization and solving the\n", + "linear system as an equation would reduce this down to\n", + "$\\mathcal{O}(n^2)$ operations." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " u, s, v = scl.svd(x)\n", + " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before passing in the data to the function we append a column with ones to the training data." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "omega = get_ols_weights(X_train_own,y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fitting with scikit-learn\n", + "\n", + "Next we fit a `LinearRegression`-model from Scikit-learn for comparison." + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Extracting the $J$-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "J_own = omega[1:].reshape(L, L)\n", + "J_sk = clf.coef_.reshape(L, L)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A way of looking at the coefficients in $J$ is to plot the matrices as images." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_own, **cmap_args)\n", + "plt.title(\"Home-made OLS\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_sk, **cmap_args)\n", + "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that our model for the least squares method performes close\n", + "to the benchmark from Scikit-learn. It is interesting to note that OLS\n", + "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n", + "valid matrix elements for $J$.\n", + "\n", + "## Ridge regression\n", + "\n", + "Having explored the ordinary least squares we move on to ridge\n", + "regression. In ridge regression we include a **regularizer**. This\n", + "involves a new cost function which leads to a new estimate for the\n", + "weights $\\omega$. This results in a penalized regression problem. The\n", + "cost function is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(X, \\omega; \\lambda) = ||X\\omega - y||^2 + \\lambda ||\\omega||^2\n", + " = (X\\omega - y)^T(X\\omega - y) + \\lambda \\omega^T\\omega.\n", + "\\label{_auto31} \\tag{42}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finding the extremum of this function yields the weights" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\omega(\\lambda) = \\frac{X^Ty}{X^TX + \\lambda} \\to \\frac{\\omega_{\\text{LS}}}{1 + \\lambda},\n", + "\\label{_auto32} \\tag{43}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\omega_{\\text{LS}}$ is the weights from ordinary least\n", + "squares. The last assumption assumes that $X$ is orthogonal, which it\n", + "is not. We will therefore resort to solving the equation as it stands\n", + "on the left hand side." + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray:\n", + " return x.T @ y @ scl.inv(\n", + " x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda\n", + " )\n", + "lambda = 0.1\n", + "omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda]))\n", + "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n", + "J_ridge_own = omega_ridge[1:].reshape(L, L)\n", + "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_ridge_own, **cmap_args)\n", + "plt.title(\"Home-made ridge regression\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_ridge_sk, **cmap_args)\n", + "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## LASSO regression\n", + "\n", + "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(X, \\omega; \\lambda) =\n", + " ||X\\omega - y||^2 + \\lambda ||\\omega||\n", + " = (X\\omega - y)^T(X\\omega - y) + \\lambda \\sqrt{\\omega^T\\omega}.\n", + "\\label{_auto33} \\tag{44}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n", + "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_lasso_sk, **cmap_args)\n", + "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is quite striking how LASSO breaks the symmetry of the coupling\n", + "constant as opposed to ridge and OLS. We get a sparse solution with\n", + "$J_{j, j + 1} = -1$.\n", + "\n", + "\n", + "## Performance of the different models\n", + "\n", + "In order to judge which model performs best at varying values of $\\lambda$ (for ridge and LASSO) we compute $R^2$ which is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " R^2 = 1 - \\frac{(y - \\hat{y})^2}{(y - \\bar{y})^2},\n", + "\\label{_auto34} \\tag{45}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $y$ is a vector with the true values of the energy, $\\hat{y}$ is the predicted values of $y$ from the models and $\\bar{y}$ is the mean of $\\hat{y}$." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def r_squared(y, y_hat):\n", + " return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is the same metric used by Scikit-learn for their regression models when scoring." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "y_hat = clf.predict(X_test)\n", + "r_test = r_squared(y_test, y_hat)\n", + "sk_r_test = clf.score(X_test, y_test)\n", + "\n", + "assert abs(r_test - sk_r_test) < 1e-2" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Performance as function of the regularization parameter\n", + "\n", + "We see how the different models perform for a different set of values for $\\lambda$." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "lambdas = np.logspace(-4, 5, 10)\n", + "\n", + "train_errors = {\n", + " \"ols_own\": np.zeros(lambdas.size),\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_own\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "test_errors = {\n", + " \"ols_own\": np.zeros(lambdas.size),\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_own\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "plot_counter = 1\n", + "\n", + "fig = plt.figure(figsize=(32, 54))\n", + "\n", + "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n", + " omega = get_ols_weights(X_train_own, y_train)\n", + " y_hat_train = X_train_own @ omega\n", + " y_hat_test = X_test_own @ omega\n", + "\n", + " train_errors[\"ols_own\"][i] = r_squared(y_train, y_hat_train)\n", + " test_errors[\"ols_own\"][i] = r_squared(y_test, y_hat_test)\n", + "\n", + " plt.subplot(10, 5, plot_counter)\n", + " plt.imshow(omega[1:].reshape(L, L), **cmap_args)\n", + " plt.title(\"Home made OLS\")\n", + " plot_counter += 1\n", + "\n", + " omega = get_ridge_weights(X_train_own, y_train, _lambda)\n", + " y_hat_train = X_train_own @ omega\n", + " y_hat_test = X_test_own @ omega\n", + "\n", + " train_errors[\"ridge_own\"][i] = r_squared(y_train, y_hat_train)\n", + " test_errors[\"ridge_own\"][i] = r_squared(y_test, y_hat_test)\n", + "\n", + " plt.subplot(10, 5, plot_counter)\n", + " plt.imshow(omega[1:].reshape(L, L), **cmap_args)\n", + " plt.title(r\"Home made ridge, $\\lambda = %.4f$\" % _lambda)\n", + " plot_counter += 1\n", + "\n", + " for key, method in zip(\n", + " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n", + " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n", + " ):\n", + " method = method.fit(X_train, y_train)\n", + "\n", + " train_errors[key][i] = method.score(X_train, y_train)\n", + " test_errors[key][i] = method.score(X_test, y_test)\n", + "\n", + " omega = method.coef_.reshape(L, L)\n", + "\n", + " plt.subplot(10, 5, plot_counter)\n", + " plt.imshow(omega, **cmap_args)\n", + " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n", + " plot_counter += 1\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see that LASSO quite fast reaches a good solution for low\n", + "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n", + "much. Ridge is more stable over a larger range of values for\n", + "$\\lambda$, but eventually also fades away.\n", + "\n", + "## Finding the optimal value of $\\lambda$\n", + "\n", + "To determine which value of $\\lambda$ is best we plot the accuracy of\n", + "the models when predicting the training and the testing set. We expect\n", + "the accuracy of the training set to be quite good, but if the accuracy\n", + "of the testing set is much lower this tells us that we might be\n", + "subject to an overfit model. The ideal scenario is an accuracy on the\n", + "testing set that is close to the accuracy of the training set." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "\n", + "colors = {\n", + " \"ols_own\": \"b\",\n", + " \"ridge_own\": \"g\",\n", + " \"ols_sk\": \"r\",\n", + " \"ridge_sk\": \"y\",\n", + " \"lasso_sk\": \"c\"\n", + "}\n", + "\n", + "for key in train_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " train_errors[key],\n", + " colors[key],\n", + " label=\"Train {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "\n", + "for key in test_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " test_errors[key],\n", + " colors[key] + \"--\",\n", + " label=\"Test {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "#plt.semilogx(lambdas, train_errors[\"ols_own\"], label=\"Train (OLS own)\")\n", + "#plt.semilogx(lambdas, test_errors[\"ols_own\"], label=\"Test (OLS own)\")\n", + "\n", + "plt.legend(loc=\"best\", fontsize=18)\n", + "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n", + "plt.ylabel(r\"$R^2$\", fontsize=18)\n", + "plt.tick_params(labelsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n", + "achieve a very good accuracy on the test set. This by far surpases the\n", + "other models for all values of $\\lambda$." ] } ], diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz index 424c6b90fd81e5eac7a322984e503b44fd04042c..0a0242504c311398d0ad42ebaa86c8eca7b3acc3 100644 GIT binary patch literal 211 zcmb2|=3qE`XkRn~^V{=hvzQD;+7hqF9o^ZaI8#1WyyW2`g-g;MJ|&O*lO}VVyIbzI zFm|i8=F<0@e_A|Ip0hb}pHAF#*L_bk)26Ay?V-Dxzn=V+U+tW`{rAK9$2C9AtUnU{eA1e0weOn5|H;M1gpPOBu#li#%Ii-oCB% zA}dW@>2v+;!>a8C?~A^D+&LrlrF!o?H_g!LOL;8UuFhWd$SnS;tKQF)Akoyh(=U5n zF$hr(JRkk?q0D5yzglm%7hc>EUtOEK`RC*Lw>68-n{Uj@o$@n$der+PKU1#Vd!LfI z?ckA&y{F%7EY8l{(|LF^OR?4M1ee`@C6{kpDgRPe!&4{yMKSGhxI7C3GWdUl_4usG J4;VBU7yvX{Wtji~ diff --git a/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf b/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf index c84b672da70b1d53e68cde67c5fa9f4d077ff052..b80d0630e68c46925732e8d462628d29327d8345 100644 GIT binary patch delta 250982 zcmZsCWmFwYvv&5z-QC^YU4y&3dvN!SI|O&PU?E6w*AU#@J-8G6%Q^SH=RK_Nu35W( zJo8j_b@lG*ny#5IH)JKhC~E=%kR}|a<`yz(02u&r{TF}{0Ho}a&57*aWmq7Y9%*Jx z5jZQ@qTrVklhZ%$_BeSLXkB4)|98wM#t?%oL!~q zO9>$pcm>0eJp`VfkSPo+CoGy7wEE3nA^X*VCW4F%8kCZhNT`(-w|Gu?*et+LlSrL;n%=$9i0NP*-52o-fqA=|G+=`K<>G&-6fKnSy(thme5Sh=oOOA#9(y1rFZA5 zTmi$Tz#jqvpYEL`75TCibh6`ovKcvf^pBjqm-k(l52)R zf)0k1EjnBD5-3N;u8v#0YcOcIxtm=Ov-J7be?6&CbT(l>OT`7ax}1rcjS|Armf z&$Svr&m{^lny0R7A?uR-R?iq+Ya;JzFhDxo_|#|{Oj>!iN*Y^T{|u_FipuYxt;3!! zD<3}E3Qwn1n!@OWAO1ZthW7(V2k~*ZDw$-|?s5{{PIq2C;G|amd!?yVVJY;H%e~`0 zPvuqNX8||yRH?cnZIc=?+I`K7r4nZFiC$;A)%CAN)5YBLDv+Y!Dht>v;_{l@wlGsn zpxY?P$I>z#HPeuK*;r6f{8BO$e&6@OtQi|TVR2@l0ILeT;|@Reb8`SNofsRw+rg|8 zFiJ!&XYKKeQVUxeWH9iBJ{4d3EUH3mUFYL%M-s31Tg1v8eb-;G&qvFwu3B4t#`)-bF7{3G96mv9!66I1 zR^f@JmWxAKXf8>f>$J(SYis&sJ}st~P0YDPeteBr_bX#Oj`BjCGVe-erOD_~aG^=s z;GOo8SEx zvku;CfE)|3h#CA#j|%7^5FL+9_;4}xnxuGKqcJ*lYpf;KNJ%kO5d!4f80zMmn0W;? 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However, if a method has high variance then small changes in the training data can result in large changes in the model. In general, more flexible statistical methods have higher variance. + +!split +===== The one-dimensional Ising model, project 2 ===== + +The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant $J$ is given by + +!bt +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\end{align} +!et +where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the low temperature limit there is no phase transition. + +We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies. + + +!bc pycod +L = 40 +n = int(1e4) + +spins = np.random.choice([-1, 1], size=(n, L)) +J = 1.0 + +energies = np.zeros(n) + +for i in range(n): + energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1)) +!ec + + + +!split +===== Example: The one-dimensional Ising model ===== + + +Here we use linear (ordinary least squares), ridge and LASSO +regression to predict the energy in the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use the linear regression models to fit a value for +the coupling constant to achieve this. +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.axes_grid1 import make_axes_locatable +import seaborn as sns +import scipy.linalg as scl +from sklearn.model_selection import train_test_split +import sklearn.linear_model as skl +import tqdm +sns.set(color_codes=True) +cmap_args=dict(vmin=-1., vmax=1., cmap='seismic') +!ec + +!split +===== Reformulating the problem to suit regression ===== + +A more general form for the one-dimensional Ising model is + +!bt +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\end{align} +!et + +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form +!bt +\begin{align} + H = X J, +\end{align} +!et + +where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the +elements $-J_{jk}$. This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +!bt +\begin{align} + y = X\omega + \epsilon, +\end{align} +!et + + + +!bc pycod +X = np.zeros((n, L ** 2)) +for i in range(n): + X[i] = np.outer(spins[i], spins[i]).ravel() +y = energies +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96) + +X_train_own = np.concatenate( + (np.ones(len(X_train))[:, np.newaxis], X_train), + axis=1 +) + +X_test_own = np.concatenate( + (np.ones(len(X_test))[:, np.newaxis], X_test), + axis=1 +) +!ec + +!split +===== Linear regression ===== + +The problem at hand is to try to fit the equation +!bt +\begin{align} + y = f(x) + \epsilon, +\end{align} +!et + +where $f(x)$ is some unknown function of the data $x$ and $\epsilon$ +is normally distributed with mean zero noise with standard deviation +$\sigma_{\epsilon}$. Our job is to try to find a predictor which +estimates the function $f(x)$. In linear regression we assume that we +can formulate the problem as + +!bt +\begin{align} + y = X\omega + \epsilon, +\end{align} +!et + +where $X$ and $\omega$ are now matrices. Our job at hand is now to +find a _cost function_ $C$, which we wish to minimize in order to find +the best estimate of $\omega$. + +!split +===== Ordinary least squares ===== + +In the ordinary least squares method we choose the cost function +!bt +\begin{align} + C(X, \omega) = ||X\omega - y||^2 + = (X\omega - y)^T(X\omega - y) +\end{align} +!et +We then find the extremal point of $C$ by taking the derivative with respect to $\omega$ and setting it to zero, i.e., + +!bt +\begin{align} + \dfrac{\mathrm{d}C}{\mathrm{d}\omega} + = 0. +\end{align} +!et +This yields the expression for $\omega$ to be +!bt +\begin{align} + \omega = \frac{X^T y}{X^T X}, +\end{align} +!et + +which immediately imposes some requirements on $X$ as there must exist +an inverse of $X^T X$. If the expression we are modelling contains an +intercept, i.e., a constant expression we must make sure that the +first column of $X$ consists of $1$. + + +!bc pycod +def get_ols_weights_naive(x: np.ndarray, y: np.ndarray) -> np.ndarray: + return scl.inv(x.T @ x) @ (x.T @ y) +omega = get_ols_weights_naive(X_train_own, y_train) +!ec + + +!split +===== Singular Value decomposition ===== +Doing the inversion directly turns out to be a bad idea as the matrix +$X^TX$ is singular. An alternative approach is to use the _singular +value decomposition_. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for $\omega$ as + +!bt +\begin{align} + \omega = X^{+}y, +\end{align} +!et +where the pseudoinverse of $X$ is given by +!bt +\begin{align} + X^{+} = \frac{X^T}{X^T X}. +\end{align} +!et + +Using singular value decomposition we have that $X = U\Sigma V^T$, +where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for +$\omega$ to +!bt +\begin{align} + \omega = V\Sigma^{+} U^T y. +\end{align} +!et + +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a +general matrix. Instead, doing $QR$-factorization and solving the +linear system as an equation would reduce this down to +$\mathcal{O}(n^2)$ operations. + + +!bc pycod +def get_ols_weights(x: np.ndarray, y: np.ndarray) -> np.ndarray: + u, s, v = scl.svd(x) + return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y +!ec +Before passing in the data to the function we append a column with ones to the training data. + + +!bc pycod +omega = get_ols_weights(X_train_own,y_train) +!ec + +!split +===== Fitting with scikit-learn ===== + +Next we fit a `LinearRegression`-model from Scikit-learn for comparison. + +!bc pycod +clf = skl.LinearRegression().fit(X_train, y_train) +!ec + +Extracting the $J$-matrix from both our own method and the Scikit-learn model where we make sure to remove the intercept. + + +!bc pycod +J_own = omega[1:].reshape(L, L) +J_sk = clf.coef_.reshape(L, L) +!ec + +A way of looking at the coefficients in $J$ is to plot the matrices as images. + + +!bc pycod +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_own, **cmap_args) +plt.title("Home-made OLS", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_sk, **cmap_args) +plt.title("LinearRegression from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) +plt.show() +!ec + +We can see that our model for the least squares method performes close +to the benchmark from Scikit-learn. It is interesting to note that OLS +considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as +valid matrix elements for $J$. + +!split +===== Ridge regression ===== + +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a _regularizer_. This +involves a new cost function which leads to a new estimate for the +weights $\omega$. This results in a penalized regression problem. The +cost function is given by + +!bt +\begin{align} + C(X, \omega; \lambda) = ||X\omega - y||^2 + \lambda ||\omega||^2 + = (X\omega - y)^T(X\omega - y) + \lambda \omega^T\omega. +\end{align} +!et +Finding the extremum of this function yields the weights + +!bt +\begin{align} + \omega(\lambda) = \frac{X^Ty}{X^TX + \lambda} \to \frac{\omega_{\text{LS}}}{1 + \lambda}, +\end{align} +!et + +where $\omega_{\text{LS}}$ is the weights from ordinary least +squares. The last assumption assumes that $X$ is orthogonal, which it +is not. We will therefore resort to solving the equation as it stands +on the left hand side. + + +!bc pycod +def get_ridge_weights(x: np.ndarray, y: np.ndarray, _lambda: float) -> np.ndarray: + return x.T @ y @ scl.inv( + x.T @ x + np.eye(x.shape[1], x.shape[1]) * _lambda + ) +lambda = 0.1 +omega_ridge = get_ridge_weights(X_train_own, y_train, np.array([_lambda])) +clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train) +J_ridge_own = omega_ridge[1:].reshape(L, L) +J_ridge_sk = clf_ridge.coef_.reshape(L, L) +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_ridge_own, **cmap_args) +plt.title("Home-made ridge regression", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_ridge_sk, **cmap_args) +plt.title("Ridge from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +plt.show() +!ec + +!split +===== LASSO regression ===== + +In the _Least Absolute Shrinkage and Selection Operator_ (LASSO)-method we get a third cost function. +!bt +\begin{align} + C(X, \omega; \lambda) = + ||X\omega - y||^2 + \lambda ||\omega|| + = (X\omega - y)^T(X\omega - y) + \lambda \sqrt{\omega^T\omega}. +\end{align} +!et +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-learn. + +!bc pycod +clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train) +J_lasso_sk = clf_lasso.coef_.reshape(L, L) +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_lasso_sk, **cmap_args) +plt.title("Lasso from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +plt.show() +!ec + +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +$J_{j, j + 1} = -1$. + + +!split +===== Performance of the different models ===== + +In order to judge which model performs best at varying values of $\lambda$ (for ridge and LASSO) we compute $R^2$ which is given by + +!bt +\begin{align} + R^2 = 1 - \frac{(y - \hat{y})^2}{(y - \bar{y})^2}, +\end{align} +!et +where $y$ is a vector with the true values of the energy, $\hat{y}$ is the predicted values of $y$ from the models and $\bar{y}$ is the mean of $\hat{y}$. + + +!bc pycod +def r_squared(y, y_hat): + return 1 - np.sum((y - y_hat) ** 2) / np.sum((y - np.mean(y_hat)) ** 2) +!ec + +This is the same metric used by Scikit-learn for their regression models when scoring. +!bc pycod +y_hat = clf.predict(X_test) +r_test = r_squared(y_test, y_hat) +sk_r_test = clf.score(X_test, y_test) + +assert abs(r_test - sk_r_test) < 1e-2 +!ec + + +!split +===== Performance as function of the regularization parameter ===== + +We see how the different models perform for a different set of values for $\lambda$. + + +!bc pycod +lambdas = np.logspace(-4, 5, 10) + +train_errors = { + "ols_own": np.zeros(lambdas.size), + "ols_sk": np.zeros(lambdas.size), + "ridge_own": np.zeros(lambdas.size), + "ridge_sk": np.zeros(lambdas.size), + "lasso_sk": np.zeros(lambdas.size) +} + +test_errors = { + "ols_own": np.zeros(lambdas.size), + "ols_sk": np.zeros(lambdas.size), + "ridge_own": np.zeros(lambdas.size), + "ridge_sk": np.zeros(lambdas.size), + "lasso_sk": np.zeros(lambdas.size) +} + +plot_counter = 1 + +fig = plt.figure(figsize=(32, 54)) + +for i, _lambda in enumerate(tqdm.tqdm(lambdas)): + omega = get_ols_weights(X_train_own, y_train) + y_hat_train = X_train_own @ omega + y_hat_test = X_test_own @ omega + + train_errors["ols_own"][i] = r_squared(y_train, y_hat_train) + test_errors["ols_own"][i] = r_squared(y_test, y_hat_test) + + plt.subplot(10, 5, plot_counter) + plt.imshow(omega[1:].reshape(L, L), **cmap_args) + plt.title("Home made OLS") + plot_counter += 1 + + omega = get_ridge_weights(X_train_own, y_train, _lambda) + y_hat_train = X_train_own @ omega + y_hat_test = X_test_own @ omega + + train_errors["ridge_own"][i] = r_squared(y_train, y_hat_train) + test_errors["ridge_own"][i] = r_squared(y_test, y_hat_test) + + plt.subplot(10, 5, plot_counter) + plt.imshow(omega[1:].reshape(L, L), **cmap_args) + plt.title(r"Home made ridge, $\lambda = %.4f$" % _lambda) + plot_counter += 1 + + for key, method in zip( + ["ols_sk", "ridge_sk", "lasso_sk"], + [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)] + ): + method = method.fit(X_train, y_train) + + train_errors[key][i] = method.score(X_train, y_train) + test_errors[key][i] = method.score(X_test, y_test) + + omega = method.coef_.reshape(L, L) + + plt.subplot(10, 5, plot_counter) + plt.imshow(omega, **cmap_args) + plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda)) + plot_counter += 1 + +plt.show() +!ec + +We can see that LASSO quite fast reaches a good solution for low +values of $\lambda$, but will "wither" when we increase $\lambda$ too +much. Ridge is more stable over a larger range of values for +$\lambda$, but eventually also fades away. + +!split +===== Finding the optimal value of $\lambda$ ===== + +To determine which value of $\lambda$ is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + + +!bc pycod +fig = plt.figure(figsize=(20, 14)) + +colors = { + "ols_own": "b", + "ridge_own": "g", + "ols_sk": "r", + "ridge_sk": "y", + "lasso_sk": "c" +} + +for key in train_errors: + plt.semilogx( + lambdas, + train_errors[key], + colors[key], + label="Train {0}".format(key), + linewidth=4.0 + ) + +for key in test_errors: + plt.semilogx( + lambdas, + test_errors[key], + colors[key] + "--", + label="Test {0}".format(key), + linewidth=4.0 + ) +#plt.semilogx(lambdas, train_errors["ols_own"], label="Train (OLS own)") +#plt.semilogx(lambdas, test_errors["ols_own"], label="Test (OLS own)") + +plt.legend(loc="best", fontsize=18) +plt.xlabel(r"$\lambda$", fontsize=18) +plt.ylabel(r"$R^2$", fontsize=18) +plt.tick_params(labelsize=18) +plt.show() +!ec + +From the above figure we can see that LASSO with $\lambda = 10^{-2}$ +achieve a very good accuracy on the test set. This by far surpases the +other models for all values of $\lambda$.

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