From 752205a9591c0420f9406cc47ed70ac91923dc76 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 16 Sep 2022 05:33:02 +0200 Subject: [PATCH] took out a slide --- doc/pub/week37/html/._week37-bs000.html | 15 +- doc/pub/week37/html/._week37-bs001.html | 15 +- doc/pub/week37/html/._week37-bs002.html | 15 +- doc/pub/week37/html/._week37-bs003.html | 15 +- doc/pub/week37/html/._week37-bs004.html | 15 +- doc/pub/week37/html/._week37-bs005.html | 15 +- doc/pub/week37/html/._week37-bs006.html | 15 +- doc/pub/week37/html/._week37-bs007.html | 15 +- doc/pub/week37/html/._week37-bs008.html | 15 +- doc/pub/week37/html/._week37-bs009.html | 15 +- doc/pub/week37/html/._week37-bs010.html | 15 +- doc/pub/week37/html/._week37-bs011.html | 15 +- doc/pub/week37/html/._week37-bs012.html | 15 +- doc/pub/week37/html/._week37-bs013.html | 15 +- doc/pub/week37/html/._week37-bs014.html | 15 +- doc/pub/week37/html/._week37-bs015.html | 15 +- doc/pub/week37/html/._week37-bs016.html | 15 +- doc/pub/week37/html/._week37-bs017.html | 15 +- doc/pub/week37/html/._week37-bs018.html | 15 +- doc/pub/week37/html/._week37-bs019.html | 15 +- doc/pub/week37/html/._week37-bs020.html | 15 +- doc/pub/week37/html/._week37-bs021.html | 15 +- doc/pub/week37/html/._week37-bs022.html | 15 +- doc/pub/week37/html/._week37-bs023.html | 15 +- doc/pub/week37/html/._week37-bs024.html | 15 +- doc/pub/week37/html/._week37-bs025.html | 15 +- doc/pub/week37/html/._week37-bs026.html | 15 +- doc/pub/week37/html/._week37-bs027.html | 15 +- doc/pub/week37/html/._week37-bs028.html | 15 +- doc/pub/week37/html/._week37-bs029.html | 15 +- doc/pub/week37/html/._week37-bs030.html | 15 +- doc/pub/week37/html/._week37-bs031.html | 15 +- doc/pub/week37/html/._week37-bs032.html | 15 +- doc/pub/week37/html/._week37-bs033.html | 15 +- doc/pub/week37/html/._week37-bs034.html | 15 +- doc/pub/week37/html/._week37-bs035.html | 15 +- doc/pub/week37/html/._week37-bs036.html | 15 +- doc/pub/week37/html/._week37-bs037.html | 15 +- doc/pub/week37/html/._week37-bs038.html | 15 +- doc/pub/week37/html/._week37-bs039.html | 14 +- doc/pub/week37/html/._week37-bs040.html | 14 +- doc/pub/week37/html/._week37-bs041.html | 14 +- doc/pub/week37/html/._week37-bs042.html | 14 +- doc/pub/week37/html/._week37-bs043.html | 14 +- doc/pub/week37/html/._week37-bs044.html | 56 +-- doc/pub/week37/html/._week37-bs045.html | 141 ++++++-- doc/pub/week37/html/._week37-bs046.html | 153 ++++----- doc/pub/week37/html/._week37-bs047.html | 57 +-- doc/pub/week37/html/week37-bs.html | 15 +- doc/pub/week37/html/week37-reveal.html | 35 -- doc/pub/week37/html/week37-solarized.html | 33 -- doc/pub/week37/html/week37.html | 33 -- doc/pub/week37/ipynb/ipynb-week37-src.tar.gz | Bin 1022692 -> 1022692 bytes doc/pub/week37/ipynb/week37.ipynb | 344 ++++++++----------- doc/src/week37/week37.do.txt | 26 -- 55 files changed, 584 insertions(+), 964 deletions(-) diff --git a/doc/pub/week37/html/._week37-bs000.html b/doc/pub/week37/html/._week37-bs000.html index da2d4f749..909297b04 100644 --- a/doc/pub/week37/html/._week37-bs000.html +++ b/doc/pub/week37/html/._week37-bs000.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week37/html/._week37-bs001.html b/doc/pub/week37/html/._week37-bs001.html index 9b7db5f9f..6ee2bea7f 100644 --- a/doc/pub/week37/html/._week37-bs001.html +++ b/doc/pub/week37/html/._week37-bs001.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -298,7 +293,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week37/html/._week37-bs002.html b/doc/pub/week37/html/._week37-bs002.html index 308a0625f..a3ea46dde 100644 --- a/doc/pub/week37/html/._week37-bs002.html +++ b/doc/pub/week37/html/._week37-bs002.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -285,7 +280,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week37/html/._week37-bs003.html b/doc/pub/week37/html/._week37-bs003.html index 1d30f26fa..974b6bd63 100644 --- a/doc/pub/week37/html/._week37-bs003.html +++ b/doc/pub/week37/html/._week37-bs003.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -306,7 +301,7 @@ $$
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    • diff --git a/doc/pub/week37/html/._week37-bs004.html b/doc/pub/week37/html/._week37-bs004.html index b7811b617..1d2b0bf1b 100644 --- a/doc/pub/week37/html/._week37-bs004.html +++ b/doc/pub/week37/html/._week37-bs004.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -318,7 +313,7 @@ $$
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    • diff --git a/doc/pub/week37/html/._week37-bs005.html b/doc/pub/week37/html/._week37-bs005.html index 7d317a45b..af20d6f2a 100644 --- a/doc/pub/week37/html/._week37-bs005.html +++ b/doc/pub/week37/html/._week37-bs005.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -313,7 +308,7 @@ is equivalent to the maximization/minimization of the function itself.
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    • diff --git a/doc/pub/week37/html/._week37-bs006.html b/doc/pub/week37/html/._week37-bs006.html index f1b195271..7664c2510 100644 --- a/doc/pub/week37/html/._week37-bs006.html +++ b/doc/pub/week37/html/._week37-bs006.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -312,7 +307,7 @@ $$
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    • diff --git a/doc/pub/week37/html/._week37-bs007.html b/doc/pub/week37/html/._week37-bs007.html index d21af49a4..91fc0f5ab 100644 --- a/doc/pub/week37/html/._week37-bs007.html +++ b/doc/pub/week37/html/._week37-bs007.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -303,7 +298,7 @@ $$
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs008.html b/doc/pub/week37/html/._week37-bs008.html index d96bf602d..c2f5d2931 100644 --- a/doc/pub/week37/html/._week37-bs008.html +++ b/doc/pub/week37/html/._week37-bs008.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -299,7 +294,7 @@ necesseraly normalized and is normally called the likelihood function.
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    • diff --git a/doc/pub/week37/html/._week37-bs009.html b/doc/pub/week37/html/._week37-bs009.html index 3dee3d9bc..356569c67 100644 --- a/doc/pub/week37/html/._week37-bs009.html +++ b/doc/pub/week37/html/._week37-bs009.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -388,7 +383,7 @@ plt.show()
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    • diff --git a/doc/pub/week37/html/._week37-bs010.html b/doc/pub/week37/html/._week37-bs010.html index a40f7d770..8a289e682 100644 --- a/doc/pub/week37/html/._week37-bs010.html +++ b/doc/pub/week37/html/._week37-bs010.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -376,7 +371,7 @@ lambdas = np.19
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    • diff --git a/doc/pub/week37/html/._week37-bs011.html b/doc/pub/week37/html/._week37-bs011.html index 322b9f459..3164dd10b 100644 --- a/doc/pub/week37/html/._week37-bs011.html +++ b/doc/pub/week37/html/._week37-bs011.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -318,7 +313,7 @@ $$
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs012.html b/doc/pub/week37/html/._week37-bs012.html index b1904a778..c3c9ea747 100644 --- a/doc/pub/week37/html/._week37-bs012.html +++ b/doc/pub/week37/html/._week37-bs012.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -327,7 +322,7 @@ $$
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs013.html b/doc/pub/week37/html/._week37-bs013.html index b501aeee3..21f70cb54 100644 --- a/doc/pub/week37/html/._week37-bs013.html +++ b/doc/pub/week37/html/._week37-bs013.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -321,7 +316,7 @@ $$
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    • diff --git a/doc/pub/week37/html/._week37-bs014.html b/doc/pub/week37/html/._week37-bs014.html index 52f9d89c7..aa425b047 100644 --- a/doc/pub/week37/html/._week37-bs014.html +++ b/doc/pub/week37/html/._week37-bs014.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -303,7 +298,7 @@ our regression analysis. In what follows we will
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    • diff --git a/doc/pub/week37/html/._week37-bs015.html b/doc/pub/week37/html/._week37-bs015.html index 66865e6eb..6f56339b8 100644 --- a/doc/pub/week37/html/._week37-bs015.html +++ b/doc/pub/week37/html/._week37-bs015.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -319,7 +314,7 @@ cross-validation and the bootstrap method.
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    • diff --git a/doc/pub/week37/html/._week37-bs016.html b/doc/pub/week37/html/._week37-bs016.html index 65d5a43b2..642b02308 100644 --- a/doc/pub/week37/html/._week37-bs016.html +++ b/doc/pub/week37/html/._week37-bs016.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -316,7 +311,7 @@ bootstrap is widely used.
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    • diff --git a/doc/pub/week37/html/._week37-bs017.html b/doc/pub/week37/html/._week37-bs017.html index 3a5a8bb66..306498b98 100644 --- a/doc/pub/week37/html/._week37-bs017.html +++ b/doc/pub/week37/html/._week37-bs017.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -305,7 +300,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week37/html/._week37-bs018.html b/doc/pub/week37/html/._week37-bs018.html index 90da25c1d..ebf599d08 100644 --- a/doc/pub/week37/html/._week37-bs018.html +++ b/doc/pub/week37/html/._week37-bs018.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week37/html/._week37-bs019.html b/doc/pub/week37/html/._week37-bs019.html index 18f0df998..3083ef510 100644 --- a/doc/pub/week37/html/._week37-bs019.html +++ b/doc/pub/week37/html/._week37-bs019.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -316,7 +311,7 @@ training error reaches a saturation.
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    • diff --git a/doc/pub/week37/html/._week37-bs020.html b/doc/pub/week37/html/._week37-bs020.html index 718f0586a..7c5f9b155 100644 --- a/doc/pub/week37/html/._week37-bs020.html +++ b/doc/pub/week37/html/._week37-bs020.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -311,7 +306,7 @@ need for bootstrapping.
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    • diff --git a/doc/pub/week37/html/._week37-bs021.html b/doc/pub/week37/html/._week37-bs021.html index a38c849bf..bfd4a531d 100644 --- a/doc/pub/week37/html/._week37-bs021.html +++ b/doc/pub/week37/html/._week37-bs021.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -307,7 +302,7 @@ number \( i \) is left out. Using this notation, define
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    • diff --git a/doc/pub/week37/html/._week37-bs022.html b/doc/pub/week37/html/._week37-bs022.html index 960cf3c48..90e56fb54 100644 --- a/doc/pub/week37/html/._week37-bs022.html +++ b/doc/pub/week37/html/._week37-bs022.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -343,7 +338,7 @@ t = jackknife(x, stat)
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    • diff --git a/doc/pub/week37/html/._week37-bs023.html b/doc/pub/week37/html/._week37-bs023.html index 65a9c40c9..f45fef1d1 100644 --- a/doc/pub/week37/html/._week37-bs023.html +++ b/doc/pub/week37/html/._week37-bs023.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -314,7 +309,7 @@ advantages:
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    • diff --git a/doc/pub/week37/html/._week37-bs024.html b/doc/pub/week37/html/._week37-bs024.html index 0b8bbf169..b7ed19bf7 100644 --- a/doc/pub/week37/html/._week37-bs024.html +++ b/doc/pub/week37/html/._week37-bs024.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ $$
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs025.html b/doc/pub/week37/html/._week37-bs025.html index 5902378fa..ea3d4ea08 100644 --- a/doc/pub/week37/html/._week37-bs025.html +++ b/doc/pub/week37/html/._week37-bs025.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ product of individual \( p(x_i) \). The independence assumption is important in
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    • diff --git a/doc/pub/week37/html/._week37-bs026.html b/doc/pub/week37/html/._week37-bs026.html index 93c7da03e..dbca8ec0a 100644 --- a/doc/pub/week37/html/._week37-bs026.html +++ b/doc/pub/week37/html/._week37-bs026.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -318,7 +313,7 @@ $$
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs027.html b/doc/pub/week37/html/._week37-bs027.html index fc38a630e..e46dbd26d 100644 --- a/doc/pub/week37/html/._week37-bs027.html +++ b/doc/pub/week37/html/._week37-bs027.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -320,7 +315,7 @@ and \( \mu \) is also the mean of the PDF \( p(x) \).
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs028.html b/doc/pub/week37/html/._week37-bs028.html index 1fadf67e3..25bdc96a8 100644 --- a/doc/pub/week37/html/._week37-bs028.html +++ b/doc/pub/week37/html/._week37-bs028.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -332,7 +327,7 @@ finite \( m \), it is not always possible to find a closed form /analytic expres
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs029.html b/doc/pub/week37/html/._week37-bs029.html index 7ec7db238..5690f28ca 100644 --- a/doc/pub/week37/html/._week37-bs029.html +++ b/doc/pub/week37/html/._week37-bs029.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -310,7 +305,7 @@ construct a confidence interval for the estimates.
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs030.html b/doc/pub/week37/html/._week37-bs030.html index ff309ba89..f615e31ec 100644 --- a/doc/pub/week37/html/._week37-bs030.html +++ b/doc/pub/week37/html/._week37-bs030.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -317,7 +312,7 @@ Bootstrap method, why it works and various theorems related to it.
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    • diff --git a/doc/pub/week37/html/._week37-bs031.html b/doc/pub/week37/html/._week37-bs031.html index fc2ed7e73..15c603f07 100644 --- a/doc/pub/week37/html/._week37-bs031.html +++ b/doc/pub/week37/html/._week37-bs031.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -304,7 +299,7 @@ estimators.
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs032.html b/doc/pub/week37/html/._week37-bs032.html index 059afe833..f2d23ea5b 100644 --- a/doc/pub/week37/html/._week37-bs032.html +++ b/doc/pub/week37/html/._week37-bs032.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ idea is to use the relative frequency of \( \widehat{\beta}^* \)
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs033.html b/doc/pub/week37/html/._week37-bs033.html index e15f99e0c..af4c13368 100644 --- a/doc/pub/week37/html/._week37-bs033.html +++ b/doc/pub/week37/html/._week37-bs033.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -306,7 +301,7 @@ result in some asymptotic sense? The answer is yes.
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs034.html b/doc/pub/week37/html/._week37-bs034.html index 8560c1d80..9ee7fcaff 100644 --- a/doc/pub/week37/html/._week37-bs034.html +++ b/doc/pub/week37/html/._week37-bs034.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -312,7 +307,7 @@ example, if you are interested in estimating the variance of \( \widehat
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs035.html b/doc/pub/week37/html/._week37-bs035.html index ff4fccac9..fd4788e5c 100644 --- a/doc/pub/week37/html/._week37-bs035.html +++ b/doc/pub/week37/html/._week37-bs035.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -357,7 +352,7 @@ t = bootstrap(x, datapoints)
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs036.html b/doc/pub/week37/html/._week37-bs036.html index a8565777e..cbce5b422 100644 --- a/doc/pub/week37/html/._week37-bs036.html +++ b/doc/pub/week37/html/._week37-bs036.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -324,7 +319,7 @@ plt.show()
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs037.html b/doc/pub/week37/html/._week37-bs037.html index 77e1dd106..73133202a 100644 --- a/doc/pub/week37/html/._week37-bs037.html +++ b/doc/pub/week37/html/._week37-bs037.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -350,7 +345,7 @@ $$
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs038.html b/doc/pub/week37/html/._week37-bs038.html index 34db4de1c..8cabf4e2e 100644 --- a/doc/pub/week37/html/._week37-bs038.html +++ b/doc/pub/week37/html/._week37-bs038.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -298,8 +293,6 @@ MathJax.Hub.Config({
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    • -
  • ...
    • -
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs039.html b/doc/pub/week37/html/._week37-bs039.html index 948e27b74..58a2b072e 100644 --- a/doc/pub/week37/html/._week37-bs039.html +++ b/doc/pub/week37/html/._week37-bs039.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -367,7 +362,6 @@ plt.show()
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    • -
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs040.html b/doc/pub/week37/html/._week37-bs040.html index 4007d3fa9..a574f0f7b 100644 --- a/doc/pub/week37/html/._week37-bs040.html +++ b/doc/pub/week37/html/._week37-bs040.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -358,7 +353,6 @@ plt.show()
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    • -
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs041.html b/doc/pub/week37/html/._week37-bs041.html index 98989f4c1..020a7450b 100644 --- a/doc/pub/week37/html/._week37-bs041.html +++ b/doc/pub/week37/html/._week37-bs041.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -320,7 +315,6 @@ flexible statistical methods have higher variance.
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs042.html b/doc/pub/week37/html/._week37-bs042.html index 93c733270..c11e44bc8 100644 --- a/doc/pub/week37/html/._week37-bs042.html +++ b/doc/pub/week37/html/._week37-bs042.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -381,7 +376,6 @@ plt.show()
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    • -
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs043.html b/doc/pub/week37/html/._week37-bs043.html index 92af9f171..3bf8f8d3e 100644 --- a/doc/pub/week37/html/._week37-bs043.html +++ b/doc/pub/week37/html/._week37-bs043.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -303,7 +298,6 @@ cross-validation (LOOCV).
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs044.html b/doc/pub/week37/html/._week37-bs044.html index 25899c0f8..dbe2a30d2 100644 --- a/doc/pub/week37/html/._week37-bs044.html +++ b/doc/pub/week37/html/._week37-bs044.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -265,35 +260,23 @@ MathJax.Hub.Config({

     

     

     

    - -

    How to set up the cross-validation for Ridge and/or Lasso

    - -
      -
    • Define a range of interest for the penalty parameter.
      • -
    • Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
      • -
    • Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
      • -
    -$$ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -$$ - - -
      -
    • Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
      • -
    • Repeat the first three steps such that each sample plays the role of the test set once.
      • -
    • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
      • -
    -$$ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -$$ + +

    Cross-validation in brief

    +

    For the various values of \( k \)

    +
      +
    1. shuffle the dataset randomly.
      2. +
    2. Split the dataset into \( k \) groups.
      3. +
    3. For each unique group: +
        +
      1. Decide which group to use as set for test data
        2. +
      2. Take the remaining groups as a training data set
        3. +
      3. Fit a model on the training set and evaluate it on the test set
        4. +
      4. Retain the evaluation score and discard the model
        5. +
      +
    4. Summarize the model using the sample of model evaluation scores
      5. +

    diff --git a/doc/pub/week37/html/._week37-bs045.html b/doc/pub/week37/html/._week37-bs045.html index 8c1d51079..be88f20c9 100644 --- a/doc/pub/week37/html/._week37-bs045.html +++ b/doc/pub/week37/html/._week37-bs045.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -266,22 +261,121 @@ MathJax.Hub.Config({

     

     

     

    -

    Cross-validation in brief

    +

    Code Example for Cross-validation and \( k \)-fold Cross-validation

    + +

    The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.

    + + +
    +
    +
    +
    +
    + 
    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+# Generate the data.
+nsamples = 100
+x = np.random.randn(nsamples)
+y = 3*x**2 + np.random.randn(nsamples)
+
+## Cross-validation on Ridge regression using KFold only
+
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 6)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+
+# Perform the cross-validation to estimate MSE
+scores_KFold = np.zeros((nlambdas, k))
+
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    j = 0
+    for train_inds, test_inds in kfold.split(x):
+        xtrain = x[train_inds]
+        ytrain = y[train_inds]
+
+        xtest = x[test_inds]
+        ytest = y[test_inds]
+
+        Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
+        ridge.fit(Xtrain, ytrain[:, np.newaxis])
+
+        Xtest = poly.fit_transform(xtest[:, np.newaxis])
+        ypred = ridge.predict(Xtest)
+
+        scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
+
+        j += 1
+    i += 1
+
+
+estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
+
+## Cross-validation using cross_val_score from sklearn along with KFold
+
+# kfold is an instance initialized above as:
+# kfold = KFold(n_splits = k)
+
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+
+    X = poly.fit_transform(x[:, np.newaxis])
+    estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
+
+    # cross_val_score return an array containing the estimated negative mse for every fold.
+    # we have to the the mean of every array in order to get an estimate of the mse of the model
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+
+    i += 1
+
+## Plot and compare the slightly different ways to perform cross-validation
+
+plt.figure()
+
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('mse')
+
+plt.legend()
+
+plt.show()
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    -

    For the various values of \( k \)

    -
      -
    1. shuffle the dataset randomly.
      2. -
    2. Split the dataset into \( k \) groups.
      3. -
    3. For each unique group: -
        -
      1. Decide which group to use as set for test data
        2. -
      2. Take the remaining groups as a training data set
        3. -
      3. Fit a model on the training set and evaluate it on the test set
        4. -
      4. Retain the evaluation score and discard the model
        5. -
      -
    4. Summarize the model using the sample of model evaluation scores
      5. -

      @@ -299,7 +393,6 @@ MathJax.Hub.Config({
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    diff --git a/doc/pub/week37/html/._week37-bs046.html b/doc/pub/week37/html/._week37-bs046.html index 48d02a96a..da9a7aba4 100644 --- a/doc/pub/week37/html/._week37-bs046.html +++ b/doc/pub/week37/html/._week37-bs046.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -266,9 +261,8 @@ MathJax.Hub.Config({

     

     

     

    -

    Code Example for Cross-validation and \( k \)-fold Cross-validation

    +

    More examples on bootstrap and cross-validation and errors

    -

    The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.

    @@ -276,94 +270,83 @@ MathJax.Hub.Config({
    - 
    import numpy as np
+  
    # Common imports
+import os
+import numpy as np
+import pandas as pd
 import matplotlib.pyplot as plt
-from sklearn.model_selection import KFold
-from sklearn.linear_model import Ridge
-from sklearn.model_selection import cross_val_score
-from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
 
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
 
-# Generate the data.
-nsamples = 100
-x = np.random.randn(nsamples)
-y = 3*x**2 + np.random.randn(nsamples)
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
 
-## Cross-validation on Ridge regression using KFold only
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
 
-# Decide degree on polynomial to fit
-poly = PolynomialFeatures(degree = 6)
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
 
-# Decide which values of lambda to use
-nlambdas = 500
-lambdas = np.logspace(-3, 5, nlambdas)
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
 
-# Initialize a KFold instance
-k = 5
-kfold = KFold(n_splits = k)
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
-# Perform the cross-validation to estimate MSE
-scores_KFold = np.zeros((nlambdas, k))
+infile = open(data_path("EoS.csv"),'r')
 
-i = 0
-for lmb in lambdas:
-    ridge = Ridge(alpha = lmb)
-    j = 0
-    for train_inds, test_inds in kfold.split(x):
-        xtrain = x[train_inds]
-        ytrain = y[train_inds]
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
 
-        xtest = x[test_inds]
-        ytest = y[test_inds]
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
 
-        Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
-        ridge.fit(Xtrain, ytrain[:, np.newaxis])
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
 
-        Xtest = poly.fit_transform(xtest[:, np.newaxis])
-        ypred = ridge.predict(Xtest)
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=False).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
 
-        scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
-
-        j += 1
-    i += 1
-
-
-estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
-
-## Cross-validation using cross_val_score from sklearn along with KFold
-
-# kfold is an instance initialized above as:
-# kfold = KFold(n_splits = k)
-
-estimated_mse_sklearn = np.zeros(nlambdas)
-i = 0
-for lmb in lambdas:
-    ridge = Ridge(alpha = lmb)
-
-    X = poly.fit_transform(x[:, np.newaxis])
-    estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
-
-    # cross_val_score return an array containing the estimated negative mse for every fold.
-    # we have to the the mean of every array in order to get an estimate of the mse of the model
-    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
-
-    i += 1
-
-## Plot and compare the slightly different ways to perform cross-validation
-
-plt.figure()
-
-plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
-plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('mse')
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
 
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
 plt.legend()
-
 plt.show()
    @@ -380,6 +363,7 @@ plt.show()
    +

    Note that we kept the intercept column in the fitting here. This means that we need to set the intercept in the call to the Scikit-Learn function as False. Alternatively, we could have set up the design matrix \( X \) without the first column of ones.

    @@ -397,7 +381,6 @@ plt.show()

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    • diff --git a/doc/pub/week37/html/._week37-bs047.html b/doc/pub/week37/html/._week37-bs047.html index ee7a0b53f..16f0a1e98 100644 --- a/doc/pub/week37/html/._week37-bs047.html +++ b/doc/pub/week37/html/._week37-bs047.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -265,9 +260,10 @@ MathJax.Hub.Config({

     

     

     

    - -

    More examples on bootstrap and cross-validation and errors

    + +

    The same example but now with cross-validation

    +

    In this example we keep the intercept column again but add cross-validation in order to estimate the best possible value of the means squared error.

    @@ -281,9 +277,11 @@ MathJax.Hub.Config({ import pandas as pd import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.model_selection import train_test_split -from sklearn.utils import resample from sklearn.metrics import mean_squared_error +from sklearn.model_selection import KFold +from sklearn.model_selection import cross_val_score + + # Where to save the figures and data files PROJECT_ROOT_DIR = "Results" FIGURE_ID = "Results/FigureFiles" @@ -320,35 +318,22 @@ Density = EoS[& Maxpolydegree = 30 X = np.zeros((len(Density),Maxpolydegree)) X[:,0] = 1.0 -testerror = np.zeros(Maxpolydegree) -trainingerror = np.zeros(Maxpolydegree) +estimated_mse_sklearn = np.zeros(Maxpolydegree) polynomial = np.zeros(Maxpolydegree) +k =5 +kfold = KFold(n_splits = k) -trials = 100 for polydegree in range(1, Maxpolydegree): polynomial[polydegree] = polydegree for degree in range(polydegree): X[:,degree] = Density**(degree/3.0) - + OLS = LinearRegression(fit_intercept=False) # loop over trials in order to estimate the expectation value of the MSE - testerror[polydegree] = 0.0 - trainingerror[polydegree] = 0.0 - for samples in range(trials): - x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) - model = LinearRegression(fit_intercept=False).fit(x_train, y_train) - ypred = model.predict(x_train) - ytilde = model.predict(x_test) - testerror[polydegree] += mean_squared_error(y_test, ytilde) - trainingerror[polydegree] += mean_squared_error(y_train, ypred) + estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold) +#[:, np.newaxis] + estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds) - testerror[polydegree] /= trials - trainingerror[polydegree] /= trials - print("Degree of polynomial: %3d"% polynomial[polydegree]) - print("Mean squared error on training data: %.8f" % trainingerror[polydegree]) - print("Mean squared error on test data: %.8f" % testerror[polydegree]) - -plt.plot(polynomial, np.log10(trainingerror), label='Training Error') -plt.plot(polynomial, np.log10(testerror), label='Test Error') +plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error') plt.xlabel('Polynomial degree') plt.ylabel('log10[MSE]') plt.legend() @@ -368,8 +353,6 @@ plt.show()
    -

    Note that we kept the intercept column in the fitting here. This means that we need to set the intercept in the call to the Scikit-Learn function as False. Alternatively, we could have set up the design matrix \( X \) without the first column of ones.

    -

      @@ -385,8 +368,6 @@ plt.show()
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    diff --git a/doc/pub/week37/html/week37-bs.html b/doc/pub/week37/html/week37-bs.html index da2d4f749..909297b04 100644 --- a/doc/pub/week37/html/week37-bs.html +++ b/doc/pub/week37/html/week37-bs.html @@ -152,10 +152,6 @@ doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=d 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -250,11 +246,10 @@ MathJax.Hub.Config({
  • Summing up
  • Another Example from Scikit-Learn's Repository
  • Various steps in cross-validation
    • -
  • How to set up the cross-validation for Ridge and/or Lasso
    • -
  • Cross-validation in brief
    • -
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • -
  • More examples on bootstrap and cross-validation and errors
    • -
  • The same example but now with cross-validation
    • +
  • Cross-validation in brief
    • +
  • Code Example for Cross-validation and \( k \)-fold Cross-validation
    • +
  • More examples on bootstrap and cross-validation and errors
    • +
  • The same example but now with cross-validation
    • @@ -309,7 +304,7 @@ MathJax.Hub.Config({
  • 9
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/week37-reveal.html b/doc/pub/week37/html/week37-reveal.html index 605886141..b70a1e09a 100644 --- a/doc/pub/week37/html/week37-reveal.html +++ b/doc/pub/week37/html/week37-reveal.html @@ -1721,41 +1721,6 @@ cross-validation (LOOCV).

    -
    -

    How to set up the cross-validation for Ridge and/or Lasso

    - -
      -

    • Define a range of interest for the penalty parameter.
      • -

    • Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
      • -

    • Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
      • -
    -

    -

     
    -$$ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -$$ -

     
    - - -

      -

    • Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
      • -

    • Repeat the first three steps such that each sample plays the role of the test set once.
      • -

    • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
      • -
    -

    -

     
    -$$ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -$$ -

     
    -

    -

    Cross-validation in brief

    diff --git a/doc/pub/week37/html/week37-solarized.html b/doc/pub/week37/html/week37-solarized.html index 13b2d648a..fbba8964d 100644 --- a/doc/pub/week37/html/week37-solarized.html +++ b/doc/pub/week37/html/week37-solarized.html @@ -179,10 +179,6 @@ div.toc p,a { 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -1643,35 +1639,6 @@ choosing \( k=n \). This particular case is referred to as leave-one-out cross-validation (LOOCV).

    - -

    How to set up the cross-validation for Ridge and/or Lasso

    - -
      -
    • Define a range of interest for the penalty parameter.
      • -
    • Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
      • -
    • Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
      • -
    -$$ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -$$ - - -
      -
    • Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
      • -
    • Repeat the first three steps such that each sample plays the role of the test set once.
      • -
    • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
      • -
    -$$ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -$$ - -









    Cross-validation in brief

    diff --git a/doc/pub/week37/html/week37.html b/doc/pub/week37/html/week37.html index b32c6c36c..851920ee8 100644 --- a/doc/pub/week37/html/week37.html +++ b/doc/pub/week37/html/week37.html @@ -256,10 +256,6 @@ div.toc p,a { 2, None, 'various-steps-in-cross-validation'), - ('How to set up the cross-validation for Ridge and/or Lasso', - 2, - None, - 'how-to-set-up-the-cross-validation-for-ridge-and-or-lasso'), ('Cross-validation in brief', 2, None, @@ -1720,35 +1716,6 @@ choosing \( k=n \). This particular case is referred to as leave-one-out cross-validation (LOOCV).

    - -

    How to set up the cross-validation for Ridge and/or Lasso

    - -
      -
    • Define a range of interest for the penalty parameter.
      • -
    • Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
      • -
    • Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
      • -
    -$$ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -$$ - - -
      -
    • Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
      • -
    • Repeat the first three steps such that each sample plays the role of the test set once.
      • -
    • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
      • -
    -$$ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -$$ - -









    Cross-validation in brief

    diff --git a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz index b2f81f95a2456a9086b7c7ab3e0cbc8f1e7ab4eb..ea57fbcbc6c7aa5ad42d0c2db6e5f82fe84a9286 100644 GIT binary patch delta 62 zcmWN_Hvxb!002Si6KW*jBxX>;@DmR35j#wo(EvAI$*JVTeue6IPo$PcTIr;he+CIM O$|SQavdWf;&;0>DI1q#Y delta 62 zcmWN_Hvxb!002Si6KWiQlXzqJ2?zLy9j2^!fE%ymRC40DLbbmqQcEMPbkfT|g9I66 Nl35m6Wy{3p{s1sX5Iz6^ diff --git a/doc/pub/week37/ipynb/week37.ipynb b/doc/pub/week37/ipynb/week37.ipynb index 2687a3a87..29c0793ab 100644 --- a/doc/pub/week37/ipynb/week37.ipynb +++ b/doc/pub/week37/ipynb/week37.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "3413b194", + "id": "83e91ce3", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "ecf8eb8c", + "id": "88f778c9", "metadata": { "editable": true }, @@ -29,7 +29,7 @@ }, { "cell_type": "markdown", - "id": "b873b4b3", + "id": "8cad2f0c", "metadata": { "editable": true }, @@ -54,7 +54,7 @@ }, { "cell_type": "markdown", - "id": "6f7dcfc4", + "id": "197e7446", "metadata": { "editable": true }, @@ -64,7 +64,7 @@ }, { "cell_type": "markdown", - "id": "f578dcee", + "id": "04951cec", "metadata": { "editable": true }, @@ -87,7 +87,7 @@ }, { "cell_type": "markdown", - "id": "7c52f047", + "id": "ce4380d8", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "eb27f88a", + "id": "206592ba", "metadata": { "editable": true }, @@ -112,7 +112,7 @@ }, { "cell_type": "markdown", - "id": "99bd0787", + "id": "4a0d88d0", "metadata": { "editable": true }, @@ -124,7 +124,7 @@ }, { "cell_type": "markdown", - "id": "323d053d", + "id": "e6c7ba75", "metadata": { "editable": true }, @@ -136,7 +136,7 @@ }, { "cell_type": "markdown", - "id": "c5bcb8bb", + "id": "665e0170", "metadata": { "editable": true }, @@ -148,7 +148,7 @@ }, { "cell_type": "markdown", - "id": "00c07681", + "id": "73636714", "metadata": { "editable": true }, @@ -159,7 +159,7 @@ }, { "cell_type": "markdown", - "id": "24fec3d9", + "id": "ad90bb50", "metadata": { "editable": true }, @@ -171,7 +171,7 @@ }, { "cell_type": "markdown", - "id": "dc03828d", + "id": "24493b42", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "ee74422b", + "id": "fd50e415", "metadata": { "editable": true }, @@ -194,7 +194,7 @@ }, { "cell_type": "markdown", - "id": "7dcdd285", + "id": "3d8e865e", "metadata": { "editable": true }, @@ -204,7 +204,7 @@ }, { "cell_type": "markdown", - "id": "39107290", + "id": "17f93455", "metadata": { "editable": true }, @@ -235,7 +235,7 @@ }, { "cell_type": "markdown", - "id": "378cc7e7", + "id": "9d6b2b2a", "metadata": { "editable": true }, @@ -247,7 +247,7 @@ }, { "cell_type": "markdown", - "id": "f55bf648", + "id": "03ca6196", "metadata": { "editable": true }, @@ -259,7 +259,7 @@ }, { "cell_type": "markdown", - "id": "982e45f8", + "id": "a9f157c0", "metadata": { "editable": true }, @@ -269,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "52050954", + "id": "7a0b2baa", "metadata": { "editable": true }, @@ -281,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "401762d1", + "id": "0500cb7f", "metadata": { "editable": true }, @@ -291,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "c39377bd", + "id": "98951cc5", "metadata": { "editable": true }, @@ -303,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "b16491e9", + "id": "a168a53a", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "fed7e2df", + "id": "40c3e355", "metadata": { "editable": true }, @@ -325,7 +325,7 @@ }, { "cell_type": "markdown", - "id": "9beafdd9", + "id": "690e5ebd", "metadata": { "editable": true }, @@ -337,7 +337,7 @@ }, { "cell_type": "markdown", - "id": "ad2ee7bf", + "id": "d5e4d063", "metadata": { "editable": true }, @@ -349,7 +349,7 @@ }, { "cell_type": "markdown", - "id": "10eb9160", + "id": "f3c99110", "metadata": { "editable": true }, @@ -359,7 +359,7 @@ }, { "cell_type": "markdown", - "id": "18cda10f", + "id": "8b94286b", "metadata": { "editable": true }, @@ -371,7 +371,7 @@ }, { "cell_type": "markdown", - "id": "fb6f2e9d", + "id": "274f358c", "metadata": { "editable": true }, @@ -381,7 +381,7 @@ }, { "cell_type": "markdown", - "id": "3d415353", + "id": "75c6245e", "metadata": { "editable": true }, @@ -398,7 +398,7 @@ }, { "cell_type": "markdown", - "id": "9dd02f04", + "id": "c813eb77", "metadata": { "editable": true }, @@ -417,7 +417,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "f6d8eab0", + "id": "6bef25fe", "metadata": { "collapsed": false, "editable": true @@ -493,7 +493,7 @@ }, { "cell_type": "markdown", - "id": "dd37b16d", + "id": "0ff10c31", "metadata": { "editable": true }, @@ -503,7 +503,7 @@ }, { "cell_type": "markdown", - "id": "cc8c954c", + "id": "c072a6dd", "metadata": { "editable": true }, @@ -533,7 +533,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "e631888e", + "id": "71231be2", "metadata": { "collapsed": false, "editable": true @@ -582,7 +582,7 @@ }, { "cell_type": "markdown", - "id": "e8a1b23b", + "id": "b9475a90", "metadata": { "editable": true }, @@ -596,7 +596,7 @@ }, { "cell_type": "markdown", - "id": "6462ab01", + "id": "1b3063f0", "metadata": { "editable": true }, @@ -608,7 +608,7 @@ }, { "cell_type": "markdown", - "id": "b1c5c4f0", + "id": "e48ff5b1", "metadata": { "editable": true }, @@ -618,7 +618,7 @@ }, { "cell_type": "markdown", - "id": "3b4b3365", + "id": "d731fabe", "metadata": { "editable": true }, @@ -630,7 +630,7 @@ }, { "cell_type": "markdown", - "id": "a87967e1", + "id": "4cfd7f0e", "metadata": { "editable": true }, @@ -640,7 +640,7 @@ }, { "cell_type": "markdown", - "id": "a995bc0d", + "id": "8f7e06da", "metadata": { "editable": true }, @@ -652,7 +652,7 @@ }, { "cell_type": "markdown", - "id": "cde16d8a", + "id": "951c18fa", "metadata": { "editable": true }, @@ -662,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "40002307", + "id": "f830a42e", "metadata": { "editable": true }, @@ -674,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "ad1f47f2", + "id": "dc3bb676", "metadata": { "editable": true }, @@ -684,7 +684,7 @@ }, { "cell_type": "markdown", - "id": "44205dca", + "id": "36b4dc39", "metadata": { "editable": true }, @@ -700,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "5911811f", + "id": "e58db2db", "metadata": { "editable": true }, @@ -712,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "1a29cb1b", + "id": "f9d2d731", "metadata": { "editable": true }, @@ -722,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "3c8baa81", + "id": "4285c782", "metadata": { "editable": true }, @@ -734,7 +734,7 @@ }, { "cell_type": "markdown", - "id": "99c31881", + "id": "860c78f8", "metadata": { "editable": true }, @@ -747,7 +747,7 @@ }, { "cell_type": "markdown", - "id": "7b0c2d78", + "id": "5f736b0d", "metadata": { "editable": true }, @@ -759,7 +759,7 @@ }, { "cell_type": "markdown", - "id": "5ab49aec", + "id": "a8336ec5", "metadata": { "editable": true }, @@ -769,7 +769,7 @@ }, { "cell_type": "markdown", - "id": "c64df7b2", + "id": "d29a5d9d", "metadata": { "editable": true }, @@ -781,7 +781,7 @@ }, { "cell_type": "markdown", - "id": "7848825a", + "id": "b33925a8", "metadata": { "editable": true }, @@ -791,7 +791,7 @@ }, { "cell_type": "markdown", - "id": "99512677", + "id": "7af8aa06", "metadata": { "editable": true }, @@ -803,7 +803,7 @@ }, { "cell_type": "markdown", - "id": "d2415ce3", + "id": "3b620f9e", "metadata": { "editable": true }, @@ -815,7 +815,7 @@ }, { "cell_type": "markdown", - "id": "ff4c9a96", + "id": "8dda9b19", "metadata": { "editable": true }, @@ -825,7 +825,7 @@ }, { "cell_type": "markdown", - "id": "32566e9b", + "id": "9094fb73", "metadata": { "editable": true }, @@ -837,7 +837,7 @@ }, { "cell_type": "markdown", - "id": "5305fbde", + "id": "3566a77d", "metadata": { "editable": true }, @@ -849,7 +849,7 @@ }, { "cell_type": "markdown", - "id": "3689be86", + "id": "afd5b520", "metadata": { "editable": true }, @@ -861,7 +861,7 @@ }, { "cell_type": "markdown", - "id": "408012c9", + "id": "fedb062e", "metadata": { "editable": true }, @@ -871,7 +871,7 @@ }, { "cell_type": "markdown", - "id": "64de98b1", + "id": "1c402711", "metadata": { "editable": true }, @@ -883,7 +883,7 @@ }, { "cell_type": "markdown", - "id": "95077ce1", + "id": "ca3ffbe8", "metadata": { "editable": true }, @@ -893,7 +893,7 @@ }, { "cell_type": "markdown", - "id": "5ae72845", + "id": "ef5649aa", "metadata": { "editable": true }, @@ -912,7 +912,7 @@ }, { "cell_type": "markdown", - "id": "37f35940", + "id": "733c5a10", "metadata": { "editable": true }, @@ -940,7 +940,7 @@ }, { "cell_type": "markdown", - "id": "cec4ccbf", + "id": "8ab3bfdd", "metadata": { "editable": true }, @@ -966,7 +966,7 @@ }, { "cell_type": "markdown", - "id": "91ebab74", + "id": "a9ec244e", "metadata": { "editable": true }, @@ -983,7 +983,7 @@ }, { "cell_type": "markdown", - "id": "f3ac7279", + "id": "c48eb948", "metadata": { "editable": true }, @@ -1003,7 +1003,7 @@ }, { "cell_type": "markdown", - "id": "ac4c5cdb", + "id": "00971958", "metadata": { "editable": true }, @@ -1032,7 +1032,7 @@ }, { "cell_type": "markdown", - "id": "83ae82a8", + "id": "65d9f81b", "metadata": { "editable": true }, @@ -1057,7 +1057,7 @@ }, { "cell_type": "markdown", - "id": "4f9d0cf9", + "id": "cadd98e1", "metadata": { "editable": true }, @@ -1071,7 +1071,7 @@ }, { "cell_type": "markdown", - "id": "bc98c140", + "id": "e652a034", "metadata": { "editable": true }, @@ -1083,7 +1083,7 @@ }, { "cell_type": "markdown", - "id": "5ed9625e", + "id": "286616bf", "metadata": { "editable": true }, @@ -1096,7 +1096,7 @@ }, { "cell_type": "markdown", - "id": "1937dce6", + "id": "62d35d9a", "metadata": { "editable": true }, @@ -1107,7 +1107,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "49b5827a", + "id": "b452c740", "metadata": { "collapsed": false, "editable": true @@ -1146,7 +1146,7 @@ }, { "cell_type": "markdown", - "id": "d59c7cb3", + "id": "4423761d", "metadata": { "editable": true }, @@ -1171,7 +1171,7 @@ }, { "cell_type": "markdown", - "id": "dc98f057", + "id": "9e042d22", "metadata": { "editable": true }, @@ -1191,7 +1191,7 @@ }, { "cell_type": "markdown", - "id": "16792734", + "id": "17dabd08", "metadata": { "editable": true }, @@ -1203,7 +1203,7 @@ }, { "cell_type": "markdown", - "id": "a9191ac1", + "id": "29decc08", "metadata": { "editable": true }, @@ -1213,7 +1213,7 @@ }, { "cell_type": "markdown", - "id": "b512964a", + "id": "74dded9c", "metadata": { "editable": true }, @@ -1228,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "ad322dc6", + "id": "ae938950", "metadata": { "editable": true }, @@ -1241,7 +1241,7 @@ }, { "cell_type": "markdown", - "id": "d1149979", + "id": "fe91bf5a", "metadata": { "editable": true }, @@ -1254,7 +1254,7 @@ }, { "cell_type": "markdown", - "id": "bae2c22e", + "id": "3e0ba223", "metadata": { "editable": true }, @@ -1266,7 +1266,7 @@ }, { "cell_type": "markdown", - "id": "7815160c", + "id": "5f44dec4", "metadata": { "editable": true }, @@ -1279,7 +1279,7 @@ }, { "cell_type": "markdown", - "id": "dac5fb75", + "id": "04f24465", "metadata": { "editable": true }, @@ -1290,7 +1290,7 @@ }, { "cell_type": "markdown", - "id": "8b5e65ba", + "id": "c4d46b5b", "metadata": { "editable": true }, @@ -1304,7 +1304,7 @@ }, { "cell_type": "markdown", - "id": "1b98fbe5", + "id": "75f7562f", "metadata": { "editable": true }, @@ -1314,7 +1314,7 @@ }, { "cell_type": "markdown", - "id": "e7409503", + "id": "1b4315d0", "metadata": { "editable": true }, @@ -1328,7 +1328,7 @@ }, { "cell_type": "markdown", - "id": "7adc92e9", + "id": "904ccefa", "metadata": { "editable": true }, @@ -1341,7 +1341,7 @@ }, { "cell_type": "markdown", - "id": "9d58e53e", + "id": "947e35c0", "metadata": { "editable": true }, @@ -1354,7 +1354,7 @@ }, { "cell_type": "markdown", - "id": "d63c1e78", + "id": "e8c626e4", "metadata": { "editable": true }, @@ -1364,7 +1364,7 @@ }, { "cell_type": "markdown", - "id": "3ab9990a", + "id": "c51cb4dd", "metadata": { "editable": true }, @@ -1377,7 +1377,7 @@ }, { "cell_type": "markdown", - "id": "385e8e52", + "id": "d5b1a803", "metadata": { "editable": true }, @@ -1387,7 +1387,7 @@ }, { "cell_type": "markdown", - "id": "4dbefb62", + "id": "1a142b5a", "metadata": { "editable": true }, @@ -1400,7 +1400,7 @@ }, { "cell_type": "markdown", - "id": "23f4e746", + "id": "0dc30002", "metadata": { "editable": true }, @@ -1412,7 +1412,7 @@ }, { "cell_type": "markdown", - "id": "2cffab06", + "id": "26f23bd8", "metadata": { "editable": true }, @@ -1431,7 +1431,7 @@ }, { "cell_type": "markdown", - "id": "04ba02de", + "id": "99e316c5", "metadata": { "editable": true }, @@ -1444,7 +1444,7 @@ }, { "cell_type": "markdown", - "id": "53de7c66", + "id": "35830d99", "metadata": { "editable": true }, @@ -1456,7 +1456,7 @@ }, { "cell_type": "markdown", - "id": "f5054463", + "id": "989bfe6a", "metadata": { "editable": true }, @@ -1469,7 +1469,7 @@ }, { "cell_type": "markdown", - "id": "07569301", + "id": "9cb57daf", "metadata": { "editable": true }, @@ -1489,7 +1489,7 @@ }, { "cell_type": "markdown", - "id": "8f6dd2ff", + "id": "311a55d9", "metadata": { "editable": true }, @@ -1512,7 +1512,7 @@ }, { "cell_type": "markdown", - "id": "18e63889", + "id": "209dfe12", "metadata": { "editable": true }, @@ -1527,7 +1527,7 @@ }, { "cell_type": "markdown", - "id": "66177c23", + "id": "9ddf4ef4", "metadata": { "editable": true }, @@ -1539,7 +1539,7 @@ }, { "cell_type": "markdown", - "id": "10312f61", + "id": "4b2b7bfc", "metadata": { "editable": true }, @@ -1559,7 +1559,7 @@ }, { "cell_type": "markdown", - "id": "12b94aae", + "id": "8999dff6", "metadata": { "editable": true }, @@ -1579,7 +1579,7 @@ }, { "cell_type": "markdown", - "id": "c3f7b485", + "id": "982d5fbc", "metadata": { "editable": true }, @@ -1603,7 +1603,7 @@ }, { "cell_type": "markdown", - "id": "e08528e6", + "id": "2ae38e48", "metadata": { "editable": true }, @@ -1624,7 +1624,7 @@ }, { "cell_type": "markdown", - "id": "c9ac007c", + "id": "bd20d60a", "metadata": { "editable": true }, @@ -1654,7 +1654,7 @@ }, { "cell_type": "markdown", - "id": "0f4a3620", + "id": "0c72fdb0", "metadata": { "editable": true }, @@ -1678,7 +1678,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "097242cb", + "id": "49104a67", "metadata": { "collapsed": false, "editable": true @@ -1715,7 +1715,7 @@ }, { "cell_type": "markdown", - "id": "751ffe74", + "id": "27642741", "metadata": { "editable": true }, @@ -1725,7 +1725,7 @@ }, { "cell_type": "markdown", - "id": "dd1cb382", + "id": "e91dfb35", "metadata": { "editable": true }, @@ -1736,7 +1736,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "c127ea22", + "id": "3f088b19", "metadata": { "collapsed": false, "editable": true @@ -1756,7 +1756,7 @@ }, { "cell_type": "markdown", - "id": "03706aaa", + "id": "1d36aae1", "metadata": { "editable": true }, @@ -1774,7 +1774,7 @@ }, { "cell_type": "markdown", - "id": "2bd565a3", + "id": "b1ed3d14", "metadata": { "editable": true }, @@ -1786,7 +1786,7 @@ }, { "cell_type": "markdown", - "id": "b5ba3c44", + "id": "d05df5cd", "metadata": { "editable": true }, @@ -1803,7 +1803,7 @@ }, { "cell_type": "markdown", - "id": "fd0d03ce", + "id": "757ce9bd", "metadata": { "editable": true }, @@ -1815,7 +1815,7 @@ }, { "cell_type": "markdown", - "id": "9a06d7b1", + "id": "0e176bde", "metadata": { "editable": true }, @@ -1825,7 +1825,7 @@ }, { "cell_type": "markdown", - "id": "f0884966", + "id": "6227b309", "metadata": { "editable": true }, @@ -1837,7 +1837,7 @@ }, { "cell_type": "markdown", - "id": "9fcfd8e4", + "id": "ad1dab0e", "metadata": { "editable": true }, @@ -1854,7 +1854,7 @@ }, { "cell_type": "markdown", - "id": "72786344", + "id": "37753004", "metadata": { "editable": true }, @@ -1866,7 +1866,7 @@ }, { "cell_type": "markdown", - "id": "9cbe3bd1", + "id": "59e6e439", "metadata": { "editable": true }, @@ -1876,7 +1876,7 @@ }, { "cell_type": "markdown", - "id": "a9f1b60c", + "id": "5bd100b3", "metadata": { "editable": true }, @@ -1888,7 +1888,7 @@ }, { "cell_type": "markdown", - "id": "df279145", + "id": "c3a847fc", "metadata": { "editable": true }, @@ -1898,7 +1898,7 @@ }, { "cell_type": "markdown", - "id": "cc5b4571", + "id": "a52ccb28", "metadata": { "editable": true }, @@ -1910,7 +1910,7 @@ }, { "cell_type": "markdown", - "id": "0b73c25e", + "id": "9d662c2b", "metadata": { "editable": true }, @@ -1920,7 +1920,7 @@ }, { "cell_type": "markdown", - "id": "160fb1bb", + "id": "fa90fc6f", "metadata": { "editable": true }, @@ -1936,7 +1936,7 @@ }, { "cell_type": "markdown", - "id": "a6436245", + "id": "7338e614", "metadata": { "editable": true }, @@ -1947,7 +1947,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "a51b5594", + "id": "af3464ee", "metadata": { "collapsed": false, "editable": true @@ -2012,7 +2012,7 @@ }, { "cell_type": "markdown", - "id": "81a7b921", + "id": "aa9520b0", "metadata": { "editable": true }, @@ -2023,7 +2023,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "83c3d0e1", + "id": "73f2eca3", "metadata": { "collapsed": false, "editable": true @@ -2080,7 +2080,7 @@ }, { "cell_type": "markdown", - "id": "a5fccc44", + "id": "e8b3e361", "metadata": { "editable": true }, @@ -2118,7 +2118,7 @@ }, { "cell_type": "markdown", - "id": "32bb1864", + "id": "122aad6f", "metadata": { "editable": true }, @@ -2129,7 +2129,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "be84f020", + "id": "f8996c67", "metadata": { "collapsed": false, "editable": true @@ -2211,7 +2211,7 @@ }, { "cell_type": "markdown", - "id": "93f4f24b", + "id": "b877b018", "metadata": { "editable": true }, @@ -2236,67 +2236,7 @@ }, { "cell_type": "markdown", - "id": "4775ea37", - "metadata": { - "editable": true - }, - "source": [ - "## How to set up the cross-validation for Ridge and/or Lasso\n", - "\n", - "* Define a range of interest for the penalty parameter.\n", - "\n", - "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", - "\n", - "* Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\boldsymbol{\\sigma}_{-i}^2(\\lambda)$, as" - ] - }, - { - "cell_type": "markdown", - "id": "d6dc5694", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\boldsymbol{\\beta}_{-i}(\\lambda) & = ( \\boldsymbol{X}_{-i, \\ast}^{T}\n", - "\\boldsymbol{X}_{-i, \\ast} + \\lambda \\boldsymbol{I}_{pp})^{-1}\n", - "\\boldsymbol{X}_{-i, \\ast}^{T} \\boldsymbol{y}_{-i}\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "6f9343ce", - "metadata": { - "editable": true - }, - "source": [ - "* Evaluate the prediction performance of these models on the test set by $[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", - "\n", - "* Repeat the first three steps such that each sample plays the role of the test set once.\n", - "\n", - "* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as" - ] - }, - { - "cell_type": "markdown", - "id": "c1180543", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[y_i, \\mathbf{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "8d7e0832", + "id": "4ba845d0", "metadata": { "editable": true }, @@ -2324,7 +2264,7 @@ }, { "cell_type": "markdown", - "id": "1236691a", + "id": "dbfb73ea", "metadata": { "editable": true }, @@ -2337,7 +2277,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "f3da8566", + "id": "67a992a9", "metadata": { "collapsed": false, "editable": true @@ -2437,7 +2377,7 @@ }, { "cell_type": "markdown", - "id": "af0e6ba4", + "id": "388aeee5", "metadata": { "editable": true }, @@ -2448,7 +2388,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "fddda52e", + "id": "e39bfca9", "metadata": { "collapsed": false, "editable": true @@ -2537,7 +2477,7 @@ }, { "cell_type": "markdown", - "id": "5fee654f", + "id": "10dd42f3", "metadata": { "editable": true }, @@ -2547,7 +2487,7 @@ }, { "cell_type": "markdown", - "id": "c24b5918", + "id": "094bf65d", "metadata": { "editable": true }, @@ -2560,7 +2500,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "974f9c58", + "id": "ea273fff", "metadata": { "collapsed": false, "editable": true diff --git a/doc/src/week37/week37.do.txt b/doc/src/week37/week37.do.txt index b0f732ace..e16d0ee9b 100644 --- a/doc/src/week37/week37.do.txt +++ b/doc/src/week37/week37.do.txt @@ -1278,32 +1278,6 @@ involves a degree of randomness. This may be fully excluded when choosing $k=n$. This particular case is referred to as leave-one-out cross-validation (LOOCV). -!split -===== How to set up the cross-validation for Ridge and/or Lasso ===== - -* Define a range of interest for the penalty parameter. - -* Divide the data set into training and test set comprising samples $\{1, \ldots, n\} \setminus i$ and $\{ i \}$, respectively. - -* Fit the linear regression model by means of ridge estimation for each $\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\bm{\sigma}_{-i}^2(\lambda)$, as -!bt -\begin{align*} -\bm{\beta}_{-i}(\lambda) & = ( \bm{X}_{-i, \ast}^{T} -\bm{X}_{-i, \ast} + \lambda \bm{I}_{pp})^{-1} -\bm{X}_{-i, \ast}^{T} \bm{y}_{-i} -\end{align*} -!et - -* Evaluate the prediction performance of these models on the test set by $[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function. - -* Repeat the first three steps such that each sample plays the role of the test set once. - -* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as -!bt -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -!et !split ===== Cross-validation in brief =====