From f2d379b2d2aec239c0603a4e6cc8b1546852443f Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 16 Sep 2021 07:10:45 +0200 Subject: [PATCH] updating --- doc/pub/week37/html/._week37-bs000.html | 56 ++-- doc/pub/week37/html/._week37-bs001.html | 56 ++-- doc/pub/week37/html/._week37-bs002.html | 56 ++-- doc/pub/week37/html/._week37-bs003.html | 56 ++-- doc/pub/week37/html/._week37-bs004.html | 56 ++-- doc/pub/week37/html/._week37-bs005.html | 56 ++-- doc/pub/week37/html/._week37-bs006.html | 56 ++-- doc/pub/week37/html/._week37-bs007.html | 56 ++-- doc/pub/week37/html/._week37-bs008.html | 56 ++-- doc/pub/week37/html/._week37-bs009.html | 56 ++-- doc/pub/week37/html/._week37-bs010.html | 56 ++-- doc/pub/week37/html/._week37-bs011.html | 56 ++-- doc/pub/week37/html/._week37-bs012.html | 56 ++-- doc/pub/week37/html/._week37-bs013.html | 56 ++-- doc/pub/week37/html/._week37-bs014.html | 56 ++-- doc/pub/week37/html/._week37-bs015.html | 56 ++-- doc/pub/week37/html/._week37-bs016.html | 56 ++-- doc/pub/week37/html/._week37-bs017.html | 56 ++-- doc/pub/week37/html/._week37-bs018.html | 56 ++-- doc/pub/week37/html/._week37-bs019.html | 56 ++-- doc/pub/week37/html/._week37-bs020.html | 56 ++-- doc/pub/week37/html/._week37-bs021.html | 56 ++-- doc/pub/week37/html/._week37-bs022.html | 56 ++-- doc/pub/week37/html/._week37-bs023.html | 56 ++-- doc/pub/week37/html/._week37-bs024.html | 82 ++++-- doc/pub/week37/html/._week37-bs025.html | 84 +++--- doc/pub/week37/html/._week37-bs026.html | 89 ++++--- doc/pub/week37/html/._week37-bs027.html | 96 ++++--- doc/pub/week37/html/._week37-bs028.html | 135 +++++----- doc/pub/week37/html/._week37-bs029.html | 119 +++------ doc/pub/week37/html/._week37-bs030.html | 124 ++++----- doc/pub/week37/html/._week37-bs031.html | 118 ++++----- doc/pub/week37/html/._week37-bs032.html | 98 +++---- doc/pub/week37/html/._week37-bs033.html | 148 +++++------ doc/pub/week37/html/._week37-bs034.html | 127 +++++++--- doc/pub/week37/html/._week37-bs035.html | 102 ++++---- doc/pub/week37/html/._week37-bs036.html | 133 +++++++--- doc/pub/week37/html/._week37-bs037.html | 186 +++++++------- doc/pub/week37/html/._week37-bs038.html | 162 ++++++------ doc/pub/week37/html/._week37-bs039.html | 156 ++++++------ doc/pub/week37/html/week37-bs.html | 56 ++-- doc/pub/week37/html/week37-reveal.html | 186 +++++++++++++- doc/pub/week37/html/week37-solarized.html | 178 ++++++++++++- doc/pub/week37/html/week37.html | 178 ++++++++++++- doc/pub/week37/ipynb/ipynb-week37-src.tar.gz | Bin 192 -> 192 bytes doc/pub/week37/ipynb/week37.ipynb | 253 ++++++++++++++++++- doc/src/week37/test.py | 38 --- doc/src/week37/week37.do.txt | 153 ++++++++++- 48 files changed, 2914 insertions(+), 1431 deletions(-) delete mode 100644 doc/src/week37/test.py diff --git a/doc/pub/week37/html/._week37-bs000.html b/doc/pub/week37/html/._week37-bs000.html index c76d88187..e847a82eb 100644 --- a/doc/pub/week37/html/._week37-bs000.html +++ b/doc/pub/week37/html/._week37-bs000.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -296,7 +316,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs001.html b/doc/pub/week37/html/._week37-bs001.html index 9ccc0f4c0..8e885a44e 100644 --- a/doc/pub/week37/html/._week37-bs001.html +++ b/doc/pub/week37/html/._week37-bs001.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -284,7 +304,7 @@ Recommended Reading:
  • 10
  • 11
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs002.html b/doc/pub/week37/html/._week37-bs002.html index 9cb66a8d0..50d44a61d 100644 --- a/doc/pub/week37/html/._week37-bs002.html +++ b/doc/pub/week37/html/._week37-bs002.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -273,7 +293,7 @@ MathJax.Hub.Config({
  • 11
  • 12
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs003.html b/doc/pub/week37/html/._week37-bs003.html index e274a724f..7295e40ea 100644 --- a/doc/pub/week37/html/._week37-bs003.html +++ b/doc/pub/week37/html/._week37-bs003.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -293,7 +313,7 @@ $$
  • 12
  • 13
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs004.html b/doc/pub/week37/html/._week37-bs004.html index 88f8f81e4..d617be471 100644 --- a/doc/pub/week37/html/._week37-bs004.html +++ b/doc/pub/week37/html/._week37-bs004.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -307,7 +327,7 @@ It is a conditional probability (see below) and reads as the likelihood of a dom
  • 13
  • 14
  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs005.html b/doc/pub/week37/html/._week37-bs005.html index 7ff12725a..3dd2675ee 100644 --- a/doc/pub/week37/html/._week37-bs005.html +++ b/doc/pub/week37/html/._week37-bs005.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -301,7 +321,7 @@ is equivalent to the maximization/minimization of the function itself.
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  • ...
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  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs006.html b/doc/pub/week37/html/._week37-bs006.html index e0c81b325..3dad1ce7b 100644 --- a/doc/pub/week37/html/._week37-bs006.html +++ b/doc/pub/week37/html/._week37-bs006.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -301,7 +321,7 @@ $$
  • 15
  • 16
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs007.html b/doc/pub/week37/html/._week37-bs007.html index 4c73ef313..5f18a7b13 100644 --- a/doc/pub/week37/html/._week37-bs007.html +++ b/doc/pub/week37/html/._week37-bs007.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -292,7 +312,7 @@ which is Bayes' theorem. It allows us to evaluate the uncertainty in in \( X \)
  • 16
  • 17
  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs008.html b/doc/pub/week37/html/._week37-bs008.html index 97dd65b8f..5a366462a 100644 --- a/doc/pub/week37/html/._week37-bs008.html +++ b/doc/pub/week37/html/._week37-bs008.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -288,7 +308,7 @@ The function \( p(X) \) on the right hand side is called the prior while the fun
  • 17
  • 18
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs009.html b/doc/pub/week37/html/._week37-bs009.html index dce65b3f3..3fb2cd4cc 100644 --- a/doc/pub/week37/html/._week37-bs009.html +++ b/doc/pub/week37/html/._week37-bs009.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -360,7 +380,7 @@ How can we understand this?
  • 18
  • 19
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs010.html b/doc/pub/week37/html/._week37-bs010.html index 9bc442eba..c035b5a94 100644 --- a/doc/pub/week37/html/._week37-bs010.html +++ b/doc/pub/week37/html/._week37-bs010.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -345,7 +365,7 @@ lambdas = np.19
  • 20
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs011.html b/doc/pub/week37/html/._week37-bs011.html index a3ed3af59..b463b2563 100644 --- a/doc/pub/week37/html/._week37-bs011.html +++ b/doc/pub/week37/html/._week37-bs011.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -311,7 +331,7 @@ We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one
  • 20
  • 21
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs012.html b/doc/pub/week37/html/._week37-bs012.html index a103da9ed..d218264d2 100644 --- a/doc/pub/week37/html/._week37-bs012.html +++ b/doc/pub/week37/html/._week37-bs012.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -317,7 +337,7 @@ which is our Ridge cost function! Nice, isn't it?
  • 21
  • 22
  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs013.html b/doc/pub/week37/html/._week37-bs013.html index 3f25daeba..a6bc3f127 100644 --- a/doc/pub/week37/html/._week37-bs013.html +++ b/doc/pub/week37/html/._week37-bs013.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -311,7 +331,7 @@ which is our Lasso cost function!
  • 22
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  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs014.html b/doc/pub/week37/html/._week37-bs014.html index 7099f3c8a..f446dec34 100644 --- a/doc/pub/week37/html/._week37-bs014.html +++ b/doc/pub/week37/html/._week37-bs014.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -293,7 +313,7 @@ and discuss how to select a given model (one of the difficult parts in machine l
  • 23
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs015.html b/doc/pub/week37/html/._week37-bs015.html index f89038534..d3d25e8a2 100644 --- a/doc/pub/week37/html/._week37-bs015.html +++ b/doc/pub/week37/html/._week37-bs015.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -310,7 +330,7 @@ cross-validation and the bootstrap method.
  • 24
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  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs016.html b/doc/pub/week37/html/._week37-bs016.html index 824aaf17d..cc156e3ef 100644 --- a/doc/pub/week37/html/._week37-bs016.html +++ b/doc/pub/week37/html/._week37-bs016.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -306,7 +326,7 @@ bootstrap is widely used.
  • 25
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  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs017.html b/doc/pub/week37/html/._week37-bs017.html index ba3745d38..4d8800b51 100644 --- a/doc/pub/week37/html/._week37-bs017.html +++ b/doc/pub/week37/html/._week37-bs017.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -293,7 +313,7 @@ MathJax.Hub.Config({
  • 26
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  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs018.html b/doc/pub/week37/html/._week37-bs018.html index 88f6119bc..f4c40e1ad 100644 --- a/doc/pub/week37/html/._week37-bs018.html +++ b/doc/pub/week37/html/._week37-bs018.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -299,7 +319,7 @@ MathJax.Hub.Config({
  • 27
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  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs019.html b/doc/pub/week37/html/._week37-bs019.html index 535dc6955..1476142a3 100644 --- a/doc/pub/week37/html/._week37-bs019.html +++ b/doc/pub/week37/html/._week37-bs019.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -305,7 +325,7 @@ training error reaches a saturation.
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs020.html b/doc/pub/week37/html/._week37-bs020.html index 3e9efadf1..cd4d3da74 100644 --- a/doc/pub/week37/html/._week37-bs020.html +++ b/doc/pub/week37/html/._week37-bs020.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -299,7 +319,7 @@ need for bootstrapping.
  • 29
  • 30
  • ...
    • -
  • 41
    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs021.html b/doc/pub/week37/html/._week37-bs021.html index 6820b3568..44f7cfb16 100644 --- a/doc/pub/week37/html/._week37-bs021.html +++ b/doc/pub/week37/html/._week37-bs021.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -295,7 +315,7 @@ number \( i \) is left out. Using this notation, define
  • 30
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs022.html b/doc/pub/week37/html/._week37-bs022.html index 45763c521..8cf95d6ed 100644 --- a/doc/pub/week37/html/._week37-bs022.html +++ b/doc/pub/week37/html/._week37-bs022.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -312,7 +332,7 @@ t = jackknife(x, stat)
  • 31
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs023.html b/doc/pub/week37/html/._week37-bs023.html index 69121a3de..977fa0311 100644 --- a/doc/pub/week37/html/._week37-bs023.html +++ b/doc/pub/week37/html/._week37-bs023.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -304,7 +324,7 @@ Before we proceed however, we need to remind ourselves about a central theorem i
  • 32
  • 33
  • ...
    • -
  • 41
    • +
  • 48
  • »
    • diff --git a/doc/pub/week37/html/._week37-bs024.html b/doc/pub/week37/html/._week37-bs024.html index 6e3118d71..13f7be910 100644 --- a/doc/pub/week37/html/._week37-bs024.html +++ b/doc/pub/week37/html/._week37-bs024.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,18 +273,24 @@ MathJax.Hub.Config({ -

    Resampling methods: Bootstrap background

    +

    The Central Limit Theorem

    -Since \( \widehat{\beta} = \widehat{\beta}(\boldsymbol{X}) \) is a function of random variables, -\( \widehat{\beta} \) itself must be a random variable. Thus it has -a pdf, call this function \( p(\boldsymbol{t}) \). The aim of the bootstrap is to -estimate \( p(\boldsymbol{t}) \) by the relative frequency of -\( \widehat{\beta} \). You can think of this as using a histogram -in the place of \( p(\boldsymbol{t}) \). If the relative frequency closely -resembles \( p(\vec{t}) \), then using numerics, it is straight forward to -estimate all the interesting parameters of \( p(\boldsymbol{t}) \) using point -estimators. +Suppose we have a PDF \( p(x) \) from which we generate a series \( N \) +of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \) +is viewed as the average of a specific measurement, e.g., throwing +dice 100 times and then taking the average value, or producing a certain +amount of random numbers. +For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion +which follows. + +

    +If we compute the mean \( z \) of \( m \) such mean values \( x_i \) +$$ + z=\frac{x_1+x_2+\dots+x_m}{m}, +$$ + +the question we pose is which is the PDF of the new variable \( z \).

    @@ -292,7 +318,7 @@ estimators.

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    • -
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    • +
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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs025.html b/doc/pub/week37/html/._week37-bs025.html index 16a9f7ac0..3c219d9e8 100644 --- a/doc/pub/week37/html/._week37-bs025.html +++ b/doc/pub/week37/html/._week37-bs025.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,24 +273,22 @@ MathJax.Hub.Config({ -

    Resampling methods: More Bootstrap background

    +

    Finding the Limit

    -In the case that \( \widehat{\beta} \) has -more than one component, and the components are independent, we use the -same estimator on each component separately. If the probability -density function of \( X_i \), \( p(x) \), had been known, then it would have -been straightforward to do this by: +The probability of obtaining an average value \( z \) is the product of the +probabilities of obtaining arbitrary individual mean values \( x_i \), +but with the constraint that the average is \( z \). We can express this through +the following expression +$$ + \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m) + \delta(z-\frac{x_1+x_2+\dots+x_m}{m}), +$$ -

      -
    1. Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).
      2. -
    2. Then using these numbers, we could compute a replica of \( \widehat{\beta} \) called \( \widehat{\beta}^* \).
      3. -
    - -By repeated use of the above two points, many -estimates of \( \widehat{\beta} \) can be obtained. The -idea is to use the relative frequency of \( \widehat{\beta}^* \) -(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \). +where the \( \delta \)-function enbodies the constraint that the mean is \( z \). +All measurements that lead to each individual \( x_i \) are expected to +be independent, which in turn means that we can express \( \tilde{p} \) as the +product of individual \( p(x_i) \). The independence assumption is important in the derivation of the central limit theorem.

    @@ -298,7 +316,7 @@ idea is to use the relative frequency of \( \widehat{\beta}^* \)

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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs026.html b/doc/pub/week37/html/._week37-bs026.html index 4f46b9cea..8f950e79c 100644 --- a/doc/pub/week37/html/._week37-bs026.html +++ b/doc/pub/week37/html/._week37-bs026.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,20 +273,31 @@ MathJax.Hub.Config({ -

    Resampling methods: Bootstrap approach

    +

    Rewriting the \( \delta \)-function

    -But -unless there is enough information available about the process that -generated \( X_1,X_2,\cdots,X_n \), \( p(x) \) is in general -unknown. Therefore, Efron in 1979 asked the -question: What if we replace \( p(x) \) by the relative frequency -of the observation \( X_i \)? +If we use the integral expression for the \( \delta \)-function -

    -If we draw observations in accordance with -the relative frequency of the observations, will we obtain the same -result in some asymptotic sense? The answer is yes. +$$ + \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty} + dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)}, +$$ + +and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value +we arrive at +$$ + \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty} + dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty} + dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m, +$$ + +with the integral over \( x \) resulting in + +$$ + \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}= + \int_{-\infty}^{\infty}dxp(x) + \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right]. +$$

    @@ -294,7 +325,7 @@ result in some asymptotic sense? The answer is yes.

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  • »
    • diff --git a/doc/pub/week37/html/._week37-bs027.html b/doc/pub/week37/html/._week37-bs027.html index 9f24bf346..62315f9cb 100644 --- a/doc/pub/week37/html/._week37-bs027.html +++ b/doc/pub/week37/html/._week37-bs027.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,27 +273,33 @@ MathJax.Hub.Config({ -

    Resampling methods: Bootstrap steps

    +

    Identifying Terms

    -The independent bootstrap works like this: +The second term on the rhs disappears since this is just the mean and +employing the definition of \( \sigma^2 \) we have +$$ + \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}= + 1-\frac{q^2\sigma^2}{2m^2}+\dots, +$$ -

      -
    1. Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).
      2. -
    2. Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).
      3. -
    3. Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\beta}^* \) by evaluating \( \widehat \beta \) under the observations \( \boldsymbol{x}^* \).
      4. -
    4. Repeat this process \( k \) times.
      5. -
    +resulting in -When you are done, you can draw a histogram of the relative frequency -of \( \widehat \beta^* \). This is your estimate of the probability -distribution \( p(t) \). Using this probability distribution you can -estimate any statistics thereof. In principle you never draw the -histogram of the relative frequency of \( \widehat{\beta}^* \). Instead -you use the estimators corresponding to the statistic of interest. For -example, if you are interested in estimating the variance of \( \widehat -\beta \), apply the etsimator \( \widehat \sigma^2 \) to the values -\( \widehat \beta^* \). +$$ + \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx + \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m, +$$ + +and in the limit \( m\rightarrow \infty \) we obtain + +$$ + \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})} + \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)}, +$$ + +which is the normal distribution with variance +\( \sigma^2_m=\sigma^2/m \), where \( \sigma \) is the variance of the PDF \( p(x) \) +and \( \mu \) is also the mean of the PDF \( p(x) \).

    @@ -301,7 +327,7 @@ example, if you are interested in estimating the variance of \( \widehat

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    • diff --git a/doc/pub/week37/html/._week37-bs028.html b/doc/pub/week37/html/._week37-bs028.html index e5ae78549..3336600e9 100644 --- a/doc/pub/week37/html/._week37-bs028.html +++ b/doc/pub/week37/html/._week37-bs028.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,67 +273,40 @@ MathJax.Hub.Config({ -

    Code example for the Bootstrap method

    +

    Wrapping it up

    -The following code starts with a Gaussian distribution with mean value -\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data -used in the bootstrap analysis. The bootstrap analysis returns a data -set after a given number of bootstrap operations (as many as we have -data points). This data set consists of estimated mean values for each -bootstrap operation. The histogram generated by the bootstrap method -shows that the distribution for these mean values is also a Gaussian, -centered around the mean value \( \mu=100 \) but with standard deviation -\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in -this case the same as the number of original data points). The value -of the standard deviation is what we expect from the central limit -theorem. +Thus, the central limit theorem states that the PDF \( \tilde{p}(z) \) of +the average of \( m \) random values corresponding to a PDF \( p(x) \) +is a normal distribution whose mean is the +mean value of the PDF \( p(x) \) and whose variance is the variance +of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).

    +The theorem is satisfied by a large class of PDFs. Note however that for a +finite \( m \), it is not always possible to find a closed expression for +\( \tilde{p}(x) \). +The central limit theorem leads then to the well-known expression for the +standard deviation, given by - -

    from numpy import *
-from numpy.random import randint, randn
-from time import time
-import matplotlib.mlab as mlab
-import matplotlib.pyplot as plt
+$$
+    \sigma_m=
+\frac{\sigma}{\sqrt{m}}.
+$$
 
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
-    return mean(data)
+
    
+The latter is true only if the average value is known exactly. This is obtained in the limit
+\( m\rightarrow \infty \)  only. Because the mean and the variance are measured quantities we obtain 
+the familiar expression in statistics
+$$
+    \sigma_m\approx 
+\frac{\sigma}{\sqrt{m-1}}.
+$$
 
-# Bootstrap algorithm
-def bootstrap(data, statistic, R):
-    t = zeros(R); n = len(data); inds = arange(n); t0 = time()
-    # non-parametric bootstrap         
-    for i in range(R):
-        t[i] = statistic(data[randint(0,n,n)])
+
    
+In many cases however the above estimate for the standard deviation,
+in particular if correlations are strong, may be too simplistic.
 
-    # analysis    
-    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
-    print("original           bias      std. error")
-    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
-    return t
-
-
-mu, sigma = 100, 15
-datapoints = 10000
-x = mu + sigma*random.randn(datapoints)
-# bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
-# the histogram of the bootstrapped  data                                                                                                    
-n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
-
-# add a 'best fit' line  
-y = mlab.normpdf( binsboot, mean(t), std(t))
-lt = plt.plot(binsboot, y, 'r--', linewidth=1)
-plt.xlabel('Smarts')
-plt.ylabel('Probability')
-plt.axis([99.5, 100.6, 0, 3.0])
-plt.grid(True)
-
-plt.show()
-

    @@ -340,7 +333,7 @@ plt.show()

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    • diff --git a/doc/pub/week37/html/._week37-bs029.html b/doc/pub/week37/html/._week37-bs029.html index 5b6079c13..00385625f 100644 --- a/doc/pub/week37/html/._week37-bs029.html +++ b/doc/pub/week37/html/._week37-bs029.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,68 +273,7 @@ MathJax.Hub.Config({ -

    The bias-variance tradeoff

    - -

    -We will discuss the bias-variance tradeoff in the context of -continuous predictions such as regression. However, many of the -intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). - -

    -Let us assume that the true data is generated from a noisy model - -$$ -\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon} -$$ - -

    -where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \). - -

    -In our derivation of the ordinary least squares method we defined then -an approximation to the function \( f \) in terms of the parameters -\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model, -that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \). - -

    -Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function -$$ -C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. -$$ - -

    -We can rewrite this as -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. -$$ - -

    -The three terms represent the square of the bias of the learning -method, which can be thought of as the error caused by the simplifying -assumptions built into the method. The second term represents the -variance of the chosen model and finally the last terms is variance of -the error \( \boldsymbol{\epsilon} \). - -

    -To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \). -We use a more compact notation in terms of the expectation value -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right], -$$ - -and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right], -$$ - -which, using the abovementioned expectation values can be rewritten as -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2, -$$ - -that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \). +

    Confidence Intervals

    @@ -342,7 +301,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model

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    • diff --git a/doc/pub/week37/html/._week37-bs030.html b/doc/pub/week37/html/._week37-bs030.html index d8cb071fd..e08c98b82 100644 --- a/doc/pub/week37/html/._week37-bs030.html +++ b/doc/pub/week37/html/._week37-bs030.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,65 +273,19 @@ MathJax.Hub.Config({ -

    Example code for Bias-Variance tradeoff

    +

    Resampling methods: Bootstrap background

    +

    +Since \( \widehat{\beta} = \widehat{\beta}(\boldsymbol{X}) \) is a function of random variables, +\( \widehat{\beta} \) itself must be a random variable. Thus it has +a pdf, call this function \( p(\boldsymbol{t}) \). The aim of the bootstrap is to +estimate \( p(\boldsymbol{t}) \) by the relative frequency of +\( \widehat{\beta} \). You can think of this as using a histogram +in the place of \( p(\boldsymbol{t}) \). If the relative frequency closely +resembles \( p(\vec{t}) \), then using numerics, it is straight forward to +estimate all the interesting parameters of \( p(\boldsymbol{t}) \) using point +estimators. - -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
-
-np.random.seed(2018)
-
-n = 500
-n_boostraps = 100
-degree = 18  # A quite high value, just to show.
-noise = 0.1
-
-# Make data set.
-x = np.linspace(-1, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)
-
-# Hold out some test data that is never used in training.
-x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-
-# Combine x transformation and model into one operation.
-# Not neccesary, but convenient.
-model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
-
-# The following (m x n_bootstraps) matrix holds the column vectors y_pred
-# for each bootstrap iteration.
-y_pred = np.empty((y_test.shape[0], n_boostraps))
-for i in range(n_boostraps):
-    x_, y_ = resample(x_train, y_train)
-
-    # Evaluate the new model on the same test data each time.
-    y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
-
-# Note: Expectations and variances taken w.r.t. different training
-# data sets, hence the axis=1. Subsequent means are taken across the test data
-# set in order to obtain a total value, but before this we have error/bias/variance
-# calculated per data point in the test set.
-# Note 2: The use of keepdims=True is important in the calculation of bias as this 
-# maintains the column vector form. Dropping this yields very unexpected results.
-error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-print('Error:', error)
-print('Bias^2:', bias)
-print('Var:', variance)
-print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
-
-plt.plot(x[::5, :], y[::5, :], label='f(x)')
-plt.scatter(x_test, y_test, label='Data points')
-plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')
-plt.legend()
-plt.show()
-

    @@ -338,7 +312,7 @@ plt.show()

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    • diff --git a/doc/pub/week37/html/._week37-bs031.html b/doc/pub/week37/html/._week37-bs031.html index 94e215c0b..bb66758f6 100644 --- a/doc/pub/week37/html/._week37-bs031.html +++ b/doc/pub/week37/html/._week37-bs031.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,57 +273,25 @@ MathJax.Hub.Config({ -

    Understanding what happens

    +

    Resampling methods: More Bootstrap background

    +

    +In the case that \( \widehat{\beta} \) has +more than one component, and the components are independent, we use the +same estimator on each component separately. If the probability +density function of \( X_i \), \( p(x) \), had been known, then it would have +been straightforward to do this by: - -

    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
+
      
+
    1.  Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).
      2.  
+
    2.  Then using these numbers, we could compute a replica of \( \widehat{\beta} \) called \( \widehat{\beta}^* \).
      3.  
+
    
 
-np.random.seed(2018)
+By repeated use of the above two points, many
+estimates of \( \widehat{\beta} \) can  be obtained. The
+idea is to use the relative frequency of \( \widehat{\beta}^* \)
+(think of a histogram) as an estimate of \( p(\boldsymbol{t}) \).
 
-n = 40
-n_boostraps = 100
-maxdegree = 14
-
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-
-for degree in range(maxdegree):
-    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
-    y_pred = np.empty((y_test.shape[0], n_boostraps))
-    for i in range(n_boostraps):
-        x_, y_ = resample(x_train, y_train)
-        y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
-
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-plt.show()
-

    @@ -329,6 +317,8 @@ plt.show()

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    • diff --git a/doc/pub/week37/html/._week37-bs032.html b/doc/pub/week37/html/._week37-bs032.html index d5abaff49..9429955bb 100644 --- a/doc/pub/week37/html/._week37-bs032.html +++ b/doc/pub/week37/html/._week37-bs032.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -251,41 +271,22 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Summing up

    +

    Resampling methods: Bootstrap approach

    -The bias-variance tradeoff summarizes the fundamental tension in -machine learning, particularly supervised learning, between the -complexity of a model and the amount of training data needed to train -it. Since data is often limited, in practice it is often useful to -use a less-complex model with higher bias, that is a model whose asymptotic -performance is worse than another model because it is easier to -train and less sensitive to sampling noise arising from having a -finite-sized training dataset (smaller variance). +But +unless there is enough information available about the process that +generated \( X_1,X_2,\cdots,X_n \), \( p(x) \) is in general +unknown. Therefore, Efron in 1979 asked the +question: What if we replace \( p(x) \) by the relative frequency +of the observation \( X_i \)?

    -The above equations tell us that in -order to minimize the expected test error, we need to select a -statistical learning method that simultaneously achieves low variance -and low bias. Note that variance is inherently a nonnegative quantity, -and squared bias is also nonnegative. Hence, we see that the expected -test MSE can never lie below \( Var(\epsilon) \), the irreducible error. - -

    -What do we mean by the variance and bias of a statistical learning -method? The variance refers to the amount by which our model would change if we -estimated it using a different training data set. Since the training -data are used to fit the statistical learning method, different -training data sets will result in a different estimate. But ideally the -estimate for our model should not vary too much between training -sets. However, if a method has high variance then small changes in -the training data can result in large changes in the model. In general, more -flexible statistical methods have higher variance. - -

    -You may also find this recent article of interest. +If we draw observations in accordance with +the relative frequency of the observations, will we obtain the same +result in some asymptotic sense? The answer is yes.

    @@ -311,6 +312,9 @@ You may also find this recent 39

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    • diff --git a/doc/pub/week37/html/._week37-bs033.html b/doc/pub/week37/html/._week37-bs033.html index 2313c3c86..e3ed9cf41 100644 --- a/doc/pub/week37/html/._week37-bs033.html +++ b/doc/pub/week37/html/._week37-bs033.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,82 +273,28 @@ MathJax.Hub.Config({ -

    Another Example from Scikit-Learn's Repository

    +

    Resampling methods: Bootstrap steps

    +

    +The independent bootstrap works like this: - -

    """
-============================
-Underfitting vs. Overfitting
-============================
+
      
+
    1.  Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).
      2.  
+
    2.  Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).
      3.  
+
    3.  Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\beta}^* \) by evaluating \( \widehat \beta \) under the observations \( \boldsymbol{x}^* \).
      4.  
+
    4.  Repeat this process \( k \) times.
      5.  
+
    
 
-This example demonstrates the problems of underfitting and overfitting and
-how we can use linear regression with polynomial features to approximate
-nonlinear functions. The plot shows the function that we want to approximate,
-which is a part of the cosine function. In addition, the samples from the
-real function and the approximations of different models are displayed. The
-models have polynomial features of different degrees. We can see that a
-linear function (polynomial with degree 1) is not sufficient to fit the
-training samples. This is called **underfitting**. A polynomial of degree 4
-approximates the true function almost perfectly. However, for higher degrees
-the model will **overfit** the training data, i.e. it learns the noise of the
-training data.
-We evaluate quantitatively **overfitting** / **underfitting** by using
-cross-validation. We calculate the mean squared error (MSE) on the validation
-set, the higher, the less likely the model generalizes correctly from the
-training data.
-"""
+When you are done, you can draw a histogram of the relative frequency
+of \( \widehat \beta^* \). This is your estimate of the probability
+distribution \( p(t) \). Using this probability distribution you can
+estimate any statistics thereof. In principle you never draw the
+histogram of the relative frequency of \( \widehat{\beta}^* \). Instead
+you use the estimators corresponding to the statistic of interest. For
+example, if you are interested in estimating the variance of \( \widehat
+\beta \), apply the etsimator \( \widehat \sigma^2 \) to the values
+\( \widehat \beta^* \).
 
-print(__doc__)
-
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.linear_model import LinearRegression
-from sklearn.model_selection import cross_val_score
-
-
-def true_fun(X):
-    return np.cos(1.5 * np.pi * X)
-
-np.random.seed(0)
-
-n_samples = 30
-degrees = [1, 4, 15]
-
-X = np.sort(np.random.rand(n_samples))
-y = true_fun(X) + np.random.randn(n_samples) * 0.1
-
-plt.figure(figsize=(14, 5))
-for i in range(len(degrees)):
-    ax = plt.subplot(1, len(degrees), i + 1)
-    plt.setp(ax, xticks=(), yticks=())
-
-    polynomial_features = PolynomialFeatures(degree=degrees[i],
-                                             include_bias=False)
-    linear_regression = LinearRegression()
-    pipeline = Pipeline([("polynomial_features", polynomial_features),
-                         ("linear_regression", linear_regression)])
-    pipeline.fit(X[:, np.newaxis], y)
-
-    # Evaluate the models using crossvalidation
-    scores = cross_val_score(pipeline, X[:, np.newaxis], y,
-                             scoring="neg_mean_squared_error", cv=10)
-
-    X_test = np.linspace(0, 1, 100)
-    plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
-    plt.plot(X_test, true_fun(X_test), label="True function")
-    plt.scatter(X, y, edgecolor='b', s=20, label="Samples")
-    plt.xlabel("x")
-    plt.ylabel("y")
-    plt.xlim((0, 1))
-    plt.ylim((-2, 2))
-    plt.legend(loc="best")
-    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
-        degrees[i], -scores.mean(), scores.std()))
-plt.show()
-

    @@ -352,6 +318,10 @@ plt.show()

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    • diff --git a/doc/pub/week37/html/._week37-bs034.html b/doc/pub/week37/html/._week37-bs034.html index 3197dd223..8005b6762 100644 --- a/doc/pub/week37/html/._week37-bs034.html +++ b/doc/pub/week37/html/._week37-bs034.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -251,25 +271,61 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Various steps in cross-validation

    +

    Code example for the Bootstrap method

    -When the repetitive splitting of the data set is done randomly, -samples may accidently end up in a fast majority of the splits in -either training or test set. Such samples may have an unbalanced -influence on either model building or prediction evaluation. To avoid -this \( k \)-fold cross-validation structures the data splitting. The -samples are divided into \( k \) more or less equally sized exhaustive and -mutually exclusive subsets. In turn (at each split) one of these -subsets plays the role of the test set while the union of the -remaining subsets constitutes the training set. Such a splitting -warrants a balanced representation of each sample in both training and -test set over the splits. Still the division into the \( k \) subsets -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). +The following code starts with a Gaussian distribution with mean value +\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data +used in the bootstrap analysis. The bootstrap analysis returns a data +set after a given number of bootstrap operations (as many as we have +data points). This data set consists of estimated mean values for each +bootstrap operation. The histogram generated by the bootstrap method +shows that the distribution for these mean values is also a Gaussian, +centered around the mean value \( \mu=100 \) but with standard deviation +\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in +this case the same as the number of original data points). The value +of the standard deviation is what we expect from the central limit +theorem. + +

    + + +

    from numpy import *
+from numpy.random import randint, randn
+from time import time
+import matplotlib.mlab as mlab
+import matplotlib.pyplot as plt
+
+# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below
+
+def statistics(data):
+    return mean(data)
+
+
+# Bootstrap algorithm
+def bootstrap(data, statistic, R):
+    t = zeros(R); n = len(data); inds = arange(n); t0 = time()
+    # non-parametric bootstrap         
+    for i in range(R):
+        t[i] = statistic(data[randint(0,n,n)])
+
+    # analysis    
+    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
+    return t
+
+
+mu, sigma = 100, 15
+datapoints = 10000
+x = mu + sigma*random.randn(datapoints)
+# bootstrap returns the data sample                                    
+t = bootstrap(x, statistics, datapoints)
+
    +

    +We see that our new variance and from that the standard deviation, agrees with the central limit theorem.

    @@ -293,6 +349,11 @@ cross-validation (LOOCV).

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    • diff --git a/doc/pub/week37/html/._week37-bs035.html b/doc/pub/week37/html/._week37-bs035.html index c2319bfa8..7a5cee7c4 100644 --- a/doc/pub/week37/html/._week37-bs035.html +++ b/doc/pub/week37/html/._week37-bs035.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -251,37 +271,25 @@ MathJax.Hub.Config({

     

     

     

    - + -

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

    +

    Plotting the Histogram

    +

    -

    + +
    # the histogram of the bootstrapped  data                                                                                                    
+n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
 
-$$
-\begin{align*}
-\boldsymbol{\beta}_{-i}(\lambda) & =  ( \boldsymbol{X}_{-i, \ast}^{T}
-\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
-\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
-\end{align*}
-$$
-
-
-
      
-
    •  Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). 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
      • 
-
    
-
-$$
-\begin{align*}
-\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
-\end{align*}
-$$
+# add a 'best fit' line  
+y = mlab.normpdf( binsboot, mean(t), std(t))
+lt = plt.plot(binsboot, y, 'r--', linewidth=1)
+plt.xlabel('Smarts')
+plt.ylabel('Probability')
+plt.axis([99.5, 100.6, 0, 3.0])
+plt.grid(True)
 
+plt.show()
+

    @@ -303,6 +311,12 @@ $$

  • 39
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    • +
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    • +
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    • +
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    • +
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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 8c99bfee7..ed5929a32 100644 --- a/doc/pub/week37/html/._week37-bs036.html +++ b/doc/pub/week37/html/._week37-bs036.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,26 +273,70 @@ MathJax.Hub.Config({ -

    Cross-validation in brief

    +

    The bias-variance tradeoff

    -For the various values of \( k \) +We will discuss the bias-variance tradeoff in the context of +continuous predictions such as regression. However, many of the +intuitions and ideas discussed here also carry over to classification +tasks. Consider a dataset \( \mathcal{L} \) consisting of the data +\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). -

      -
    1. shuffle the dataset randomly.
      2. -
    2. Split the dataset into \( k \) groups.
      3. -
    3. For each unique group: +

      +Let us assume that the true data is generated from a noisy model -

        -
      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. -
      +$$ +\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon} +$$ -
    4. Summarize the model using the sample of model evaluation scores
      5. -
    +

    +where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \). +

    +In our derivation of the ordinary least squares method we defined then +an approximation to the function \( f \) in terms of the parameters +\( \boldsymbol{\beta} \) and the design matrix \( \boldsymbol{X} \) which embody our model, +that is \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \). + +

    +Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function +$$ +C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. +$$ + +

    +We can rewrite this as +$$ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. +$$ + +

    +The three terms represent the square of the bias of the learning +method, which can be thought of as the error caused by the simplifying +assumptions built into the method. The second term represents the +variance of the chosen model and finally the last terms is variance of +the error \( \boldsymbol{\epsilon} \). + +

    +To derive this equation, we need to recall that the variance of \( \boldsymbol{y} \) and \( \boldsymbol{\epsilon} \) are both equal to \( \sigma^2 \). The mean value of \( \boldsymbol{\epsilon} \) is by definition equal to zero. Furthermore, the function \( f \) is not a stochastics variable, idem for \( \boldsymbol{\tilde{y}} \). +We use a more compact notation in terms of the expectation value +$$ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right], +$$ + +and adding and subtracting \( \mathbb{E}\left[\boldsymbol{\tilde{y}}\right] \) we get +$$ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right], +$$ + +which, using the abovementioned expectation values can be rewritten as +$$ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2, +$$ + +that is the rewriting in terms of the so-called bias, the variance of the model \( \boldsymbol{\tilde{y}} \) and the variance of \( \boldsymbol{\epsilon} \). + +

    diff --git a/doc/pub/week37/html/._week37-bs037.html b/doc/pub/week37/html/._week37-bs037.html index 6e8fb56b3..5492b8212 100644 --- a/doc/pub/week37/html/._week37-bs037.html +++ b/doc/pub/week37/html/._week37-bs037.html @@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source 2, None, 'resampling-methods-bootstrap'), + ('The Central Limit Theorem', + 2, + None, + 'the-central-limit-theorem'), + ('Finding the Limit', 2, None, 'finding-the-limit'), + ('Rewriting the $\\delta$-function', + 2, + None, + 'rewriting-the-delta-function'), + ('Identifying Terms', 2, None, 'identifying-terms'), + ('Wrapping it up', 2, None, 'wrapping-it-up'), + ('Confidence Intervals', 2, None, 'confidence-intervals'), ('Resampling methods: Bootstrap background', 2, None, @@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source 2, None, 'code-example-for-the-bootstrap-method'), + ('Plotting the Histogram', 2, None, 'plotting-the-histogram'), ('The bias-variance tradeoff', 2, None, @@ -221,23 +234,30 @@ MathJax.Hub.Config({
  • Resampling methods: Jackknife
  • Jackknife code example
  • Resampling methods: Bootstrap
    • -
  • Resampling methods: Bootstrap background
    • -
  • Resampling methods: More Bootstrap background
    • -
  • Resampling methods: Bootstrap approach
    • -
  • Resampling methods: Bootstrap steps
    • -
  • Code example for the Bootstrap method
    • -
  • The bias-variance tradeoff
    • -
  • Example code for Bias-Variance tradeoff
    • -
  • Understanding what happens
    • -
  • 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 with Ridge
    • +
  • The Central Limit Theorem
    • +
  • Finding the Limit
    • +
  • Rewriting the \( \delta \)-function
    • +
  • Identifying Terms
    • +
  • Wrapping it up
    • +
  • Confidence Intervals
    • +
  • Resampling methods: Bootstrap background
    • +
  • Resampling methods: More Bootstrap background
    • +
  • Resampling methods: Bootstrap approach
    • +
  • Resampling methods: Bootstrap steps
    • +
  • Code example for the Bootstrap method
    • +
  • Plotting the Histogram
    • +
  • The bias-variance tradeoff
    • +
  • Example code for Bias-Variance tradeoff
    • +
  • Understanding what happens
    • +
  • 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 with Ridge
    • @@ -253,101 +273,63 @@ MathJax.Hub.Config({ -

    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. +

    Example code for Bias-Variance tradeoff

    -

    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
+
    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
 from sklearn.preprocessing import PolynomialFeatures
+from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
+from sklearn.utils import resample
 
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
+np.random.seed(2018)
 
-# Generate the data.
-nsamples = 100
-x = np.random.randn(nsamples)
-y = 3*x**2 + np.random.randn(nsamples)
+n = 500
+n_boostraps = 100
+degree = 18  # A quite high value, just to show.
+noise = 0.1
 
-## Cross-validation on Ridge regression using KFold only
+# Make data set.
+x = np.linspace(-1, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)
 
-# Decide degree on polynomial to fit
-poly = PolynomialFeatures(degree = 6)
+# Hold out some test data that is never used in training.
+x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
 
-# Decide which values of lambda to use
-nlambdas = 500
-lambdas = np.logspace(-3, 5, nlambdas)
+# Combine x transformation and model into one operation.
+# Not neccesary, but convenient.
+model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
 
-# Initialize a KFold instance
-k = 5
-kfold = KFold(n_splits = k)
+# The following (m x n_bootstraps) matrix holds the column vectors y_pred
+# for each bootstrap iteration.
+y_pred = np.empty((y_test.shape[0], n_boostraps))
+for i in range(n_boostraps):
+    x_, y_ = resample(x_train, y_train)
 
-# Perform the cross-validation to estimate MSE
-scores_KFold = np.zeros((nlambdas, k))
+    # Evaluate the new model on the same test data each time.
+    y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
 
-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')
+# Note: Expectations and variances taken w.r.t. different training
+# data sets, hence the axis=1. Subsequent means are taken across the test data
+# set in order to obtain a total value, but before this we have error/bias/variance
+# calculated per data point in the test set.
+# Note 2: The use of keepdims=True is important in the calculation of bias as this 
+# maintains the column vector form. Dropping this yields very unexpected results.
+error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+print('Error:', error)
+print('Bias^2:', bias)
+print('Var:', variance)
+print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
 
+plt.plot(x[::5, :], y[::5, :], label='f(x)')
+plt.scatter(x_test, y_test, label='Data points')
+plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')
 plt.legend()
-
 plt.show()
 
    
 
    
@@ -369,6 +351,14 @@ plt.show()
   
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    • 
 
 
diff --git a/doc/pub/week37/html/._week37-bs038.html b/doc/pub/week37/html/._week37-bs038.html
index d27c4bdb2..765c37a2d 100644
--- a/doc/pub/week37/html/._week37-bs038.html
+++ b/doc/pub/week37/html/._week37-bs038.html
@@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'resampling-methods-bootstrap'),
+              ('The Central Limit Theorem',
+               2,
+               None,
+               'the-central-limit-theorem'),
+              ('Finding the Limit', 2, None, 'finding-the-limit'),
+              ('Rewriting the $\\delta$-function',
+               2,
+               None,
+               'rewriting-the-delta-function'),
+              ('Identifying Terms', 2, None, 'identifying-terms'),
+              ('Wrapping it up', 2, None, 'wrapping-it-up'),
+              ('Confidence Intervals', 2, None, 'confidence-intervals'),
               ('Resampling methods: Bootstrap background',
                2,
                None,
@@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'code-example-for-the-bootstrap-method'),
+              ('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
               ('The bias-variance tradeoff',
                2,
                None,
@@ -221,23 +234,30 @@ MathJax.Hub.Config({
       
  • Resampling methods: Jackknife
    • 
       
  • Jackknife code example
    • 
       
  • Resampling methods: Bootstrap
    • 
-      
  • Resampling methods: Bootstrap background
    • 
-      
  • Resampling methods: More Bootstrap background
    • 
-      
  • Resampling methods: Bootstrap approach
    • 
-      
  • Resampling methods: Bootstrap steps
    • 
-      
  • Code example for the Bootstrap method
    • 
-      
  • The bias-variance tradeoff
    • 
-      
  • Example code for Bias-Variance tradeoff
    • 
-      
  • Understanding what happens
    • 
-      
  • 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 with Ridge
    • 
+      
  • The Central Limit Theorem
    • 
+      
  • Finding the Limit
    • 
+      
  • Rewriting the \( \delta \)-function
    • 
+      
  • Identifying Terms
    • 
+      
  • Wrapping it up
    • 
+      
  • Confidence Intervals
    • 
+      
  • Resampling methods: Bootstrap background
    • 
+      
  • Resampling methods: More Bootstrap background
    • 
+      
  • Resampling methods: Bootstrap approach
    • 
+      
  • Resampling methods: Bootstrap steps
    • 
+      
  • Code example for the Bootstrap method
    • 
+      
  • Plotting the Histogram
    • 
+      
  • The bias-variance tradeoff
    • 
+      
  • Example code for Bias-Variance tradeoff
    • 
+      
  • Understanding what happens
    • 
+      
  • 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 with Ridge
    • 
 
         
       
@@ -253,87 +273,54 @@ MathJax.Hub.Config({
 
 
 
-
    More examples on bootstrap and cross-validation and errors 
    
-
+
    Understanding what happens 
    
 
    
 
 
-
    # Common imports
-import os
+
    import matplotlib.pyplot as plt
 import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
 from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.preprocessing import PolynomialFeatures
 from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
 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/"
 
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
+np.random.seed(2018)
 
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
+n = 40
+n_boostraps = 100
+maxdegree = 14
 
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
 
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+error = np.zeros(maxdegree)
+bias = np.zeros(maxdegree)
+variance = np.zeros(maxdegree)
+polydegree = np.zeros(maxdegree)
+x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
 
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
+for degree in range(maxdegree):
+    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+    y_pred = np.empty((y_test.shape[0], n_boostraps))
+    for i in range(n_boostraps):
+        x_, y_ = resample(x_train, y_train)
+        y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
 
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    polydegree[degree] = degree
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organize the data into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-
-Maxpolydegree = 30
-X = np.zeros((len(Density),Maxpolydegree))
-X[:,0] = 1.0
-testerror = np.zeros(Maxpolydegree)
-trainingerror = np.zeros(Maxpolydegree)
-polynomial = np.zeros(Maxpolydegree)
-
-trials = 100
-for polydegree in range(1, Maxpolydegree):
-    polynomial[polydegree] = polydegree
-    for degree in range(polydegree):
-        X[:,degree] = Density**(degree/3.0)
-
-# 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=True).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) 
-
-    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.plot(polydegree, error, label='Error')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
 plt.legend()
 plt.show()
 
    
@@ -355,6 +342,13 @@ plt.show()
   
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    • 
 
 
diff --git a/doc/pub/week37/html/._week37-bs039.html b/doc/pub/week37/html/._week37-bs039.html
index 3b8e683ba..c44309d4d 100644
--- a/doc/pub/week37/html/._week37-bs039.html
+++ b/doc/pub/week37/html/._week37-bs039.html
@@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'resampling-methods-bootstrap'),
+              ('The Central Limit Theorem',
+               2,
+               None,
+               'the-central-limit-theorem'),
+              ('Finding the Limit', 2, None, 'finding-the-limit'),
+              ('Rewriting the $\\delta$-function',
+               2,
+               None,
+               'rewriting-the-delta-function'),
+              ('Identifying Terms', 2, None, 'identifying-terms'),
+              ('Wrapping it up', 2, None, 'wrapping-it-up'),
+              ('Confidence Intervals', 2, None, 'confidence-intervals'),
               ('Resampling methods: Bootstrap background',
                2,
                None,
@@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'code-example-for-the-bootstrap-method'),
+              ('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
               ('The bias-variance tradeoff',
                2,
                None,
@@ -221,23 +234,30 @@ MathJax.Hub.Config({
       
  • Resampling methods: Jackknife
    • 
       
  • Jackknife code example
    • 
       
  • Resampling methods: Bootstrap
    • 
-      
  • Resampling methods: Bootstrap background
    • 
-      
  • Resampling methods: More Bootstrap background
    • 
-      
  • Resampling methods: Bootstrap approach
    • 
-      
  • Resampling methods: Bootstrap steps
    • 
-      
  • Code example for the Bootstrap method
    • 
-      
  • The bias-variance tradeoff
    • 
-      
  • Example code for Bias-Variance tradeoff
    • 
-      
  • Understanding what happens
    • 
-      
  • 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 with Ridge
    • 
+      
  • The Central Limit Theorem
    • 
+      
  • Finding the Limit
    • 
+      
  • Rewriting the \( \delta \)-function
    • 
+      
  • Identifying Terms
    • 
+      
  • Wrapping it up
    • 
+      
  • Confidence Intervals
    • 
+      
  • Resampling methods: Bootstrap background
    • 
+      
  • Resampling methods: More Bootstrap background
    • 
+      
  • Resampling methods: Bootstrap approach
    • 
+      
  • Resampling methods: Bootstrap steps
    • 
+      
  • Code example for the Bootstrap method
    • 
+      
  • Plotting the Histogram
    • 
+      
  • The bias-variance tradeoff
    • 
+      
  • Example code for Bias-Variance tradeoff
    • 
+      
  • Understanding what happens
    • 
+      
  • 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 with Ridge
    • 
 
         
       
@@ -253,79 +273,40 @@ MathJax.Hub.Config({
 
 
 
-
    The same example but now with cross-validation 
    
+
    Summing up 
    
 
 
    
+The bias-variance tradeoff summarizes the fundamental tension in
+machine learning, particularly supervised learning, between the
+complexity of a model and the amount of training data needed to train
+it.  Since data is often limited, in practice it is often useful to
+use a less-complex model with higher bias, that is  a model whose asymptotic
+performance is worse than another model because it is easier to
+train and less sensitive to sampling noise arising from having a
+finite-sized training dataset (smaller variance).
 
-
-
    # Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.metrics import mean_squared_error
-from sklearn.model_selection import KFold
-from sklearn.model_selection import cross_val_score
+
    
+The above equations tell us that in
+order to minimize the expected test error, we need to select a
+statistical learning method that simultaneously achieves low variance
+and low bias. Note that variance is inherently a nonnegative quantity,
+and squared bias is also nonnegative. Hence, we see that the expected
+test MSE can never lie below \( Var(\epsilon) \), the irreducible error.
 
+
    
+What do we mean by the variance and bias of a statistical learning
+method? The variance refers to the amount by which our model would change if we
+estimated it using a different training data set. Since the training
+data are used to fit the statistical learning method, different
+training data sets  will result in a different estimate. But ideally the
+estimate for our model should not vary too much between training
+sets. However, if a method has high variance  then small changes in
+the training data can result in large changes in the model. In general, more
+flexible statistical methods have higher variance.
 
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
+
    
+You may also find this recent article of interest.
 
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organize the data into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-
-Maxpolydegree = 30
-X = np.zeros((len(Density),Maxpolydegree))
-X[:,0] = 1.0
-estimated_mse_sklearn = np.zeros(Maxpolydegree)
-polynomial = np.zeros(Maxpolydegree)
-k =5
-kfold = KFold(n_splits = k)
-
-for polydegree in range(1, Maxpolydegree):
-    polynomial[polydegree] = polydegree
-    for degree in range(polydegree):
-        X[:,degree] = Density**(degree/3.0)
-        OLS = LinearRegression()
-# loop over trials in order to estimate the expectation value of the MSE
-    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)
-
-plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
-plt.xlabel('Polynomial degree')
-plt.ylabel('log10[MSE]')
-plt.legend()
-plt.show()
-
    
 
    
 
    
 
@@ -343,6 +324,13 @@ 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 c76d88187..e847a82eb 100644
--- a/doc/pub/week37/html/week37-bs.html
+++ b/doc/pub/week37/html/week37-bs.html
@@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'resampling-methods-bootstrap'),
+              ('The Central Limit Theorem',
+               2,
+               None,
+               'the-central-limit-theorem'),
+              ('Finding the Limit', 2, None, 'finding-the-limit'),
+              ('Rewriting the $\\delta$-function',
+               2,
+               None,
+               'rewriting-the-delta-function'),
+              ('Identifying Terms', 2, None, 'identifying-terms'),
+              ('Wrapping it up', 2, None, 'wrapping-it-up'),
+              ('Confidence Intervals', 2, None, 'confidence-intervals'),
               ('Resampling methods: Bootstrap background',
                2,
                None,
@@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'code-example-for-the-bootstrap-method'),
+              ('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
               ('The bias-variance tradeoff',
                2,
                None,
@@ -221,23 +234,30 @@ MathJax.Hub.Config({
       
  • Resampling methods: Jackknife
    • 
       
  • Jackknife code example
    • 
       
  • Resampling methods: Bootstrap
    • 
-      
  • Resampling methods: Bootstrap background
    • 
-      
  • Resampling methods: More Bootstrap background
    • 
-      
  • Resampling methods: Bootstrap approach
    • 
-      
  • Resampling methods: Bootstrap steps
    • 
-      
  • Code example for the Bootstrap method
    • 
-      
  • The bias-variance tradeoff
    • 
-      
  • Example code for Bias-Variance tradeoff
    • 
-      
  • Understanding what happens
    • 
-      
  • 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 with Ridge
    • 
+      
  • The Central Limit Theorem
    • 
+      
  • Finding the Limit
    • 
+      
  • Rewriting the \( \delta \)-function
    • 
+      
  • Identifying Terms
    • 
+      
  • Wrapping it up
    • 
+      
  • Confidence Intervals
    • 
+      
  • Resampling methods: Bootstrap background
    • 
+      
  • Resampling methods: More Bootstrap background
    • 
+      
  • Resampling methods: Bootstrap approach
    • 
+      
  • Resampling methods: Bootstrap steps
    • 
+      
  • Code example for the Bootstrap method
    • 
+      
  • Plotting the Histogram
    • 
+      
  • The bias-variance tradeoff
    • 
+      
  • Example code for Bias-Variance tradeoff
    • 
+      
  • Understanding what happens
    • 
+      
  • 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 with Ridge
    • 
 
         
       
@@ -296,7 +316,7 @@ MathJax.Hub.Config({
   
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    • 
   
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    • 
   
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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 024e972a1..36b78168e 100644
--- a/doc/pub/week37/html/week37-reveal.html
+++ b/doc/pub/week37/html/week37-reveal.html
@@ -896,6 +896,171 @@ Before we proceed however, we need to remind ourselves about a central theorem i
 
 
 
+
    
+
    The Central Limit Theorem 
    
+
+
    
+Suppose we have a PDF \( p(x) \) from which we generate  a series \( N \)
+of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
+is viewed as the average of a specific measurement, e.g., throwing 
+dice 100 times and then taking the average value, or producing a certain
+amount of random numbers. 
+For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
+which follows.
+
+
    
+If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
+
+
     
    
+$$
+   z=\frac{x_1+x_2+\dots+x_m}{m},
+$$
+
     
    
+
+the question we pose is which is the PDF of the new variable \( z \).
+
    
+
+
+
    
+
    Finding the Limit 
    
+
+
    
+The probability of obtaining an average value \( z \) is the product of the 
+probabilities of obtaining arbitrary individual mean values \( x_i \),
+but with the constraint that the average is \( z \). We can express this through
+the following expression
+
     
    
+$$
+    \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
+    \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
+$$
+
     
    
+
+where the \( \delta \)-function enbodies the constraint that the mean is \( z \).
+All measurements that lead to each individual \( x_i \) are expected to
+be independent, which in turn means that we can express \( \tilde{p} \) as the 
+product of individual \( p(x_i) \).  The independence assumption is important in the derivation of the central limit theorem.
+
    
+
+
+
    
+
    Rewriting the \( \delta \)-function 
    
+
+
    
+If we use the integral expression for the \( \delta \)-function
+
+
     
    
+$$
+   \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
+$$
+
     
    
+
+and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value
+we arrive at
+
     
    
+$$
+   \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
+   dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
+$$
+
     
    
+
+with the integral over \( x \) resulting in
+
+
     
    
+$$
+  \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
+  \int_{-\infty}^{\infty}dxp(x)
+   \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
+$$
+
     
    
+
    
+
+
+
    
+
    Identifying Terms 
    
+
+
    
+The second term on the rhs disappears since this is just the mean and 
+employing the definition of \( \sigma^2 \) we have 
+
     
    
+$$
+  \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
+  1-\frac{q^2\sigma^2}{2m^2}+\dots,
+$$
+
     
    
+
+resulting in
+
+
     
    
+$$
+  \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
+  \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
+$$
+
     
    
+
+and in the limit \( m\rightarrow \infty \) we obtain
+
+
     
    
+$$
+   \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
+    \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
+$$
+
     
    
+
+which is the normal distribution with variance
+\( \sigma^2_m=\sigma^2/m \), where \( \sigma \) is the variance of the PDF \( p(x) \)
+and \( \mu \) is also the mean of the PDF \( p(x) \).
+
    
+
+
+
    
+
    Wrapping it up 
    
+
+
    
+Thus, the central limit theorem states that the PDF \( \tilde{p}(z) \) of
+the average of \( m \) random values corresponding to a PDF \( p(x) \) 
+is a normal distribution whose mean is the 
+mean value of the PDF \( p(x) \) and whose variance is the variance
+of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
+
+
    
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed expression for
+\( \tilde{p}(x) \).
+The central limit theorem leads then to the well-known expression for the
+standard deviation, given by
+
+
     
    
+$$
+    \sigma_m=
+\frac{\sigma}{\sqrt{m}}.
+$$
+
     
    
+
+
    
+The latter is true only if the average value is known exactly. This is obtained in the limit
+\( m\rightarrow \infty \)  only. Because the mean and the variance are measured quantities we obtain 
+the familiar expression in statistics
+
     
    
+$$
+    \sigma_m\approx 
+\frac{\sigma}{\sqrt{m-1}}.
+$$
+
     
    
+
+
    
+In many cases however the above estimate for the standard deviation,
+in particular if correlations are strong, may be too simplistic.
+
    
+
+
+
    
+
    Confidence Intervals 
    
+
    
+
+
 
    
 
    Resampling methods: Bootstrap background 
    
 
@@ -1005,10 +1170,12 @@ theorem.
 import matplotlib.mlab as mlab
 import matplotlib.pyplot as plt
 
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
+# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below
+
+def statistics(data):
     return mean(data)
 
+
 # Bootstrap algorithm
 def bootstrap(data, statistic, R):
     t = zeros(R); n = len(data); inds = arange(n); t0 = time()
@@ -1027,8 +1194,19 @@ mu, sigma = 100, 10000
 x = mu + sigma*random.randn(datapoints)
 # bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
-# the histogram of the bootstrapped  data                                                                                                    
+t = bootstrap(x, statistics, datapoints)
+
    
+
    
+We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
+
+
+
+
    
+
    Plotting the Histogram 
    
+
    
+
+
+
    # the histogram of the bootstrapped  data                                                                                                    
 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
 
 # add a 'best fit' line  
diff --git a/doc/pub/week37/html/week37-solarized.html b/doc/pub/week37/html/week37-solarized.html
index 1a7823d9f..baf9dd49b 100644
--- a/doc/pub/week37/html/week37-solarized.html
+++ b/doc/pub/week37/html/week37-solarized.html
@@ -115,6 +115,18 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'resampling-methods-bootstrap'),
+              ('The Central Limit Theorem',
+               2,
+               None,
+               'the-central-limit-theorem'),
+              ('Finding the Limit', 2, None, 'finding-the-limit'),
+              ('Rewriting the $\\delta$-function',
+               2,
+               None,
+               'rewriting-the-delta-function'),
+              ('Identifying Terms', 2, None, 'identifying-terms'),
+              ('Wrapping it up', 2, None, 'wrapping-it-up'),
+              ('Confidence Intervals', 2, None, 'confidence-intervals'),
               ('Resampling methods: Bootstrap background',
                2,
                None,
@@ -135,6 +147,7 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'code-example-for-the-bootstrap-method'),
+              ('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
               ('The bias-variance tradeoff',
                2,
                None,
@@ -918,6 +931,150 @@ Before we proceed however, we need to remind ourselves about a central theorem i
 
    
 









    
 
+
    The Central Limit Theorem 
    
+
+
    
+Suppose we have a PDF \( p(x) \) from which we generate  a series \( N \)
+of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
+is viewed as the average of a specific measurement, e.g., throwing 
+dice 100 times and then taking the average value, or producing a certain
+amount of random numbers. 
+For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
+which follows.
+
+
    
+If we compute the mean \( z \) of \( m \) such mean values \( x_i \)   
+$$
+   z=\frac{x_1+x_2+\dots+x_m}{m},
+$$
+
+the question we pose is which is the PDF of the new variable \( z \).
+
+
    
+









    
+
+
    Finding the Limit 
    
+
+
    
+The probability of obtaining an average value \( z \) is the product of the 
+probabilities of obtaining arbitrary individual mean values \( x_i \),
+but with the constraint that the average is \( z \). We can express this through
+the following expression
+$$
+    \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
+    \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
+$$
+
+where the \( \delta \)-function enbodies the constraint that the mean is \( z \).
+All measurements that lead to each individual \( x_i \) are expected to
+be independent, which in turn means that we can express \( \tilde{p} \) as the 
+product of individual \( p(x_i) \).  The independence assumption is important in the derivation of the central limit theorem.
+
+
    
+









    
+
+
    Rewriting the \( \delta \)-function 
    
+
+
    
+If we use the integral expression for the \( \delta \)-function
+
+$$
+   \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
+$$
+
+and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value
+we arrive at
+$$
+   \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
+   dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
+$$
+
+with the integral over \( x \) resulting in
+
+$$
+  \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
+  \int_{-\infty}^{\infty}dxp(x)
+   \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
+$$
+
+
    
+









    
+
+
    Identifying Terms 
    
+
+
    
+The second term on the rhs disappears since this is just the mean and 
+employing the definition of \( \sigma^2 \) we have 
+$$
+  \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
+  1-\frac{q^2\sigma^2}{2m^2}+\dots,
+$$
+
+resulting in 
+
+$$
+  \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
+  \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
+$$
+
+and in the limit \( m\rightarrow \infty \) we obtain 
+
+$$
+   \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
+    \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
+$$
+
+which is the normal distribution with variance
+\( \sigma^2_m=\sigma^2/m \), where \( \sigma \) is the variance of the PDF \( p(x) \)
+and \( \mu \) is also the mean of the PDF \( p(x) \).
+
+
    
+









    
+
+
    Wrapping it up 
    
+
+
    
+Thus, the central limit theorem states that the PDF \( \tilde{p}(z) \) of
+the average of \( m \) random values corresponding to a PDF \( p(x) \) 
+is a normal distribution whose mean is the 
+mean value of the PDF \( p(x) \) and whose variance is the variance
+of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
+
+
    
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed expression for
+\( \tilde{p}(x) \).
+The central limit theorem leads then to the well-known expression for the
+standard deviation, given by 
+
+$$
+    \sigma_m=
+\frac{\sigma}{\sqrt{m}}.
+$$
+
+
    
+The latter is true only if the average value is known exactly. This is obtained in the limit
+\( m\rightarrow \infty \)  only. Because the mean and the variance are measured quantities we obtain 
+the familiar expression in statistics
+$$
+    \sigma_m\approx 
+\frac{\sigma}{\sqrt{m-1}}.
+$$
+
+
    
+In many cases however the above estimate for the standard deviation,
+in particular if correlations are strong, may be too simplistic.
+
+
    
+









    
+
+
    Confidence Intervals 
    
+
+
    
+









    
+
 
    Resampling methods: Bootstrap background 
    
 
 
    
@@ -1024,10 +1181,12 @@ theorem.
 import matplotlib.mlab as mlab
 import matplotlib.pyplot as plt
 
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
+# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below
+
+def statistics(data):
     return mean(data)
 
+
 # Bootstrap algorithm
 def bootstrap(data, statistic, R):
     t = zeros(R); n = len(data); inds = arange(n); t0 = time()
@@ -1046,8 +1205,19 @@ mu, sigma = 100, 10000
 x = mu + sigma*random.randn(datapoints)
 # bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
-# the histogram of the bootstrapped  data                                                                                                    
+t = bootstrap(x, statistics, datapoints)
+
    
+
    
+We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
+
+
    
+









    
+
+
    Plotting the Histogram 
    
+
    
+
+
+
    # the histogram of the bootstrapped  data                                                                                                    
 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
 
 # add a 'best fit' line  
diff --git a/doc/pub/week37/html/week37.html b/doc/pub/week37/html/week37.html
index 55b22f101..4155098ea 100644
--- a/doc/pub/week37/html/week37.html
+++ b/doc/pub/week37/html/week37.html
@@ -120,6 +120,18 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'resampling-methods-bootstrap'),
+              ('The Central Limit Theorem',
+               2,
+               None,
+               'the-central-limit-theorem'),
+              ('Finding the Limit', 2, None, 'finding-the-limit'),
+              ('Rewriting the $\\delta$-function',
+               2,
+               None,
+               'rewriting-the-delta-function'),
+              ('Identifying Terms', 2, None, 'identifying-terms'),
+              ('Wrapping it up', 2, None, 'wrapping-it-up'),
+              ('Confidence Intervals', 2, None, 'confidence-intervals'),
               ('Resampling methods: Bootstrap background',
                2,
                None,
@@ -140,6 +152,7 @@ div { text-align: justify; text-justify: inter-word; }
                2,
                None,
                'code-example-for-the-bootstrap-method'),
+              ('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
               ('The bias-variance tradeoff',
                2,
                None,
@@ -923,6 +936,150 @@ Before we proceed however, we need to remind ourselves about a central theorem i
 
    
 









    
 
+
    The Central Limit Theorem 
    
+
+
    
+Suppose we have a PDF \( p(x) \) from which we generate  a series \( N \)
+of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
+is viewed as the average of a specific measurement, e.g., throwing 
+dice 100 times and then taking the average value, or producing a certain
+amount of random numbers. 
+For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
+which follows.
+
+
    
+If we compute the mean \( z \) of \( m \) such mean values \( x_i \)   
+$$
+   z=\frac{x_1+x_2+\dots+x_m}{m},
+$$
+
+the question we pose is which is the PDF of the new variable \( z \).
+
+
    
+









    
+
+
    Finding the Limit 
    
+
+
    
+The probability of obtaining an average value \( z \) is the product of the 
+probabilities of obtaining arbitrary individual mean values \( x_i \),
+but with the constraint that the average is \( z \). We can express this through
+the following expression
+$$
+    \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
+    \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
+$$
+
+where the \( \delta \)-function enbodies the constraint that the mean is \( z \).
+All measurements that lead to each individual \( x_i \) are expected to
+be independent, which in turn means that we can express \( \tilde{p} \) as the 
+product of individual \( p(x_i) \).  The independence assumption is important in the derivation of the central limit theorem.
+
+
    
+









    
+
+
    Rewriting the \( \delta \)-function 
    
+
+
    
+If we use the integral expression for the \( \delta \)-function
+
+$$
+   \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
+$$
+
+and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value
+we arrive at
+$$
+   \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
+   dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
+$$
+
+with the integral over \( x \) resulting in
+
+$$
+  \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
+  \int_{-\infty}^{\infty}dxp(x)
+   \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
+$$
+
+
    
+









    
+
+
    Identifying Terms 
    
+
+
    
+The second term on the rhs disappears since this is just the mean and 
+employing the definition of \( \sigma^2 \) we have 
+$$
+  \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
+  1-\frac{q^2\sigma^2}{2m^2}+\dots,
+$$
+
+resulting in 
+
+$$
+  \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
+  \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
+$$
+
+and in the limit \( m\rightarrow \infty \) we obtain 
+
+$$
+   \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
+    \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
+$$
+
+which is the normal distribution with variance
+\( \sigma^2_m=\sigma^2/m \), where \( \sigma \) is the variance of the PDF \( p(x) \)
+and \( \mu \) is also the mean of the PDF \( p(x) \).
+
+
    
+









    
+
+
    Wrapping it up 
    
+
+
    
+Thus, the central limit theorem states that the PDF \( \tilde{p}(z) \) of
+the average of \( m \) random values corresponding to a PDF \( p(x) \) 
+is a normal distribution whose mean is the 
+mean value of the PDF \( p(x) \) and whose variance is the variance
+of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
+
+
    
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite \( m \), it is not always possible to find a closed expression for
+\( \tilde{p}(x) \).
+The central limit theorem leads then to the well-known expression for the
+standard deviation, given by 
+
+$$
+    \sigma_m=
+\frac{\sigma}{\sqrt{m}}.
+$$
+
+
    
+The latter is true only if the average value is known exactly. This is obtained in the limit
+\( m\rightarrow \infty \)  only. Because the mean and the variance are measured quantities we obtain 
+the familiar expression in statistics
+$$
+    \sigma_m\approx 
+\frac{\sigma}{\sqrt{m-1}}.
+$$
+
+
    
+In many cases however the above estimate for the standard deviation,
+in particular if correlations are strong, may be too simplistic.
+
+
    
+









    
+
+
    Confidence Intervals 
    
+
+
    
+









    
+
 
    Resampling methods: Bootstrap background 
    
 
 
    
@@ -1029,10 +1186,12 @@ theorem.
 import matplotlib.mlab as mlab
 import matplotlib.pyplot as plt
 
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
+# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below
+
+def statistics(data):
     return mean(data)
 
+
 # Bootstrap algorithm
 def bootstrap(data, statistic, R):
     t = zeros(R); n = len(data); inds = arange(n); t0 = time()
@@ -1051,8 +1210,19 @@ mu, sigma = 100
 datapoints = 10000
 x = mu + sigma*random.randn(datapoints)
 # bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
-# the histogram of the bootstrapped  data                                                                                                    
+t = bootstrap(x, statistics, datapoints)
+
    
+
    
+We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
+
+
    
+









    
+
+
    Plotting the Histogram 
    
+
    
+
+
+
    # the histogram of the bootstrapped  data                                                                                                    
 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
 
 # add a 'best fit' line  
diff --git a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz
index 788a81b6af88d0b76beb66c2d1e679f2ab738c18..bc36c717dd8313633496fa3e41eca5ddc50b0cb2 100644
GIT binary patch
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zWVw@GI_rcPPHCz$DvRpf+%Q&_ANI_zz%&2Ev62?H``%Srfzl3ht!ua;)=3u0wpTe6
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diff --git a/doc/pub/week37/ipynb/week37.ipynb b/doc/pub/week37/ipynb/week37.ipynb
index ba7ad38df..e18b4bcde 100644
--- a/doc/pub/week37/ipynb/week37.ipynb
+++ b/doc/pub/week37/ipynb/week37.ipynb
@@ -881,6 +881,231 @@
     "\n",
     "Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called **central limit theorem**.\n",
     "\n",
+    "## The Central Limit Theorem\n",
+    "\n",
+    "\n",
+    "Suppose we have a PDF $p(x)$ from which we generate  a series $N$\n",
+    "of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n",
+    "is viewed as the average of a specific measurement, e.g., throwing \n",
+    "dice 100 times and then taking the average value, or producing a certain\n",
+    "amount of random numbers. \n",
+    "For notational ease, we set $\\langle x_i \\rangle=x_i$ in the discussion\n",
+    "which follows. \n",
+    "\n",
+    "If we compute the mean $z$ of $m$ such mean values $x_i$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "z=\\frac{x_1+x_2+\\dots+x_m}{m},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "the question we pose is which is the PDF of the new variable $z$.\n",
+    "\n",
+    "## Finding the Limit\n",
+    "\n",
+    "The probability of obtaining an average value $z$ is the product of the \n",
+    "probabilities of obtaining arbitrary individual mean values $x_i$,\n",
+    "but with the constraint that the average is $z$. We can express this through\n",
+    "the following expression"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\tilde{p}(z)=\\int dx_1p(x_1)\\int dx_2p(x_2)\\dots\\int dx_mp(x_m)\n",
+    "    \\delta(z-\\frac{x_1+x_2+\\dots+x_m}{m}),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where the $\\delta$-function enbodies the constraint that the mean is $z$.\n",
+    "All measurements that lead to each individual $x_i$ are expected to\n",
+    "be independent, which in turn means that we can express $\\tilde{p}$ as the \n",
+    "product of individual $p(x_i)$.  The independence assumption is important in the derivation of the central limit theorem.\n",
+    "\n",
+    "\n",
+    "## Rewriting the $\\delta$-function\n",
+    "\n",
+    "If we use the integral expression for the $\\delta$-function"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\delta(z-\\frac{x_1+x_2+\\dots+x_m}{m})=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
+    "   dq\\exp{\\left(iq(z-\\frac{x_1+x_2+\\dots+x_m}{m})\\right)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "and inserting $e^{i\\mu q-i\\mu q}$ where $\\mu$ is the mean value\n",
+    "we arrive at"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\tilde{p}(z)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\n",
+    "   dq\\exp{\\left(iq(z-\\mu)\\right)}\\left[\\int_{-\\infty}^{\\infty}\n",
+    "   dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}\\right]^m,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "with the integral over $x$ resulting in"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}=\n",
+    "  \\int_{-\\infty}^{\\infty}dxp(x)\n",
+    "   \\left[1+\\frac{iq(\\mu-x)}{m}-\\frac{q^2(\\mu-x)^2}{2m^2}+\\dots\\right].\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Identifying Terms\n",
+    "\n",
+    "The second term on the rhs disappears since this is just the mean and \n",
+    "employing the definition of $\\sigma^2$ we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\int_{-\\infty}^{\\infty}dxp(x)e^{\\left(iq(\\mu-x)/m\\right)}=\n",
+    "  1-\\frac{q^2\\sigma^2}{2m^2}+\\dots,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "resulting in"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\left[\\int_{-\\infty}^{\\infty}dxp(x)\\exp{\\left(iq(\\mu-x)/m\\right)}\\right]^m\\approx\n",
+    "  \\left[1-\\frac{q^2\\sigma^2}{2m^2}+\\dots \\right]^m,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "and in the limit $m\\rightarrow \\infty$ we obtain"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\tilde{p}(z)=\\frac{1}{\\sqrt{2\\pi}(\\sigma/\\sqrt{m})}\n",
+    "    \\exp{\\left(-\\frac{(z-\\mu)^2}{2(\\sigma/\\sqrt{m})^2}\\right)},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "which is the normal distribution with variance\n",
+    "$\\sigma^2_m=\\sigma^2/m$, where $\\sigma$ is the variance of the PDF $p(x)$\n",
+    "and $\\mu$ is also the mean of the PDF $p(x)$. \n",
+    "\n",
+    "## Wrapping it up\n",
+    "\n",
+    "Thus, the central limit theorem states that the PDF $\\tilde{p}(z)$ of\n",
+    "the average of $m$ random values corresponding to a PDF $p(x)$ \n",
+    "is a normal distribution whose mean is the \n",
+    "mean value of the PDF $p(x)$ and whose variance is the variance\n",
+    "of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.\n",
+    "\n",
+    "The theorem is satisfied by a large class of PDFs. Note however that for a\n",
+    "finite $m$, it is not always possible to find a closed expression for\n",
+    "$\\tilde{p}(x)$.\n",
+    "The central limit theorem leads then to the well-known expression for the\n",
+    "standard deviation, given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\sigma_m=\n",
+    "\\frac{\\sigma}{\\sqrt{m}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The latter is true only if the average value is known exactly. This is obtained in the limit\n",
+    "$m\\rightarrow \\infty$  only. Because the mean and the variance are measured quantities we obtain \n",
+    "the familiar expression in statistics"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\sigma_m\\approx \n",
+    "\\frac{\\sigma}{\\sqrt{m-1}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In many cases however the above estimate for the standard deviation,\n",
+    "in particular if correlations are strong, may be too simplistic.\n",
+    "\n",
+    "## Confidence Intervals\n",
+    "\n",
+    "\n",
     "## Resampling methods: Bootstrap background\n",
     "\n",
     "Since $\\widehat{\\beta} = \\widehat{\\beta}(\\boldsymbol{X})$ is a function of random variables,\n",
@@ -979,10 +1204,12 @@
     "import matplotlib.mlab as mlab\n",
     "import matplotlib.pyplot as plt\n",
     "\n",
-    "# Returns mean of bootstrap samples                                                                                                                                                \n",
-    "def stat(data):\n",
+    "# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below\n",
+    "\n",
+    "def statistics(data):\n",
     "    return mean(data)\n",
     "\n",
+    "\n",
     "# Bootstrap algorithm\n",
     "def bootstrap(data, statistic, R):\n",
     "    t = zeros(R); n = len(data); inds = arange(n); t0 = time()\n",
@@ -1001,7 +1228,27 @@
     "datapoints = 10000\n",
     "x = mu + sigma*random.randn(datapoints)\n",
     "# bootstrap returns the data sample                                    \n",
-    "t = bootstrap(x, stat, datapoints)\n",
+    "t = bootstrap(x, statistics, datapoints)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We see that our new variance and from that the standard deviation, agrees with the central limit theorem.\n",
+    "\n",
+    "## Plotting the Histogram"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false,
+    "editable": true
+   },
+   "outputs": [],
+   "source": [
     "# the histogram of the bootstrapped  data                                                                                                    \n",
     "n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)\n",
     "\n",
diff --git a/doc/src/week37/test.py b/doc/src/week37/test.py
deleted file mode 100644
index 958bceb5f..000000000
--- a/doc/src/week37/test.py
+++ /dev/null
@@ -1,38 +0,0 @@
-import numpy as np
-import pandas as pd
-from IPython.display import display
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-
-# Make data set.
-n = 1000
-x = np.random.rand(n)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)#+ np.random.randn(n)
-
-Maxpolydegree = 5
-X = np.zeros((len(x),Maxpolydegree))
-X[:,0] = 1.0
-
-for polydegree in range(1, Maxpolydegree):
-    for degree in range(polydegree):
-        X[:,degree] = x**(degree)
-
-
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
-
-# Decide which values of lambda to use
-nlambdas = 2
-lambdas = np.logspace(-3, -1, nlambdas)
-for i in range(nlambdas):
-    lmb = lambdas[i]
-    # Make the fit using Ridge only
-    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
-    RegRidge.fit(X_train,y_train)
-    # and then make the prediction
-    ypredictRidge = RegRidge.predict(X_test)
-    Coeffs = np.array(RegRidge.coef_)
-    BetaValues = pd.DataFrame(Coeffs)
-    BetaValues.columns = ['beta']
-    display(BetaValues)    
diff --git a/doc/src/week37/week37.do.txt b/doc/src/week37/week37.do.txt
index 74f5eae2f..2b8043666 100644
--- a/doc/src/week37/week37.do.txt
+++ b/doc/src/week37/week37.do.txt
@@ -636,6 +636,145 @@ The textbook by "Davison on the Bootstrap Methods and their Applications":"https
 
 Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called _central limit theorem_.
 
+!split
+===== The Central Limit Theorem =====
+
+
+Suppose we have a PDF $p(x)$ from which we generate  a series $N$
+of averages $\langle x_i \rangle$. Each mean value $\langle x_i \rangle$
+is viewed as the average of a specific measurement, e.g., throwing 
+dice 100 times and then taking the average value, or producing a certain
+amount of random numbers. 
+For notational ease, we set $\langle x_i \rangle=x_i$ in the discussion
+which follows. 
+
+If we compute the mean $z$ of $m$ such mean values $x_i$   
+!bt
+\[
+   z=\frac{x_1+x_2+\dots+x_m}{m},
+\]
+!et
+the question we pose is which is the PDF of the new variable $z$.
+
+!split
+===== Finding the Limit =====
+
+The probability of obtaining an average value $z$ is the product of the 
+probabilities of obtaining arbitrary individual mean values $x_i$,
+but with the constraint that the average is $z$. We can express this through
+the following expression
+!bt
+\[
+    \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
+    \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
+\]
+!et
+where the $\delta$-function enbodies the constraint that the mean is $z$.
+All measurements that lead to each individual $x_i$ are expected to
+be independent, which in turn means that we can express $\tilde{p}$ as the 
+product of individual $p(x_i)$.  The independence assumption is important in the derivation of the central limit theorem.
+
+
+!split
+===== Rewriting the $\delta$-function =====
+
+If we use the integral expression for the $\delta$-function
+
+!bt
+\[
+   \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
+\]
+!et
+and inserting $e^{i\mu q-i\mu q}$ where $\mu$ is the mean value
+we arrive at
+!bt
+\[
+   \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
+   dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
+   dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
+\]
+!et
+with the integral over $x$ resulting in
+
+!bt
+\[
+  \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
+  \int_{-\infty}^{\infty}dxp(x)
+   \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
+\]
+!et
+
+!split
+===== Identifying Terms =====
+
+The second term on the rhs disappears since this is just the mean and 
+employing the definition of $\sigma^2$ we have 
+!bt
+\[
+  \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
+  1-\frac{q^2\sigma^2}{2m^2}+\dots,
+\]
+!et
+resulting in 
+
+!bt
+\[
+  \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
+  \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
+\]
+!et
+and in the limit $m\rightarrow \infty$ we obtain 
+
+!bt
+\[
+   \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
+    \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
+\]
+!et
+which is the normal distribution with variance
+$\sigma^2_m=\sigma^2/m$, where $\sigma$ is the variance of the PDF $p(x)$
+and $\mu$ is also the mean of the PDF $p(x)$. 
+
+!split
+===== Wrapping it up =====
+
+Thus, the central limit theorem states that the PDF $\tilde{p}(z)$ of
+the average of $m$ random values corresponding to a PDF $p(x)$ 
+is a normal distribution whose mean is the 
+mean value of the PDF $p(x)$ and whose variance is the variance
+of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.
+
+The theorem is satisfied by a large class of PDFs. Note however that for a
+finite $m$, it is not always possible to find a closed expression for
+$\tilde{p}(x)$.
+The central limit theorem leads then to the well-known expression for the
+standard deviation, given by 
+
+!bt
+\[
+    \sigma_m=
+\frac{\sigma}{\sqrt{m}}.
+\]
+!et
+
+The latter is true only if the average value is known exactly. This is obtained in the limit
+$m\rightarrow \infty$  only. Because the mean and the variance are measured quantities we obtain 
+the familiar expression in statistics
+!bt
+\[
+    \sigma_m\approx 
+\frac{\sigma}{\sqrt{m-1}}.
+\]
+!et
+
+In many cases however the above estimate for the standard deviation,
+in particular if correlations are strong, may be too simplistic.
+
+!split
+===== Confidence Intervals =====
+
+
 !split
 ===== Resampling methods: Bootstrap background =====
 
@@ -727,10 +866,12 @@ from time import time
 import matplotlib.mlab as mlab
 import matplotlib.pyplot as plt
 
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
+# Returns mean of bootstrap samples                                                                              # Alternatively, we can run it using Scikit-Learn's function resample                                            # See the examples below
+
+def statistics(data):
     return mean(data)
 
+
 # Bootstrap algorithm
 def bootstrap(data, statistic, R):
     t = zeros(R); n = len(data); inds = arange(n); t0 = time()
@@ -749,7 +890,13 @@ mu, sigma = 100, 15
 datapoints = 10000
 x = mu + sigma*random.randn(datapoints)
 # bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
+t = bootstrap(x, statistics, datapoints)
+!ec
+We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
+
+!split
+===== Plotting the Histogram =====
+!bc pycod 
 # the histogram of the bootstrapped  data                                                                                                    
 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)