updating
This commit is contained in:
Morten Hjorth-Jensen
@@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
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||||
None,
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'resampling-methods-bootstrap'),
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||||
('The Central Limit Theorem',
|
||||
2,
|
||||
None,
|
||||
'the-central-limit-theorem'),
|
||||
('Finding the Limit', 2, None, 'finding-the-limit'),
|
||||
('Rewriting the $\\delta$-function',
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||||
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'),
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||||
('Resampling methods: Bootstrap background',
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||||
2,
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None,
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||||
@@ -115,6 +127,7 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'code-example-for-the-bootstrap-method'),
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||||
('Plotting the Histogram', 2, None, 'plotting-the-histogram'),
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||||
('The bias-variance tradeoff',
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||||
2,
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||||
None,
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||||
@@ -221,23 +234,30 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
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||||
</li>
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||||
@@ -296,7 +316,7 @@ MathJax.Hub.Config({
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<li><a href="._week37-bs008.html">9</a></li>
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||||
<li><a href="._week37-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
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||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
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||||
<li><a href="._week37-bs001.html">»</a></li>
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||||
</ul>
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||||
<!-- ------------------- end of main content --------------- -->
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@@ -95,6 +95,18 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
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||||
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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -284,7 +304,7 @@ Recommended Reading:
|
||||
<li><a href="._week37-bs009.html">10</a></li>
|
||||
<li><a href="._week37-bs010.html">11</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs002.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -273,7 +293,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week37-bs010.html">11</a></li>
|
||||
<li><a href="._week37-bs011.html">12</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs003.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -293,7 +313,7 @@ $$
|
||||
<li><a href="._week37-bs011.html">12</a></li>
|
||||
<li><a href="._week37-bs012.html">13</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,7 +327,7 @@ It is a conditional probability (see below) and reads as the likelihood of a dom
|
||||
<li><a href="._week37-bs012.html">13</a></li>
|
||||
<li><a href="._week37-bs013.html">14</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -301,7 +321,7 @@ is equivalent to the maximization/minimization of the function itself.
|
||||
<li><a href="._week37-bs013.html">14</a></li>
|
||||
<li><a href="._week37-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -301,7 +321,7 @@ $$
|
||||
<li><a href="._week37-bs014.html">15</a></li>
|
||||
<li><a href="._week37-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -292,7 +312,7 @@ which is Bayes' theorem. It allows us to evaluate the uncertainty in in \( X \)
|
||||
<li><a href="._week37-bs015.html">16</a></li>
|
||||
<li><a href="._week37-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -288,7 +308,7 @@ The function \( p(X) \) on the right hand side is called the prior while the fun
|
||||
<li><a href="._week37-bs016.html">17</a></li>
|
||||
<li><a href="._week37-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -360,7 +380,7 @@ How can we understand this?
|
||||
<li><a href="._week37-bs017.html">18</a></li>
|
||||
<li><a href="._week37-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -345,7 +365,7 @@ lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</
|
||||
<li><a href="._week37-bs018.html">19</a></li>
|
||||
<li><a href="._week37-bs019.html">20</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs011.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -311,7 +331,7 @@ We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one
|
||||
<li><a href="._week37-bs019.html">20</a></li>
|
||||
<li><a href="._week37-bs020.html">21</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs012.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -317,7 +337,7 @@ which is our Ridge cost function! Nice, isn't it?
|
||||
<li><a href="._week37-bs020.html">21</a></li>
|
||||
<li><a href="._week37-bs021.html">22</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs013.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -311,7 +331,7 @@ which is our Lasso cost function!
|
||||
<li><a href="._week37-bs021.html">22</a></li>
|
||||
<li><a href="._week37-bs022.html">23</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs014.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -293,7 +313,7 @@ and discuss how to select a given model (one of the difficult parts in machine l
|
||||
<li><a href="._week37-bs022.html">23</a></li>
|
||||
<li><a href="._week37-bs023.html">24</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs015.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -310,7 +330,7 @@ cross-validation and the bootstrap method.
|
||||
<li><a href="._week37-bs023.html">24</a></li>
|
||||
<li><a href="._week37-bs024.html">25</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -306,7 +326,7 @@ bootstrap is widely used.
|
||||
<li><a href="._week37-bs024.html">25</a></li>
|
||||
<li><a href="._week37-bs025.html">26</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs017.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -293,7 +313,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week37-bs025.html">26</a></li>
|
||||
<li><a href="._week37-bs026.html">27</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs018.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -299,7 +319,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week37-bs026.html">27</a></li>
|
||||
<li><a href="._week37-bs027.html">28</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -305,7 +325,7 @@ training error reaches a saturation.
|
||||
<li><a href="._week37-bs027.html">28</a></li>
|
||||
<li><a href="._week37-bs028.html">29</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs020.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -299,7 +319,7 @@ need for bootstrapping.
|
||||
<li><a href="._week37-bs028.html">29</a></li>
|
||||
<li><a href="._week37-bs029.html">30</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs021.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -295,7 +315,7 @@ number \( i \) is left out. Using this notation, define
|
||||
<li><a href="._week37-bs029.html">30</a></li>
|
||||
<li><a href="._week37-bs030.html">31</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -312,7 +332,7 @@ t <span style="color: #666666">=</span> jackknife(x, stat)
|
||||
<li><a href="._week37-bs030.html">31</a></li>
|
||||
<li><a href="._week37-bs031.html">32</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -304,7 +324,7 @@ Before we proceed however, we need to remind ourselves about a central theorem i
|
||||
<li><a href="._week37-bs031.html">32</a></li>
|
||||
<li><a href="._week37-bs032.html">33</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs024.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,18 +273,24 @@ MathJax.Hub.Config({
|
||||
<a name="part0024"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="resampling-methods-bootstrap-background" class="anchor">Resampling methods: Bootstrap background </h2>
|
||||
<h2 id="the-central-limit-theorem" class="anchor">The Central Limit Theorem </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -292,7 +318,7 @@ estimators.
|
||||
<li><a href="._week37-bs032.html">33</a></li>
|
||||
<li><a href="._week37-bs033.html">34</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,24 +273,22 @@ MathJax.Hub.Config({
|
||||
<a name="part0025"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="resampling-methods-more-bootstrap-background" class="anchor">Resampling methods: More Bootstrap background </h2>
|
||||
<h2 id="finding-the-limit" class="anchor">Finding the Limit </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
|
||||
\delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
|
||||
$$
|
||||
|
||||
<ol>
|
||||
<li> Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).</li>
|
||||
<li> Then using these numbers, we could compute a replica of \( \widehat{\beta} \) called \( \widehat{\beta}^* \).</li>
|
||||
</ol>
|
||||
|
||||
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.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -298,7 +316,7 @@ idea is to use the relative frequency of \( \widehat{\beta}^* \)
|
||||
<li><a href="._week37-bs033.html">34</a></li>
|
||||
<li><a href="._week37-bs034.html">35</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,20 +273,31 @@ MathJax.Hub.Config({
|
||||
<a name="part0026"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="resampling-methods-bootstrap-approach" class="anchor">Resampling methods: Bootstrap approach </h2>
|
||||
<h2 id="rewriting-the-delta-function" class="anchor">Rewriting the \( \delta \)-function </h2>
|
||||
|
||||
<p>
|
||||
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, <a href="https://projecteuclid.org/euclid.aos/1176344552" target="_self">Efron in 1979</a> 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
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
|
||||
\int_{-\infty}^{\infty}dxp(x)
|
||||
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -294,7 +325,7 @@ result in some asymptotic sense? The answer is yes.
|
||||
<li><a href="._week37-bs034.html">35</a></li>
|
||||
<li><a href="._week37-bs035.html">36</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,27 +273,33 @@ MathJax.Hub.Config({
|
||||
<a name="part0027"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="resampling-methods-bootstrap-steps" class="anchor">Resampling methods: Bootstrap steps </h2>
|
||||
<h2 id="identifying-terms" class="anchor">Identifying Terms </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
|
||||
1-\frac{q^2\sigma^2}{2m^2}+\dots,
|
||||
$$
|
||||
|
||||
<ol>
|
||||
<li> Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).</li>
|
||||
<li> Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).</li>
|
||||
<li> Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\beta}^* \) by evaluating \( \widehat \beta \) under the observations \( \boldsymbol{x}^* \).</li>
|
||||
<li> Repeat this process \( k \) times.</li>
|
||||
</ol>
|
||||
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 \( p(x) \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -301,7 +327,7 @@ example, if you are interested in estimating the variance of \( \widehat
|
||||
<li><a href="._week37-bs035.html">36</a></li>
|
||||
<li><a href="._week37-bs036.html">37</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs028.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,67 +273,40 @@ MathJax.Hub.Config({
|
||||
<a name="part0028"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="code-example-for-the-bootstrap-method" class="anchor">Code example for the Bootstrap method </h2>
|
||||
<h2 id="wrapping-it-up" class="anchor">Wrapping it up </h2>
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
$$
|
||||
\sigma_m=
|
||||
\frac{\sigma}{\sqrt{m}}.
|
||||
$$
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
|
||||
<span style="color: #008000; font-weight: bold">return</span> mean(data)
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\sigma_m\approx
|
||||
\frac{\sigma}{\sqrt{m-1}}.
|
||||
$$
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
|
||||
t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
|
||||
<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
|
||||
t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
|
||||
<p>
|
||||
In many cases however the above estimate for the standard deviation,
|
||||
in particular if correlations are strong, may be too simplistic.
|
||||
|
||||
<span style="color: #408080; font-style: italic"># analysis </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec"</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000">print</span>(<span style="color: #BA2121">"Bootstrap Statistics :"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"original bias std. error"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
|
||||
<span style="color: #008000; font-weight: bold">return</span> t
|
||||
|
||||
|
||||
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
|
||||
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
|
||||
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
|
||||
<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
|
||||
t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
|
||||
<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
|
||||
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, alpha<span style="color: #666666">=0.75</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># add a 'best fit' line </span>
|
||||
y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
|
||||
lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">'r--'</span>, linewidth<span style="color: #666666">=1</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Smarts'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'Probability'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -340,7 +333,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs036.html">37</a></li>
|
||||
<li><a href="._week37-bs037.html">38</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs029.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,68 +273,7 @@ MathJax.Hub.Config({
|
||||
<a name="part0029"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="the-bias-variance-tradeoff" class="anchor">The bias-variance tradeoff </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
|
||||
|
||||
<p>
|
||||
Let us assume that the true data is generated from a noisy model
|
||||
|
||||
$$
|
||||
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
|
||||
|
||||
<p>
|
||||
Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
|
||||
$$
|
||||
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
|
||||
$$
|
||||
|
||||
<p>
|
||||
We can rewrite this as
|
||||
$$
|
||||
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
|
||||
$$
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\epsilon} \).
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\epsilon} \).
|
||||
<h2 id="confidence-intervals" class="anchor">Confidence Intervals </h2>
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -342,7 +301,7 @@ that is the rewriting in terms of the so-called bias, the variance of the model
|
||||
<li><a href="._week37-bs037.html">38</a></li>
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs030.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,65 +273,19 @@ MathJax.Hub.Config({
|
||||
<a name="part0030"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="example-code-for-bias-variance-tradeoff" class="anchor">Example code for Bias-Variance tradeoff </h2>
|
||||
<h2 id="resampling-methods-bootstrap-background" class="anchor">Resampling methods: Bootstrap background </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
|
||||
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
|
||||
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">500</span>
|
||||
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
degree <span style="color: #666666">=</span> <span style="color: #666666">18</span> <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
|
||||
noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-1</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
|
||||
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
|
||||
<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
|
||||
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
|
||||
<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</span>
|
||||
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
|
||||
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
|
||||
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
|
||||
<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
|
||||
<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
|
||||
<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
|
||||
<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
|
||||
<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
|
||||
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
|
||||
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">'f(x)'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data points'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">'Pred'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -338,7 +312,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs031.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,57 +273,25 @@ MathJax.Hub.Config({
|
||||
<a name="part0031"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="understanding-what-happens" class="anchor">Understanding what happens </h2>
|
||||
<h2 id="resampling-methods-more-bootstrap-background" class="anchor">Resampling methods: More Bootstrap background </h2>
|
||||
|
||||
<p>
|
||||
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:
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
|
||||
<ol>
|
||||
<li> Drawing lots of numbers from \( p(x) \), suppose we call one such set of numbers \( (X_1^*, X_2^*, \cdots, X_n^*) \).</li>
|
||||
<li> Then using these numbers, we could compute a replica of \( \widehat{\beta} \) called \( \widehat{\beta}^* \).</li>
|
||||
</ol>
|
||||
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
|
||||
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 <span style="color: #666666">=</span> <span style="color: #666666">40</span>
|
||||
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
|
||||
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
|
||||
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
|
||||
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
|
||||
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
|
||||
|
||||
polydegree[degree] <span style="color: #666666">=</span> degree
|
||||
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
|
||||
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Polynomial degree:'</span>, degree)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">'Error'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">'bias'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">'Variance'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -329,6 +317,8 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs032.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -251,41 +271,22 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0032"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="summing-up" class="anchor">Summing up </h2>
|
||||
<h2 id="resampling-methods-bootstrap-approach" class="anchor">Resampling methods: Bootstrap approach </h2>
|
||||
|
||||
<p>
|
||||
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, <a href="https://projecteuclid.org/euclid.aos/1176344552" target="_self">Efron in 1979</a> asked the
|
||||
question: What if we replace \( p(x) \) by the relative frequency
|
||||
of the observation \( X_i \)?
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
You may also find this recent <a href="https://www.pnas.org/content/116/32/15849" target="_self">article</a> 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.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -311,6 +312,9 @@ You may also find this recent <a href="https://www.pnas.org/content/116/32/15849
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs033.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,82 +273,28 @@ MathJax.Hub.Config({
|
||||
<a name="part0033"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="another-example-from-scikit-learn-s-repository" class="anchor">Another Example from Scikit-Learn's Repository </h2>
|
||||
<h2 id="resampling-methods-bootstrap-steps" class="anchor">Resampling methods: Bootstrap steps </h2>
|
||||
|
||||
<p>
|
||||
The independent bootstrap works like this:
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">"""</span>
|
||||
<span style="color: #BA2121; font-style: italic">============================</span>
|
||||
<span style="color: #BA2121; font-style: italic">Underfitting vs. Overfitting</span>
|
||||
<span style="color: #BA2121; font-style: italic">============================</span>
|
||||
<ol>
|
||||
<li> Draw with replacement \( n \) numbers for the observed variables \( \boldsymbol{x} = (x_1,x_2,\cdots,x_n) \).</li>
|
||||
<li> Define a vector \( \boldsymbol{x}^* \) containing the values which were drawn from \( \boldsymbol{x} \).</li>
|
||||
<li> Using the vector \( \boldsymbol{x}^* \) compute \( \widehat{\beta}^* \) by evaluating \( \widehat \beta \) under the observations \( \boldsymbol{x}^* \).</li>
|
||||
<li> Repeat this process \( k \) times.</li>
|
||||
</ol>
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">This example demonstrates the problems of underfitting and overfitting and</span>
|
||||
<span style="color: #BA2121; font-style: italic">how we can use linear regression with polynomial features to approximate</span>
|
||||
<span style="color: #BA2121; font-style: italic">nonlinear functions. The plot shows the function that we want to approximate,</span>
|
||||
<span style="color: #BA2121; font-style: italic">which is a part of the cosine function. In addition, the samples from the</span>
|
||||
<span style="color: #BA2121; font-style: italic">real function and the approximations of different models are displayed. The</span>
|
||||
<span style="color: #BA2121; font-style: italic">models have polynomial features of different degrees. We can see that a</span>
|
||||
<span style="color: #BA2121; font-style: italic">linear function (polynomial with degree 1) is not sufficient to fit the</span>
|
||||
<span style="color: #BA2121; font-style: italic">training samples. This is called **underfitting**. A polynomial of degree 4</span>
|
||||
<span style="color: #BA2121; font-style: italic">approximates the true function almost perfectly. However, for higher degrees</span>
|
||||
<span style="color: #BA2121; font-style: italic">the model will **overfit** the training data, i.e. it learns the noise of the</span>
|
||||
<span style="color: #BA2121; font-style: italic">training data.</span>
|
||||
<span style="color: #BA2121; font-style: italic">We evaluate quantitatively **overfitting** / **underfitting** by using</span>
|
||||
<span style="color: #BA2121; font-style: italic">cross-validation. We calculate the mean squared error (MSE) on the validation</span>
|
||||
<span style="color: #BA2121; font-style: italic">set, the higher, the less likely the model generalizes correctly from the</span>
|
||||
<span style="color: #BA2121; font-style: italic">training data.</span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
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
|
||||
\( \widehat \beta^* \).
|
||||
|
||||
<span style="color: #008000">print</span>(<span style="color: #19177C">__doc__</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">true_fun</span>(X):
|
||||
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>cos(<span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>pi <span style="color: #666666">*</span> X)
|
||||
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
|
||||
|
||||
n_samples <span style="color: #666666">=</span> <span style="color: #666666">30</span>
|
||||
degrees <span style="color: #666666">=</span> [<span style="color: #666666">1</span>, <span style="color: #666666">4</span>, <span style="color: #666666">15</span>]
|
||||
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n_samples))
|
||||
y <span style="color: #666666">=</span> true_fun(X) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(n_samples) <span style="color: #666666">*</span> <span style="color: #666666">0.1</span>
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">14</span>, <span style="color: #666666">5</span>))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(degrees)):
|
||||
ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(degrees), i <span style="color: #666666">+</span> <span style="color: #666666">1</span>)
|
||||
plt<span style="color: #666666">.</span>setp(ax, xticks<span style="color: #666666">=</span>(), yticks<span style="color: #666666">=</span>())
|
||||
|
||||
polynomial_features <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=</span>degrees[i],
|
||||
include_bias<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
|
||||
linear_regression <span style="color: #666666">=</span> LinearRegression()
|
||||
pipeline <span style="color: #666666">=</span> Pipeline([(<span style="color: #BA2121">"polynomial_features"</span>, polynomial_features),
|
||||
(<span style="color: #BA2121">"linear_regression"</span>, linear_regression)])
|
||||
pipeline<span style="color: #666666">.</span>fit(X[:, np<span style="color: #666666">.</span>newaxis], y)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Evaluate the models using crossvalidation</span>
|
||||
scores <span style="color: #666666">=</span> cross_val_score(pipeline, X[:, np<span style="color: #666666">.</span>newaxis], y,
|
||||
scoring<span style="color: #666666">=</span><span style="color: #BA2121">"neg_mean_squared_error"</span>, cv<span style="color: #666666">=10</span>)
|
||||
|
||||
X_test <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">100</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X_test, pipeline<span style="color: #666666">.</span>predict(X_test[:, np<span style="color: #666666">.</span>newaxis]), label<span style="color: #666666">=</span><span style="color: #BA2121">"Model"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X_test, true_fun(X_test), label<span style="color: #666666">=</span><span style="color: #BA2121">"True function"</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(X, y, edgecolor<span style="color: #666666">=</span><span style="color: #BA2121">'b'</span>, s<span style="color: #666666">=20</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Samples"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"x"</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"y"</span>)
|
||||
plt<span style="color: #666666">.</span>xlim((<span style="color: #666666">0</span>, <span style="color: #666666">1</span>))
|
||||
plt<span style="color: #666666">.</span>ylim((<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>))
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"best"</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Degree </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">MSE = </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">(+/- </span><span style="color: #BB6688; font-weight: bold">{:.2e}</span><span style="color: #BA2121">)"</span><span style="color: #666666">.</span>format(
|
||||
degrees[i], <span style="color: #666666">-</span>scores<span style="color: #666666">.</span>mean(), scores<span style="color: #666666">.</span>std()))
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -352,6 +318,10 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs034.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -251,25 +271,61 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0034"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="various-steps-in-cross-validation" class="anchor">Various steps in cross-validation </h2>
|
||||
<h2 id="code-example-for-the-bootstrap-method" class="anchor">Code example for the Bootstrap method </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples # Alternatively, we can run it using Scikit-Learn's function resample # See the examples below</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">statistics</span>(data):
|
||||
<span style="color: #008000; font-weight: bold">return</span> mean(data)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
|
||||
t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
|
||||
<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(R):
|
||||
t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
|
||||
|
||||
<span style="color: #408080; font-style: italic"># analysis </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec"</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000">print</span>(<span style="color: #BA2121">"Bootstrap Statistics :"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"original bias std. error"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
|
||||
<span style="color: #008000; font-weight: bold">return</span> t
|
||||
|
||||
|
||||
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
|
||||
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
|
||||
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
|
||||
<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
|
||||
t <span style="color: #666666">=</span> bootstrap(x, statistics, datapoints)
|
||||
</pre></div>
|
||||
<p>
|
||||
We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -293,6 +349,11 @@ cross-validation (LOOCV).
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs035.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -251,37 +271,25 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0035"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" class="anchor">How to set up the cross-validation for Ridge and/or Lasso </h2>
|
||||
<h2 id="plotting-the-histogram" class="anchor">Plotting the Histogram </h2>
|
||||
<p>
|
||||
|
||||
<ul>
|
||||
<li> Define a range of interest for the penalty parameter.</li>
|
||||
<li> Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.</li>
|
||||
<li> Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as</li>
|
||||
</ul>
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
|
||||
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, alpha<span style="color: #666666">=0.75</span>)
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T}
|
||||
\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
|
||||
\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
|
||||
<ul>
|
||||
<li> 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.</li>
|
||||
<li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
|
||||
<li> 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</li>
|
||||
</ul>
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
|
||||
\end{align*}
|
||||
$$
|
||||
<span style="color: #408080; font-style: italic"># add a 'best fit' line </span>
|
||||
y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
|
||||
lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">'r--'</span>, linewidth<span style="color: #666666">=1</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Smarts'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'Probability'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -303,6 +311,12 @@ $$
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="._week37-bs044.html">45</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs036.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,26 +273,70 @@ MathJax.Hub.Config({
|
||||
<a name="part0036"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="cross-validation-in-brief" class="anchor">Cross-validation in brief </h2>
|
||||
<h2 id="the-bias-variance-tradeoff" class="anchor">The bias-variance tradeoff </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
|
||||
|
||||
<ol>
|
||||
<li> shuffle the dataset randomly.</li>
|
||||
<li> Split the dataset into \( k \) groups.</li>
|
||||
<li> For each unique group:
|
||||
<p>
|
||||
Let us assume that the true data is generated from a noisy model
|
||||
|
||||
<ol type="a"></li>
|
||||
<li> Decide which group to use as set for test data</li>
|
||||
<li> Take the remaining groups as a training data set</li>
|
||||
<li> Fit a model on the training set and evaluate it on the test set</li>
|
||||
<li> Retain the evaluation score and discard the model</li>
|
||||
</ol>
|
||||
$$
|
||||
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}
|
||||
$$
|
||||
|
||||
<li> Summarize the model using the sample of model evaluation scores</li>
|
||||
</ol>
|
||||
<p>
|
||||
where \( \epsilon \) is normally distributed with mean zero and standard deviation \( \sigma^2 \).
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta} \).
|
||||
|
||||
<p>
|
||||
Thereafter we found the parameters \( \boldsymbol{\beta} \) by optimizing the means squared error via the so-called cost function
|
||||
$$
|
||||
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
|
||||
$$
|
||||
|
||||
<p>
|
||||
We can rewrite this as
|
||||
$$
|
||||
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
|
||||
$$
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\epsilon} \).
|
||||
|
||||
<p>
|
||||
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 \( \boldsymbol{\epsilon} \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -292,6 +356,13 @@ For the various values of \( k \)
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="._week37-bs044.html">45</a></li>
|
||||
<li><a href="._week37-bs045.html">46</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs037.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,101 +273,63 @@ MathJax.Hub.Config({
|
||||
<a name="part0037"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="code-example-for-cross-validation-and-k-fold-cross-validation" class="anchor">Code Example for Cross-validation and \( k \)-fold Cross-validation </h2>
|
||||
|
||||
<p>
|
||||
The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.
|
||||
<h2 id="example-code-for-bias-variance-tradeoff" class="anchor">Example code for Bias-Variance tradeoff </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
|
||||
|
||||
<span style="color: #408080; font-style: italic"># A seed just to ensure that the random numbers are the same for every run.</span>
|
||||
<span style="color: #408080; font-style: italic"># Useful for eventual debugging.</span>
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">3155</span>)
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Generate the data.</span>
|
||||
nsamples <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">3*</span>x<span style="color: #666666">**2</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(nsamples)
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">500</span>
|
||||
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
degree <span style="color: #666666">=</span> <span style="color: #666666">18</span> <span style="color: #408080; font-style: italic"># A quite high value, just to show.</span>
|
||||
noise <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic">## Cross-validation on Ridge regression using KFold only</span>
|
||||
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-1</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Decide degree on polynomial to fit</span>
|
||||
poly <span style="color: #666666">=</span> PolynomialFeatures(degree <span style="color: #666666">=</span> <span style="color: #666666">6</span>)
|
||||
<span style="color: #408080; font-style: italic"># Hold out some test data that is never used in training.</span>
|
||||
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Decide which values of lambda to use</span>
|
||||
nlambdas <span style="color: #666666">=</span> <span style="color: #666666">500</span>
|
||||
lambdas <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-3</span>, <span style="color: #666666">5</span>, nlambdas)
|
||||
<span style="color: #408080; font-style: italic"># Combine x transformation and model into one operation.</span>
|
||||
<span style="color: #408080; font-style: italic"># Not neccesary, but convenient.</span>
|
||||
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Initialize a KFold instance</span>
|
||||
k <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
|
||||
<span style="color: #408080; font-style: italic"># The following (m x n_bootstraps) matrix holds the column vectors y_pred</span>
|
||||
<span style="color: #408080; font-style: italic"># for each bootstrap iteration.</span>
|
||||
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
|
||||
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Perform the cross-validation to estimate MSE</span>
|
||||
scores_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((nlambdas, k))
|
||||
<span style="color: #408080; font-style: italic"># Evaluate the new model on the same test data each time.</span>
|
||||
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
|
||||
|
||||
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
|
||||
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
|
||||
j <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> train_inds, test_inds <span style="color: #AA22FF; font-weight: bold">in</span> kfold<span style="color: #666666">.</span>split(x):
|
||||
xtrain <span style="color: #666666">=</span> x[train_inds]
|
||||
ytrain <span style="color: #666666">=</span> y[train_inds]
|
||||
|
||||
xtest <span style="color: #666666">=</span> x[test_inds]
|
||||
ytest <span style="color: #666666">=</span> y[test_inds]
|
||||
|
||||
Xtrain <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtrain[:, np<span style="color: #666666">.</span>newaxis])
|
||||
ridge<span style="color: #666666">.</span>fit(Xtrain, ytrain[:, np<span style="color: #666666">.</span>newaxis])
|
||||
|
||||
Xtest <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(xtest[:, np<span style="color: #666666">.</span>newaxis])
|
||||
ypred <span style="color: #666666">=</span> ridge<span style="color: #666666">.</span>predict(Xtest)
|
||||
|
||||
scores_KFold[i,j] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum((ypred <span style="color: #666666">-</span> ytest[:, np<span style="color: #666666">.</span>newaxis])<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>np<span style="color: #666666">.</span>size(ypred)
|
||||
|
||||
j <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
||||
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
||||
|
||||
|
||||
estimated_mse_KFold <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(scores_KFold, axis <span style="color: #666666">=</span> <span style="color: #666666">1</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic">## Cross-validation using cross_val_score from sklearn along with KFold</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># kfold is an instance initialized above as:</span>
|
||||
<span style="color: #408080; font-style: italic"># kfold = KFold(n_splits = k)</span>
|
||||
|
||||
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(nlambdas)
|
||||
i <span style="color: #666666">=</span> <span style="color: #666666">0</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> lmb <span style="color: #AA22FF; font-weight: bold">in</span> lambdas:
|
||||
ridge <span style="color: #666666">=</span> Ridge(alpha <span style="color: #666666">=</span> lmb)
|
||||
|
||||
X <span style="color: #666666">=</span> poly<span style="color: #666666">.</span>fit_transform(x[:, np<span style="color: #666666">.</span>newaxis])
|
||||
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(ridge, X, y[:, np<span style="color: #666666">.</span>newaxis], scoring<span style="color: #666666">=</span><span style="color: #BA2121">'neg_mean_squared_error'</span>, cv<span style="color: #666666">=</span>kfold)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># cross_val_score return an array containing the estimated negative mse for every fold.</span>
|
||||
<span style="color: #408080; font-style: italic"># we have to the the mean of every array in order to get an estimate of the mse of the model</span>
|
||||
estimated_mse_sklearn[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
|
||||
|
||||
i <span style="color: #666666">+=</span> <span style="color: #666666">1</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic">## Plot and compare the slightly different ways to perform cross-validation</span>
|
||||
|
||||
plt<span style="color: #666666">.</span>figure()
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_sklearn, label <span style="color: #666666">=</span> <span style="color: #BA2121">'cross_val_score'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(np<span style="color: #666666">.</span>log10(lambdas), estimated_mse_KFold, <span style="color: #BA2121">'r--'</span>, label <span style="color: #666666">=</span> <span style="color: #BA2121">'KFold'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'log10(lambda)'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'mse'</span>)
|
||||
<span style="color: #408080; font-style: italic"># Note: Expectations and variances taken w.r.t. different training</span>
|
||||
<span style="color: #408080; font-style: italic"># data sets, hence the axis=1. Subsequent means are taken across the test data</span>
|
||||
<span style="color: #408080; font-style: italic"># set in order to obtain a total value, but before this we have error/bias/variance</span>
|
||||
<span style="color: #408080; font-style: italic"># calculated per data point in the test set.</span>
|
||||
<span style="color: #408080; font-style: italic"># Note 2: The use of keepdims=True is important in the calculation of bias as this </span>
|
||||
<span style="color: #408080; font-style: italic"># maintains the column vector form. Dropping this yields very unexpected results.</span>
|
||||
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
|
||||
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error, bias, variance, bias<span style="color: #666666">+</span>variance))
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(x[::<span style="color: #666666">5</span>, :], y[::<span style="color: #666666">5</span>, :], label<span style="color: #666666">=</span><span style="color: #BA2121">'f(x)'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x_test, y_test, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data points'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x_test, np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>), label<span style="color: #666666">=</span><span style="color: #BA2121">'Pred'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
@@ -369,6 +351,14 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="._week37-bs044.html">45</a></li>
|
||||
<li><a href="._week37-bs045.html">46</a></li>
|
||||
<li><a href="._week37-bs046.html">47</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs038.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,87 +273,54 @@ MathJax.Hub.Config({
|
||||
<a name="part0038"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="more-examples-on-bootstrap-and-cross-validation-and-errors" class="anchor">More examples on bootstrap and cross-validation and errors </h2>
|
||||
|
||||
<h2 id="understanding-what-happens" class="anchor">Understanding what happens </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.utils</span> <span style="color: #008000; font-weight: bold">import</span> resample
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
|
||||
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
||||
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
||||
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
||||
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
||||
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">2018</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">40</span>
|
||||
n_boostraps <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
||||
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
||||
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
||||
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
|
||||
model <span style="color: #666666">=</span> make_pipeline(PolynomialFeatures(degree<span style="color: #666666">=</span>degree), LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>))
|
||||
y_pred <span style="color: #666666">=</span> np<span style="color: #666666">.</span>empty((y_test<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], n_boostraps))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n_boostraps):
|
||||
x_, y_ <span style="color: #666666">=</span> resample(x_train, y_train)
|
||||
y_pred[:, i] <span style="color: #666666">=</span> model<span style="color: #666666">.</span>fit(x_, y_)<span style="color: #666666">.</span>predict(x_test)<span style="color: #666666">.</span>ravel()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
polydegree[degree] <span style="color: #666666">=</span> degree
|
||||
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>))<span style="color: #666666">**2</span> )
|
||||
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred, axis<span style="color: #666666">=1</span>, keepdims<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>) )
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Polynomial degree:'</span>, degree)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance[degree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
|
||||
|
||||
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"EoS.csv"</span>),<span style="color: #BA2121">'r'</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
|
||||
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">'Density'</span>, <span style="color: #BA2121">'Energy'</span>))
|
||||
EoS[<span style="color: #BA2121">'Energy'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">'Energy'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
||||
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
|
||||
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Energy'</span>]
|
||||
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Density'</span>]
|
||||
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
|
||||
|
||||
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">30</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),Maxpolydegree))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
testerror <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
trainingerror <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
polynomial <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
|
||||
trials <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> polydegree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>, Maxpolydegree):
|
||||
polynomial[polydegree] <span style="color: #666666">=</span> polydegree
|
||||
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(polydegree):
|
||||
X[:,degree] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(degree<span style="color: #666666">/3.0</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># loop over trials in order to estimate the expectation value of the MSE</span>
|
||||
testerror[polydegree] <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
|
||||
trainingerror[polydegree] <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
|
||||
<span style="color: #008000; font-weight: bold">for</span> samples <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(trials):
|
||||
x_train, x_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, Energies, test_size<span style="color: #666666">=0.2</span>)
|
||||
model <span style="color: #666666">=</span> LinearRegression(fit_intercept<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)<span style="color: #666666">.</span>fit(x_train, y_train)
|
||||
ypred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(x_train)
|
||||
ytilde <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(x_test)
|
||||
testerror[polydegree] <span style="color: #666666">+=</span> mean_squared_error(y_test, ytilde)
|
||||
trainingerror[polydegree] <span style="color: #666666">+=</span> mean_squared_error(y_train, ypred)
|
||||
|
||||
testerror[polydegree] <span style="color: #666666">/=</span> trials
|
||||
trainingerror[polydegree] <span style="color: #666666">/=</span> trials
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Degree of polynomial: </span><span style="color: #BB6688; font-weight: bold">%3d</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span> polynomial[polydegree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error on training data: </span><span style="color: #BB6688; font-weight: bold">%.8f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> trainingerror[polydegree])
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error on test data: </span><span style="color: #BB6688; font-weight: bold">%.8f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> testerror[polydegree])
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(trainingerror), label<span style="color: #666666">=</span><span style="color: #BA2121">'Training Error'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(testerror), label<span style="color: #666666">=</span><span style="color: #BA2121">'Test Error'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Polynomial degree'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'log10[MSE]'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">'Error'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">'bias'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">'Variance'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
@@ -355,6 +342,13 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li class="active"><a href="._week37-bs038.html">39</a></li>
|
||||
<li><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="._week37-bs044.html">45</a></li>
|
||||
<li><a href="._week37-bs045.html">46</a></li>
|
||||
<li><a href="._week37-bs046.html">47</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs039.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -253,79 +273,40 @@ MathJax.Hub.Config({
|
||||
<a name="part0039"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="the-same-example-but-now-with-cross-validation" class="anchor">The same example but now with cross-validation </h2>
|
||||
<h2 id="summing-up" class="anchor">Summing up </h2>
|
||||
|
||||
<p>
|
||||
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).
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression, Ridge, Lasso
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> KFold
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_val_score
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
||||
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
||||
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
||||
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
||||
<p>
|
||||
You may also find this recent <a href="https://www.pnas.org/content/116/32/15849" target="_self">article</a> of interest.
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
||||
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
|
||||
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"EoS.csv"</span>),<span style="color: #BA2121">'r'</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
|
||||
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">'Density'</span>, <span style="color: #BA2121">'Energy'</span>))
|
||||
EoS[<span style="color: #BA2121">'Energy'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">'Energy'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
||||
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
|
||||
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Energy'</span>]
|
||||
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Density'</span>]
|
||||
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
|
||||
|
||||
Maxpolydegree <span style="color: #666666">=</span> <span style="color: #666666">30</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),Maxpolydegree))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
estimated_mse_sklearn <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
polynomial <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Maxpolydegree)
|
||||
k <span style="color: #666666">=5</span>
|
||||
kfold <span style="color: #666666">=</span> KFold(n_splits <span style="color: #666666">=</span> k)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> polydegree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>, Maxpolydegree):
|
||||
polynomial[polydegree] <span style="color: #666666">=</span> polydegree
|
||||
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(polydegree):
|
||||
X[:,degree] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(degree<span style="color: #666666">/3.0</span>)
|
||||
OLS <span style="color: #666666">=</span> LinearRegression()
|
||||
<span style="color: #408080; font-style: italic"># loop over trials in order to estimate the expectation value of the MSE</span>
|
||||
estimated_mse_folds <span style="color: #666666">=</span> cross_val_score(OLS, X, Energies, scoring<span style="color: #666666">=</span><span style="color: #BA2121">'neg_mean_squared_error'</span>, cv<span style="color: #666666">=</span>kfold)
|
||||
<span style="color: #408080; font-style: italic">#[:, np.newaxis]</span>
|
||||
estimated_mse_sklearn[polydegree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean(<span style="color: #666666">-</span>estimated_mse_folds)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(polynomial, np<span style="color: #666666">.</span>log10(estimated_mse_sklearn), label<span style="color: #666666">=</span><span style="color: #BA2121">'Test Error'</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Polynomial degree'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'log10[MSE]'</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -343,6 +324,13 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week37-bs038.html">39</a></li>
|
||||
<li class="active"><a href="._week37-bs039.html">40</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs041.html">42</a></li>
|
||||
<li><a href="._week37-bs042.html">43</a></li>
|
||||
<li><a href="._week37-bs043.html">44</a></li>
|
||||
<li><a href="._week37-bs044.html">45</a></li>
|
||||
<li><a href="._week37-bs045.html">46</a></li>
|
||||
<li><a href="._week37-bs046.html">47</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs040.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -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({
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs030.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs034.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs035.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs036.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs037.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs038.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs039.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs040.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs041.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs042.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs043.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs044.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs045.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs046.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week37-bs047.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -296,7 +316,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week37-bs008.html">9</a></li>
|
||||
<li><a href="._week37-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week37-bs040.html">41</a></li>
|
||||
<li><a href="._week37-bs047.html">48</a></li>
|
||||
<li><a href="._week37-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -896,6 +896,171 @@ Before we proceed however, we need to remind ourselves about a central theorem i
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="the-central-limit-theorem">The Central Limit Theorem </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
z=\frac{x_1+x_2+\dots+x_m}{m},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
the question we pose is which is the PDF of the new variable \( z \).
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="finding-the-limit">Finding the Limit </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
<p> <br>
|
||||
$$
|
||||
\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
|
||||
\delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
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.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="rewriting-the-delta-function">Rewriting the \( \delta \)-function </h2>
|
||||
|
||||
<p>
|
||||
If we use the integral expression for the \( \delta \)-function
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
|
||||
dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value
|
||||
we arrive at
|
||||
<p> <br>
|
||||
$$
|
||||
\tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
|
||||
dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
|
||||
dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
with the integral over \( x \) resulting in
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
|
||||
\int_{-\infty}^{\infty}dxp(x)
|
||||
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
|
||||
$$
|
||||
<p> <br>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="identifying-terms">Identifying Terms </h2>
|
||||
|
||||
<p>
|
||||
The second term on the rhs disappears since this is just the mean and
|
||||
employing the definition of \( \sigma^2 \) we have
|
||||
<p> <br>
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
|
||||
1-\frac{q^2\sigma^2}{2m^2}+\dots,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
resulting in
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx
|
||||
\left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and in the limit \( m\rightarrow \infty \) we obtain
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})}
|
||||
\exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
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 \( p(x) \).
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="wrapping-it-up">Wrapping it up </h2>
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\sigma_m=
|
||||
\frac{\sigma}{\sqrt{m}}.
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p>
|
||||
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
|
||||
<p> <br>
|
||||
$$
|
||||
\sigma_m\approx
|
||||
\frac{\sigma}{\sqrt{m-1}}.
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p>
|
||||
In many cases however the above estimate for the standard deviation,
|
||||
in particular if correlations are strong, may be too simplistic.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="confidence-intervals">Confidence Intervals </h2>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="resampling-methods-bootstrap-background">Resampling methods: Bootstrap background </h2>
|
||||
|
||||
@@ -1005,10 +1170,12 @@ theorem.
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.mlab</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mlab</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
|
||||
|
||||
<span style="color: #228B22"># Returns mean of bootstrap samples </span>
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">stat</span>(data):
|
||||
<span style="color: #228B22"># Returns mean of bootstrap samples # Alternatively, we can run it using Scikit-Learn's function resample # See the examples below</span>
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">statistics</span>(data):
|
||||
<span style="color: #8B008B; font-weight: bold">return</span> mean(data)
|
||||
|
||||
|
||||
<span style="color: #228B22"># Bootstrap algorithm</span>
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">bootstrap</span>(data, statistic, R):
|
||||
t = zeros(R); n = <span style="color: #658b00">len</span>(data); inds = arange(n); t0 = time()
|
||||
@@ -1027,8 +1194,19 @@ mu, sigma = <span style="color: #B452CD">100</span>, <span style="color: #B452CD
|
||||
datapoints = <span style="color: #B452CD">10000</span>
|
||||
x = mu + sigma*random.randn(datapoints)
|
||||
<span style="color: #228B22"># bootstrap returns the data sample </span>
|
||||
t = bootstrap(x, stat, datapoints)
|
||||
<span style="color: #228B22"># the histogram of the bootstrapped data </span>
|
||||
t = bootstrap(x, statistics, datapoints)
|
||||
</pre></div>
|
||||
<p>
|
||||
We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="plotting-the-histogram">Plotting the Histogram </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%;"><span></span><span style="color: #228B22"># the histogram of the bootstrapped data </span>
|
||||
n, binsboot, patches = plt.hist(t, <span style="color: #B452CD">50</span>, normed=<span style="color: #B452CD">1</span>, facecolor=<span style="color: #CD5555">'red'</span>, alpha=<span style="color: #B452CD">0.75</span>)
|
||||
|
||||
<span style="color: #228B22"># add a 'best fit' line </span>
|
||||
|
||||
@@ -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
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="the-central-limit-theorem">The Central Limit Theorem </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="finding-the-limit">Finding the Limit </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="rewriting-the-delta-function">Rewriting the \( \delta \)-function </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
|
||||
\int_{-\infty}^{\infty}dxp(x)
|
||||
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="identifying-terms">Identifying Terms </h2>
|
||||
|
||||
<p>
|
||||
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 \( p(x) \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="wrapping-it-up">Wrapping it up </h2>
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\sigma_m=
|
||||
\frac{\sigma}{\sqrt{m}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\sigma_m\approx
|
||||
\frac{\sigma}{\sqrt{m-1}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
In many cases however the above estimate for the standard deviation,
|
||||
in particular if correlations are strong, may be too simplistic.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="confidence-intervals">Confidence Intervals </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="resampling-methods-bootstrap-background">Resampling methods: Bootstrap background </h2>
|
||||
|
||||
<p>
|
||||
@@ -1024,10 +1181,12 @@ theorem.
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.mlab</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mlab</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
|
||||
|
||||
<span style="color: #228B22"># Returns mean of bootstrap samples </span>
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">stat</span>(data):
|
||||
<span style="color: #228B22"># Returns mean of bootstrap samples # Alternatively, we can run it using Scikit-Learn's function resample # See the examples below</span>
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">statistics</span>(data):
|
||||
<span style="color: #8B008B; font-weight: bold">return</span> mean(data)
|
||||
|
||||
|
||||
<span style="color: #228B22"># Bootstrap algorithm</span>
|
||||
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">bootstrap</span>(data, statistic, R):
|
||||
t = zeros(R); n = <span style="color: #658b00">len</span>(data); inds = arange(n); t0 = time()
|
||||
@@ -1046,8 +1205,19 @@ mu, sigma = <span style="color: #B452CD">100</span>, <span style="color: #B452CD
|
||||
datapoints = <span style="color: #B452CD">10000</span>
|
||||
x = mu + sigma*random.randn(datapoints)
|
||||
<span style="color: #228B22"># bootstrap returns the data sample </span>
|
||||
t = bootstrap(x, stat, datapoints)
|
||||
<span style="color: #228B22"># the histogram of the bootstrapped data </span>
|
||||
t = bootstrap(x, statistics, datapoints)
|
||||
</pre></div>
|
||||
<p>
|
||||
We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="plotting-the-histogram">Plotting the Histogram </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%;"><span></span><span style="color: #228B22"># the histogram of the bootstrapped data </span>
|
||||
n, binsboot, patches = plt.hist(t, <span style="color: #B452CD">50</span>, normed=<span style="color: #B452CD">1</span>, facecolor=<span style="color: #CD5555">'red'</span>, alpha=<span style="color: #B452CD">0.75</span>)
|
||||
|
||||
<span style="color: #228B22"># add a 'best fit' line </span>
|
||||
|
||||
@@ -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
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="the-central-limit-theorem">The Central Limit Theorem </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="finding-the-limit">Finding the Limit </h2>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="rewriting-the-delta-function">Rewriting the \( \delta \)-function </h2>
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
|
||||
\int_{-\infty}^{\infty}dxp(x)
|
||||
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="identifying-terms">Identifying Terms </h2>
|
||||
|
||||
<p>
|
||||
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 \( p(x) \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="wrapping-it-up">Wrapping it up </h2>
|
||||
|
||||
<p>
|
||||
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 \( z \).
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\sigma_m=
|
||||
\frac{\sigma}{\sqrt{m}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\sigma_m\approx
|
||||
\frac{\sigma}{\sqrt{m-1}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
In many cases however the above estimate for the standard deviation,
|
||||
in particular if correlations are strong, may be too simplistic.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="confidence-intervals">Confidence Intervals </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="resampling-methods-bootstrap-background">Resampling methods: Bootstrap background </h2>
|
||||
|
||||
<p>
|
||||
@@ -1029,10 +1186,12 @@ theorem.
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
|
||||
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples # Alternatively, we can run it using Scikit-Learn's function resample # See the examples below</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">statistics</span>(data):
|
||||
<span style="color: #008000; font-weight: bold">return</span> mean(data)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
|
||||
t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
|
||||
@@ -1051,8 +1210,19 @@ mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100
|
||||
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
|
||||
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
|
||||
<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
|
||||
t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
|
||||
<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
|
||||
t <span style="color: #666666">=</span> bootstrap(x, statistics, datapoints)
|
||||
</pre></div>
|
||||
<p>
|
||||
We see that our new variance and from that the standard deviation, agrees with the central limit theorem.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="plotting-the-histogram">Plotting the Histogram </h2>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
|
||||
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, alpha<span style="color: #666666">=0.75</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># add a 'best fit' line </span>
|
||||
|
||||
Binary file not shown.
@@ -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",
|
||||
|
||||
@@ -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)
|
||||
@@ -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)
|
||||
|
||||
|
||||
Reference in New Issue
Block a user