small change
This commit is contained in:
mhjensen
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
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('Proof of the PCA Theorem', 2, None, '___sec18'),
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('PCA Proof continued', 2, None, '___sec19'),
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||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
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||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
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||||
('Kernel PCA', 2, None, '___sec27'),
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||||
('LLE', 2, None, '___sec28'),
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||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
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||||
('LLE', 2, None, '___sec27'),
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('Other techniques', 2, None, '___sec28')]}
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end of tocinfo -->
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||||
<body>
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@@ -160,15 +159,14 @@ MathJax.Hub.Config({
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
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</ul>
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</li>
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@@ -227,7 +225,7 @@ MathJax.Hub.Config({
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<li><a href="._DimRed-bs008.html">9</a></li>
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<li><a href="._DimRed-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._DimRed-bs030.html">31</a></li>
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<li><a href="._DimRed-bs029.html">30</a></li>
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<li><a href="._DimRed-bs001.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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||||
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@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
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('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
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('LLE', 2, None, '___sec27'),
|
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('Other techniques', 2, None, '___sec28')]}
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||||
end of tocinfo -->
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||||
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||||
<body>
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@@ -160,15 +159,14 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
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||||
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||||
</ul>
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||||
</li>
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@@ -227,7 +225,7 @@ data.
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<li><a href="._DimRed-bs009.html">10</a></li>
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<li><a href="._DimRed-bs010.html">11</a></li>
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||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
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||||
<li><a href="._DimRed-bs029.html">30</a></li>
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||||
<li><a href="._DimRed-bs002.html">»</a></li>
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||||
</ul>
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||||
<!-- ------------------- end of main content --------------- -->
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||||
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||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -226,7 +224,7 @@ ensures that all features are exactly between \( 0 \) and \( 1 \). The
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<li><a href="._DimRed-bs010.html">11</a></li>
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||||
<li><a href="._DimRed-bs011.html">12</a></li>
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||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
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||||
<li><a href="._DimRed-bs003.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
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||||
@@ -229,7 +227,7 @@ techniques.
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||||
<li><a href="._DimRed-bs011.html">12</a></li>
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||||
<li><a href="._DimRed-bs012.html">13</a></li>
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||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -304,7 +302,7 @@ svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<li><a href="._DimRed-bs012.html">13</a></li>
|
||||
<li><a href="._DimRed-bs013.html">14</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -254,7 +252,7 @@ svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<li><a href="._DimRed-bs013.html">14</a></li>
|
||||
<li><a href="._DimRed-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -233,7 +231,7 @@ logreg<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<li><a href="._DimRed-bs014.html">15</a></li>
|
||||
<li><a href="._DimRed-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -288,7 +286,7 @@ applications.
|
||||
<li><a href="._DimRed-bs015.html">16</a></li>
|
||||
<li><a href="._DimRed-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -218,7 +216,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see
|
||||
<li><a href="._DimRed-bs016.html">17</a></li>
|
||||
<li><a href="._DimRed-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -267,7 +265,7 @@ In the above example this is the function we constructed using <b>pandas</b>.
|
||||
<li><a href="._DimRed-bs017.html">18</a></li>
|
||||
<li><a href="._DimRed-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -262,7 +260,7 @@ $$
|
||||
<li><a href="._DimRed-bs018.html">19</a></li>
|
||||
<li><a href="._DimRed-bs019.html">20</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs011.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -250,7 +248,7 @@ C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c
|
||||
<li><a href="._DimRed-bs019.html">20</a></li>
|
||||
<li><a href="._DimRed-bs020.html">21</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs012.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -252,7 +250,7 @@ The above procedure with <b>numpy</b> can be made more compact if we use <b>pand
|
||||
<li><a href="._DimRed-bs020.html">21</a></li>
|
||||
<li><a href="._DimRed-bs021.html">22</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs013.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -234,7 +232,7 @@ We expand this model to the Franke function discussed above.
|
||||
<li><a href="._DimRed-bs021.html">22</a></li>
|
||||
<li><a href="._DimRed-bs022.html">23</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs014.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -270,7 +268,7 @@ matrix.
|
||||
<li><a href="._DimRed-bs022.html">23</a></li>
|
||||
<li><a href="._DimRed-bs023.html">24</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs015.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -250,7 +248,7 @@ It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\t
|
||||
<li><a href="._DimRed-bs023.html">24</a></li>
|
||||
<li><a href="._DimRed-bs024.html">25</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
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|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -258,7 +256,7 @@ features/predictors.
|
||||
<li><a href="._DimRed-bs024.html">25</a></li>
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs017.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
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|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
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end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -241,7 +239,7 @@ After this we ask ourselves how do we prove the link between the maximum varianc
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs018.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
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|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
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|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
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|
||||
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|
||||
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|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -228,7 +226,7 @@ The PCA theorem states that minimizing the above reconstruction error correspond
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
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|
||||
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|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -230,7 +228,7 @@ where the vectors on the rhs are known.
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs020.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
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|
||||
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|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
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|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -244,8 +242,6 @@ We are almost there, we have obtained a relation between minimizing the reconstr
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs021.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
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|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
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||||
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|
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('Randomized PCA', 2, None, '___sec26'),
|
||||
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|
||||
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|
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|
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|
||||
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||||
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|
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|
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|
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|
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|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -256,7 +254,6 @@ chapter 12.4 and discussion therein.
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,17 +182,58 @@ MathJax.Hub.Config({
|
||||
<a name="part0022"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec21" class="anchor">PCA and Scikit-Learn Functionality </h2>
|
||||
<h2 id="___sec21" class="anchor">Principal Component Analysis </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
|
||||
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
|
||||
|
||||
<p>
|
||||
The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data using either <b>pandas</b> or our own code
|
||||
<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"># Now add PCA</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
||||
pca<span style="color: #666666">.</span>fit(X_train_scaled)
|
||||
<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">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">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
|
||||
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 vanilla matrix </span>
|
||||
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
|
||||
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
|
||||
<span style="color: #408080; font-style: italic"># Pandas does the centering for us</span>
|
||||
df <span style="color: #666666">=</span> df <span style="color: #666666">-</span>df<span style="color: #666666">.</span>mean()
|
||||
display(df)
|
||||
|
||||
X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>transform(X_train_scaled)
|
||||
<span style="color: #408080; font-style: italic"># we center it ourselves</span>
|
||||
X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</span>)
|
||||
<span style="color: #408080; font-style: italic"># Then check the difference between pandas and our own set up</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_centered<span style="color: #666666">-</span>df)
|
||||
<span style="color: #408080; font-style: italic">#Now we do an SVD</span>
|
||||
U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
|
||||
c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
|
||||
c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
|
||||
W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
|
||||
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
|
||||
forget to center the data first.
|
||||
|
||||
<p>
|
||||
Once you have identified all the principal components, you can reduce the dimensionality of the dataset
|
||||
down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
|
||||
Selecting this hyperplane ensures that the projection will preserve as much variance as possible.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
|
||||
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
@@ -219,7 +258,6 @@ X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</s
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -182,41 +180,38 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0023"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec22" class="anchor">Principal Component Analysis </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
|
||||
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
|
||||
<h2 id="___sec22" class="anchor">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
|
||||
that it automatically takes care of centering the data):
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</span>)
|
||||
U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
|
||||
c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
|
||||
c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
||||
X2D <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
|
||||
forget to center the data first.
|
||||
|
||||
<p>
|
||||
Once you have identified all the principal components, you can reduce the dimensionality of the dataset
|
||||
down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
|
||||
Selecting this hyperplane ensures that the projection will preserve as much variance as possible.
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
|
||||
principal component is equal to
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
|
||||
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
|
||||
variance that lies along the axis of each principal component.
|
||||
More material to come here.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -239,7 +234,6 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs024.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -182,36 +180,35 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0024"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23" class="anchor">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec23" class="anchor">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
|
||||
that it automatically takes care of centering the data):
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
|
||||
Unless, of course, you are reducing dimensionality for data visualization — in that case you will
|
||||
generally want to reduce the dimensionality down to 2 or 3.
|
||||
The following code computes PCA without reducing dimensionality, then computes the minimum number
|
||||
of dimensions required to preserve 95% of the training set’s variance:
|
||||
<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">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
||||
X2D <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca <span style="color: #666666">=</span> PCA()
|
||||
pca<span style="color: #666666">.</span>fit(X)
|
||||
cumsum <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cumsum(pca<span style="color: #666666">.</span>explained_variance_ratio_)
|
||||
d <span style="color: #666666">=</span> np<span style="color: #666666">.</span>argmax(cumsum <span style="color: #666666">>=</span> <span style="color: #666666">0.95</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
|
||||
principal component is equal to
|
||||
You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
|
||||
of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
|
||||
a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>])<span style="color: #666666">.</span>
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca <span style="color: #666666">=</span> PCA(n_components<span style="color: #666666">=0.95</span>)
|
||||
X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
|
||||
variance that lies along the axis of each principal component.
|
||||
More material to come here.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -233,7 +230,6 @@ More material to come here.
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,33 +182,15 @@ MathJax.Hub.Config({
|
||||
<a name="part0025"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec24" class="anchor">More on the PCA </h2>
|
||||
<h2 id="___sec24" class="anchor">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
|
||||
Unless, of course, you are reducing dimensionality for data visualization — in that case you will
|
||||
generally want to reduce the dimensionality down to 2 or 3.
|
||||
The following code computes PCA without reducing dimensionality, then computes the minimum number
|
||||
of dimensions required to preserve 95% of the training set’s variance:
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
|
||||
been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
|
||||
at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
|
||||
instances arrive).
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca <span style="color: #666666">=</span> PCA()
|
||||
pca<span style="color: #666666">.</span>fit(X)
|
||||
cumsum <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cumsum(pca<span style="color: #666666">.</span>explained_variance_ratio_)
|
||||
d <span style="color: #666666">=</span> np<span style="color: #666666">.</span>argmax(cumsum <span style="color: #666666">>=</span> <span style="color: #666666">0.95</span>) <span style="color: #666666">+</span> <span style="color: #666666">1</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
|
||||
of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
|
||||
a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca <span style="color: #666666">=</span> PCA(n_components<span style="color: #666666">=0.95</span>)
|
||||
X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -231,7 +211,6 @@ X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,14 +182,18 @@ MathJax.Hub.Config({
|
||||
<a name="part0026"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec25" class="anchor">Incremental PCA </h2>
|
||||
<h2 id="___sec25" class="anchor">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
|
||||
been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
|
||||
at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
|
||||
instances arrive).
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
algorithm that quickly finds an approximation of the first d principal components. Its computational
|
||||
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
|
||||
previous algorithms when \( d \) is much smaller than \( n \).
|
||||
|
||||
<p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -212,7 +214,6 @@ instances arrive).
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,14 +182,28 @@ MathJax.Hub.Config({
|
||||
<a name="part0027"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec26" class="anchor">Randomized PCA </h2>
|
||||
<h2 id="___sec26" class="anchor">Kernel PCA </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
algorithm that quickly finds an approximation of the first d principal components. Its computational
|
||||
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
|
||||
previous algorithms when \( d \) is much smaller than \( n \).
|
||||
The kernel trick is a mathematical technique that implicitly maps instances into a
|
||||
very high-dimensional space (called the feature space), enabling nonlinear classification and regression
|
||||
with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
|
||||
space corresponds to a complex nonlinear decision boundary in the original space.
|
||||
It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
|
||||
projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
|
||||
preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
|
||||
twisted manifold.
|
||||
For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
|
||||
<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">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> KernelPCA
|
||||
rbf_pca <span style="color: #666666">=</span> KernelPCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>, kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=0.04</span>)
|
||||
X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
</div>
|
||||
</div>
|
||||
@@ -215,7 +227,6 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<li class="active"><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs028.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,32 +182,14 @@ MathJax.Hub.Config({
|
||||
<a name="part0028"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec27" class="anchor">Kernel PCA </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
<h2 id="___sec27" class="anchor">LLE </h2>
|
||||
|
||||
<p>
|
||||
The kernel trick is a mathematical technique that implicitly maps instances into a
|
||||
very high-dimensional space (called the feature space), enabling nonlinear classification and regression
|
||||
with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
|
||||
space corresponds to a complex nonlinear decision boundary in the original space.
|
||||
It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
|
||||
projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
|
||||
preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
|
||||
twisted manifold.
|
||||
For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
|
||||
<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">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> KernelPCA
|
||||
rbf_pca <span style="color: #666666">=</span> KernelPCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>, kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=0.04</span>)
|
||||
X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
|
||||
algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
|
||||
closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
|
||||
these local relationships are best preserved (more details shortly).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -228,7 +208,6 @@ X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #6666
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li class="active"><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -184,16 +182,22 @@ MathJax.Hub.Config({
|
||||
<a name="part0029"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec28" class="anchor">LLE </h2>
|
||||
<h2 id="___sec28" class="anchor">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
|
||||
algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
|
||||
closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
|
||||
these local relationships are best preserved (more details shortly).
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
<p>
|
||||
Here are some of the most popular:
|
||||
|
||||
<ul>
|
||||
<li> <b>Multidimensional Scaling (MDS)</b> reduces dimensionality while trying to preserve the distances between the instances.</li>
|
||||
<li> <b>Isomap</b> creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.</li>
|
||||
<li> <b>t-Distributed Stochastic Neighbor Embedding</b> (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).</li>
|
||||
<li> Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -209,8 +213,6 @@ these local relationships are best preserved (more details shortly).
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li class="active"><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs030.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
|
||||
@@ -93,15 +93,14 @@ Automatically generated HTML file from DocOnce source
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -160,15 +159,14 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and Scikit-Learn Functionality</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -227,7 +225,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs030.html">31</a></li>
|
||||
<li><a href="._DimRed-bs029.html">30</a></li>
|
||||
<li><a href="._DimRed-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -1148,23 +1148,7 @@ chapter 12.4 and discussion therein.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec21">PCA and Scikit-Learn Functionality </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"># Now add PCA</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
|
||||
pca.fit(X_train_scaled)
|
||||
|
||||
X_pca = pca.transform(X_train_scaled)
|
||||
</pre></div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec22">Principal Component Analysis </h2>
|
||||
<h2 id="___sec21">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1173,14 +1157,34 @@ First it identifies the hyperplane that lies closest to the data, and then it pr
|
||||
|
||||
<p>
|
||||
The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data
|
||||
training set, then extracts the first two principal components. First we center the data using either <b>pandas</b> or our own code
|
||||
<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>X_centered = X - X.mean(axis=<span style="color: #B452CD">0</span>)
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> display
|
||||
np.random.seed(<span style="color: #B452CD">100</span>)
|
||||
<span style="color: #228B22"># setting up a 10 x 5 vanilla matrix </span>
|
||||
rows = <span style="color: #B452CD">10</span>
|
||||
cols = <span style="color: #B452CD">5</span>
|
||||
X = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(X)
|
||||
<span style="color: #228B22"># Pandas does the centering for us</span>
|
||||
df = df -df.mean()
|
||||
display(df)
|
||||
|
||||
<span style="color: #228B22"># we center it ourselves</span>
|
||||
X_centered = X - X.mean(axis=<span style="color: #B452CD">0</span>)
|
||||
<span style="color: #228B22"># Then check the difference between pandas and our own set up</span>
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_centered-df)
|
||||
<span style="color: #228B22">#Now we do an SVD</span>
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, <span style="color: #B452CD">0</span>]
|
||||
c2 = V.T[:, <span style="color: #B452CD">1</span>]
|
||||
W2 = V.T[:, :<span style="color: #B452CD">2</span>]
|
||||
X2D = X_centered.dot(W2)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
@@ -1201,7 +1205,7 @@ X2D = X_centered.dot(W2)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec23">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec22">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -1210,9 +1214,11 @@ that it automatically takes care of centering the data):
|
||||
<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: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
|
||||
X2D = pca.fit_transform(X)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
@@ -1221,7 +1227,7 @@ principal component is equal to
|
||||
<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>pca.components_.T[:, <span style="color: #B452CD">0</span>]).
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>].
|
||||
</pre></div>
|
||||
<p>
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
@@ -1232,7 +1238,7 @@ More material to come here.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec24">More on the PCA </h2>
|
||||
<h2 id="___sec23">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1263,7 +1269,7 @@ X_reduced = pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec25">Incremental PCA </h2>
|
||||
<h2 id="___sec24">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1275,7 +1281,7 @@ instances arrive).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec26">Randomized PCA </h2>
|
||||
<h2 id="___sec25">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1289,7 +1295,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec27">Kernel PCA </h2>
|
||||
<h2 id="___sec26">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1315,7 +1321,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec28">LLE </h2>
|
||||
<h2 id="___sec27">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1327,7 +1333,7 @@ these local relationships are best preserved (more details shortly).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec29">Other techniques </h2>
|
||||
<h2 id="___sec28">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -113,15 +113,14 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -1088,22 +1087,7 @@ chapter 12.4 and discussion therein.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">PCA and Scikit-Learn Functionality </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"># Now add PCA</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
|
||||
pca.fit(X_train_scaled)
|
||||
|
||||
X_pca = pca.transform(X_train_scaled)
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec22">Principal Component Analysis </h2>
|
||||
<h2 id="___sec21">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1112,14 +1096,34 @@ First it identifies the hyperplane that lies closest to the data, and then it pr
|
||||
|
||||
<p>
|
||||
The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data
|
||||
training set, then extracts the first two principal components. First we center the data using either <b>pandas</b> or our own code
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>X_centered = X - X.mean(axis=<span style="color: #B452CD">0</span>)
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> display
|
||||
np.random.seed(<span style="color: #B452CD">100</span>)
|
||||
<span style="color: #228B22"># setting up a 10 x 5 vanilla matrix </span>
|
||||
rows = <span style="color: #B452CD">10</span>
|
||||
cols = <span style="color: #B452CD">5</span>
|
||||
X = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(X)
|
||||
<span style="color: #228B22"># Pandas does the centering for us</span>
|
||||
df = df -df.mean()
|
||||
display(df)
|
||||
|
||||
<span style="color: #228B22"># we center it ourselves</span>
|
||||
X_centered = X - X.mean(axis=<span style="color: #B452CD">0</span>)
|
||||
<span style="color: #228B22"># Then check the difference between pandas and our own set up</span>
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_centered-df)
|
||||
<span style="color: #228B22">#Now we do an SVD</span>
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, <span style="color: #B452CD">0</span>]
|
||||
c2 = V.T[:, <span style="color: #B452CD">1</span>]
|
||||
W2 = V.T[:, :<span style="color: #B452CD">2</span>]
|
||||
X2D = X_centered.dot(W2)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
@@ -1139,7 +1143,7 @@ X2D = X_centered.dot(W2)
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec22">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -1148,9 +1152,11 @@ that it automatically takes care of centering the data):
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span><span style="color: #228B22">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
|
||||
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
|
||||
X2D = pca.fit_transform(X)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
@@ -1159,7 +1165,7 @@ principal component is equal to
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>]).
|
||||
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>].
|
||||
</pre></div>
|
||||
<p>
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
@@ -1170,7 +1176,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">More on the PCA </h2>
|
||||
<h2 id="___sec23">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1200,7 +1206,7 @@ X_reduced = pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec25">Incremental PCA </h2>
|
||||
<h2 id="___sec24">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1212,7 +1218,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Randomized PCA </h2>
|
||||
<h2 id="___sec25">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1227,7 +1233,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec27">Kernel PCA </h2>
|
||||
<h2 id="___sec26">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1256,7 +1262,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">LLE </h2>
|
||||
<h2 id="___sec27">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1268,7 +1274,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec29">Other techniques </h2>
|
||||
<h2 id="___sec28">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -118,15 +118,14 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Proof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('PCA Proof continued', 2, None, '___sec19'),
|
||||
('The final step', 2, None, '___sec20'),
|
||||
('PCA and Scikit-Learn Functionality', 2, None, '___sec21'),
|
||||
('Principal Component Analysis', 2, None, '___sec22'),
|
||||
('PCA and scikit-learn', 2, None, '___sec23'),
|
||||
('More on the PCA', 2, None, '___sec24'),
|
||||
('Incremental PCA', 2, None, '___sec25'),
|
||||
('Randomized PCA', 2, None, '___sec26'),
|
||||
('Kernel PCA', 2, None, '___sec27'),
|
||||
('LLE', 2, None, '___sec28'),
|
||||
('Other techniques', 2, None, '___sec29')]}
|
||||
('Principal Component Analysis', 2, None, '___sec21'),
|
||||
('PCA and scikit-learn', 2, None, '___sec22'),
|
||||
('More on the PCA', 2, None, '___sec23'),
|
||||
('Incremental PCA', 2, None, '___sec24'),
|
||||
('Randomized PCA', 2, None, '___sec25'),
|
||||
('Kernel PCA', 2, None, '___sec26'),
|
||||
('LLE', 2, None, '___sec27'),
|
||||
('Other techniques', 2, None, '___sec28')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -1093,22 +1092,7 @@ chapter 12.4 and discussion therein.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">PCA and Scikit-Learn Functionality </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"># Now add PCA</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
||||
pca<span style="color: #666666">.</span>fit(X_train_scaled)
|
||||
|
||||
X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>transform(X_train_scaled)
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec22">Principal Component Analysis </h2>
|
||||
<h2 id="___sec21">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1117,14 +1101,34 @@ First it identifies the hyperplane that lies closest to the data, and then it pr
|
||||
|
||||
<p>
|
||||
The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data
|
||||
training set, then extracts the first two principal components. First we center the data using either <b>pandas</b> or our own code
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</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">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">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
|
||||
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 vanilla matrix </span>
|
||||
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
|
||||
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
|
||||
<span style="color: #408080; font-style: italic"># Pandas does the centering for us</span>
|
||||
df <span style="color: #666666">=</span> df <span style="color: #666666">-</span>df<span style="color: #666666">.</span>mean()
|
||||
display(df)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># we center it ourselves</span>
|
||||
X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</span>)
|
||||
<span style="color: #408080; font-style: italic"># Then check the difference between pandas and our own set up</span>
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_centered<span style="color: #666666">-</span>df)
|
||||
<span style="color: #408080; font-style: italic">#Now we do an SVD</span>
|
||||
U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
|
||||
c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
|
||||
c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
|
||||
W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
|
||||
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
@@ -1144,7 +1148,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec22">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -1153,9 +1157,11 @@ that it automatically takes care of centering the data):
|
||||
<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">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
||||
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
||||
X2D <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X2D)
|
||||
</pre></div>
|
||||
<p>
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
@@ -1164,7 +1170,7 @@ principal component is equal to
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>])<span style="color: #666666">.</span>
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
@@ -1175,7 +1181,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">More on the PCA </h2>
|
||||
<h2 id="___sec23">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1205,7 +1211,7 @@ X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec25">Incremental PCA </h2>
|
||||
<h2 id="___sec24">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1217,7 +1223,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Randomized PCA </h2>
|
||||
<h2 id="___sec25">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1232,7 +1238,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec27">Kernel PCA </h2>
|
||||
<h2 id="___sec26">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1261,7 +1267,7 @@ X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #6666
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">LLE </h2>
|
||||
<h2 id="___sec27">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1273,7 +1279,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec29">Other techniques </h2>
|
||||
<h2 id="___sec28">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -1308,7 +1308,14 @@
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## PCA and Scikit-Learn Functionality"
|
||||
"\n",
|
||||
"\n",
|
||||
"## Principal Component Analysis\n",
|
||||
"Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n",
|
||||
"First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n",
|
||||
"\n",
|
||||
"The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the\n",
|
||||
"training set, then extracts the first two principal components. First we center the data using either **pandas** or our own code"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1319,38 +1326,30 @@
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Now add PCA\n",
|
||||
"from sklearn.decomposition import PCA\n",
|
||||
"pca = PCA(n_components = 2)\n",
|
||||
"pca.fit(X_train_scaled)\n",
|
||||
"import numpy as np\n",
|
||||
"import pandas as pd\n",
|
||||
"from IPython.display import display\n",
|
||||
"np.random.seed(100)\n",
|
||||
"# setting up a 10 x 5 vanilla matrix \n",
|
||||
"rows = 10\n",
|
||||
"cols = 5\n",
|
||||
"X = np.random.randn(rows,cols)\n",
|
||||
"df = pd.DataFrame(X)\n",
|
||||
"# Pandas does the centering for us\n",
|
||||
"df = df -df.mean()\n",
|
||||
"display(df)\n",
|
||||
"\n",
|
||||
"X_pca = pca.transform(X_train_scaled)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Principal Component Analysis\n",
|
||||
"Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n",
|
||||
"First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n",
|
||||
"\n",
|
||||
"The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the\n",
|
||||
"training set, then extracts the first two principal components. First we center the data"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# we center it ourselves\n",
|
||||
"X_centered = X - X.mean(axis=0)\n",
|
||||
"# Then check the difference between pandas and our own set up\n",
|
||||
"print(X_centered-df)\n",
|
||||
"#Now we do an SVD\n",
|
||||
"U, s, V = np.linalg.svd(X_centered)\n",
|
||||
"c1 = V.T[:, 0]\n",
|
||||
"c2 = V.T[:, 1]"
|
||||
"c2 = V.T[:, 1]\n",
|
||||
"W2 = V.T[:, :2]\n",
|
||||
"X2D = X_centered.dot(W2)\n",
|
||||
"print(X2D)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1368,7 +1367,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"execution_count": 12,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1392,15 +1391,17 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"execution_count": 13,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"#thereafter we do a PCA with Scikit-learn\n",
|
||||
"from sklearn.decomposition import PCA\n",
|
||||
"pca = PCA(n_components = 2)\n",
|
||||
"X2D = pca.fit_transform(X)"
|
||||
"X2D = pca.fit_transform(X)\n",
|
||||
"print(X2D)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1414,13 +1415,13 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"execution_count": 14,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"pca.components_.T[:, 0])."
|
||||
"pca.components_.T[:, 0]."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1444,7 +1445,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"execution_count": 15,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1467,7 +1468,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"execution_count": 16,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -1514,7 +1515,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"execution_count": 17,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -877,19 +877,6 @@ chapter 12.4 and discussion therein.
|
||||
|
||||
|
||||
|
||||
!split
|
||||
===== PCA and Scikit-Learn Functionality =====
|
||||
|
||||
|
||||
!bc pycod
|
||||
# Now add PCA
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
pca.fit(X_train_scaled)
|
||||
|
||||
X_pca = pca.transform(X_train_scaled)
|
||||
!ec
|
||||
|
||||
|
||||
|
||||
!split
|
||||
@@ -899,12 +886,32 @@ Principal Component Analysis (PCA) is by far the most popular dimensionality red
|
||||
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
|
||||
|
||||
The following Python code uses NumPy’s _svd()_ function to obtain all the principal components of the
|
||||
training set, then extracts the first two principal components. First we center the data
|
||||
training set, then extracts the first two principal components. First we center the data using either _pandas_ or our own code
|
||||
!bc pycod
|
||||
import numpy as np
|
||||
import pandas as pd
|
||||
from IPython.display import display
|
||||
np.random.seed(100)
|
||||
# setting up a 10 x 5 vanilla matrix
|
||||
rows = 10
|
||||
cols = 5
|
||||
X = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(X)
|
||||
# Pandas does the centering for us
|
||||
df = df -df.mean()
|
||||
display(df)
|
||||
|
||||
# we center it ourselves
|
||||
X_centered = X - X.mean(axis=0)
|
||||
# Then check the difference between pandas and our own set up
|
||||
print(X_centered-df)
|
||||
#Now we do an SVD
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, 0]
|
||||
c2 = V.T[:, 1]
|
||||
W2 = V.T[:, :2]
|
||||
X2D = X_centered.dot(W2)
|
||||
print(X2D)
|
||||
!ec
|
||||
|
||||
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
|
||||
@@ -926,15 +933,17 @@ Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we d
|
||||
following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
|
||||
that it automatically takes care of centering the data):
|
||||
!bc pycod
|
||||
#thereafter we do a PCA with Scikit-learn
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
X2D = pca.fit_transform(X)
|
||||
print(X2D)
|
||||
!ec
|
||||
After fitting the PCA transformer to the dataset, you can access the principal components using the
|
||||
components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
|
||||
principal component is equal to
|
||||
!bc pycod
|
||||
pca.components_.T[:, 0]).
|
||||
pca.components_.T[:, 0].
|
||||
!ec
|
||||
Another very useful piece of information is the explained variance ratio of each principal component,
|
||||
available via the $explained\_variance\_ratio$ variable. It indicates the proportion of the dataset’s
|
||||
|
||||
@@ -1,31 +1,3 @@
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
pca.fit(X_train_scaled)
|
||||
|
||||
X_pca = pca.transform(X_train_scaled)
|
||||
|
||||
X_centered = X - X.mean(axis=0)
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, 0]
|
||||
c2 = V.T[:, 1]
|
||||
|
||||
W2 = V.T[:, :2]
|
||||
X2D = X_centered.dot(W2)
|
||||
|
||||
pca = PCA(n_components = 2)
|
||||
X2D = pca.fit_transform(X)
|
||||
|
||||
pca.components_.T[:, 0]).
|
||||
|
||||
pca = PCA()
|
||||
pca.fit(X)
|
||||
cumsum = np.cumsum(pca.explained_variance_ratio_)
|
||||
d = np.argmax(cumsum >= 0.95) + 1
|
||||
|
||||
pca = PCA(n_components=0.95)
|
||||
X_reduced = pca.fit_transform(X)
|
||||
|
||||
|
||||
import numpy as np
|
||||
import pandas as pd
|
||||
from IPython.display import display
|
||||
@@ -33,11 +5,27 @@ np.random.seed(100)
|
||||
# setting up a 10 x 5 matrix
|
||||
rows = 10
|
||||
cols = 5
|
||||
a = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(a)
|
||||
X = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(X)
|
||||
# Pandas does the centering for us
|
||||
df = df -df.mean()
|
||||
display(df)
|
||||
print(df.mean())
|
||||
print(df.std())
|
||||
display(df**2)
|
||||
|
||||
# we center it ourselves
|
||||
X_centered = X - X.mean(axis=0)
|
||||
print(X_centered-df)
|
||||
#Now we do an SVD
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, 0]
|
||||
c2 = V.T[:, 1]
|
||||
W2 = V.T[:, :2]
|
||||
X2D = X_centered.dot(W2)
|
||||
print(X2D)
|
||||
#thereafter we do a PCA with Scikit-learn
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
X2D = pca.fit_transform(X)
|
||||
print(X2D)
|
||||
|
||||
print(pca.components_.T[:, 0])
|
||||
|
||||
|
||||
Reference in New Issue
Block a user