further update, added also
This commit is contained in:
mhjensen
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -221,7 +223,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -221,7 +223,7 @@ data.
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs002.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -220,7 +222,7 @@ ensures that all features are exactly between \( 0 \) and \( 1 \). The
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs003.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -223,7 +225,7 @@ techniques.
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs012.html">13</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -298,7 +300,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -248,7 +250,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -227,7 +229,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -282,7 +284,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -212,7 +214,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -261,7 +263,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -256,7 +258,7 @@ $$
|
||||
<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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs011.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -244,7 +246,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs012.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -246,7 +248,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs013.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -228,7 +230,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs014.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -264,7 +266,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs015.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -244,7 +246,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -213,6 +215,19 @@ $$
|
||||
In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
|
||||
\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
|
||||
|
||||
<p>
|
||||
The eigenvalues tell us then how much we need to stretch the
|
||||
corresponding eigenvectors. Dimensions with large eigenvalues have
|
||||
thus large variations (large variance) and define therefore useful
|
||||
dimensions. The data points are more spread out in the direction of
|
||||
these eigenvectors. Smaller eigenvalues mean on the other hand that
|
||||
the corresponding eigenvectors are shrunk accordingly and the data
|
||||
points are tightly bunched together and there is not much variation in
|
||||
these specific directions. Hopefully then we could leave it out
|
||||
dimensions where the eigenvalues are very small. If \( p \) is very large,
|
||||
we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
|
||||
features/predictors.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -239,7 +254,7 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord
|
||||
<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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs017.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,7 +180,36 @@ MathJax.Hub.Config({
|
||||
<a name="part0017"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec16" class="anchor">Classical PCA Theorem </h2>
|
||||
<h2 id="___sec16" class="anchor">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
|
||||
<ul>
|
||||
<li> Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.</li>
|
||||
</ul>
|
||||
|
||||
$$
|
||||
\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
|
||||
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
|
||||
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
|
||||
\dots & \dots & \dots & \dots \dots & \dots \\
|
||||
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
|
||||
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
|
||||
<ul>
|
||||
<li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
|
||||
<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
|
||||
<li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
|
||||
<li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
|
||||
<li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
|
||||
</ul>
|
||||
|
||||
After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -206,7 +237,7 @@ MathJax.Hub.Config({
|
||||
<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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs018.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,7 +180,7 @@ MathJax.Hub.Config({
|
||||
<a name="part0018"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec17" class="anchor">Prof of the PCA Theorem </h2>
|
||||
<h2 id="___sec17" class="anchor">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -205,6 +207,8 @@ MathJax.Hub.Config({
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<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-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,18 +180,8 @@ MathJax.Hub.Config({
|
||||
<a name="part0019"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec18" class="anchor">Getting started with PCA </h2>
|
||||
<h2 id="___sec18" class="anchor">Prof of the PCA Theorem </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>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -214,6 +206,7 @@ X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</s
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs020.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,38 +180,17 @@ MathJax.Hub.Config({
|
||||
<a name="part0020"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec19" 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="___sec19" class="anchor">Getting started with PCA </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
|
||||
<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>]
|
||||
</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.
|
||||
<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)
|
||||
|
||||
<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)
|
||||
X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>transform(X_train_scaled)
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
@@ -234,6 +215,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs021.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -176,36 +178,41 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0021"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec20" class="anchor">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec20" 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>
|
||||
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):
|
||||
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
|
||||
<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>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>]
|
||||
</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
|
||||
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>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>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>
|
||||
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 -->
|
||||
@@ -228,6 +235,7 @@ More material to come here.
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -176,35 +178,36 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0022"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec21" class="anchor">More on the PCA </h2>
|
||||
<h2 id="___sec21" class="anchor">PCA and scikit-learn </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:
|
||||
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>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>
|
||||
<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)
|
||||
</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:
|
||||
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>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)
|
||||
<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 -->
|
||||
@@ -226,6 +229,7 @@ X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,15 +180,33 @@ MathJax.Hub.Config({
|
||||
<a name="part0023"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec22" class="anchor">Incremental PCA </h2>
|
||||
<h2 id="___sec22" class="anchor">More on the 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).
|
||||
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>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 -->
|
||||
@@ -207,6 +227,7 @@ instances arrive).
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs024.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,18 +180,14 @@ MathJax.Hub.Config({
|
||||
<a name="part0024"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec23" class="anchor">Randomized PCA </h2>
|
||||
<h2 id="___sec23" class="anchor">Incremental PCA </h2>
|
||||
|
||||
<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 \( n \).
|
||||
|
||||
<p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -210,6 +208,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,28 +180,14 @@ MathJax.Hub.Config({
|
||||
<a name="part0025"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec24" 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="___sec24" class="anchor">Randomized PCA </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>
|
||||
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 \( n \).
|
||||
|
||||
<!-- 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>
|
||||
@@ -223,6 +211,7 @@ X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #6666
|
||||
<li class="active"><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,14 +180,32 @@ MathJax.Hub.Config({
|
||||
<a name="part0026"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec25" class="anchor">LLE </h2>
|
||||
<h2 id="___sec25" 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>
|
||||
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).
|
||||
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>
|
||||
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -204,6 +224,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li class="active"><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -178,77 +180,16 @@ MathJax.Hub.Config({
|
||||
<a name="part0027"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec26" class="anchor">Other techniques </h2>
|
||||
<h2 id="___sec26" class="anchor">LLE </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
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>
|
||||
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>
|
||||
|
||||
Here are other examples where we use the <b>DataFrame</b> functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
|
||||
of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">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 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>
|
||||
a <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(a)
|
||||
display(df)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>mean())
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>std())
|
||||
display(df<span style="color: #666666">**2</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Thereafter we can select specific columns only and plot final results
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>df<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'First'</span>, <span style="color: #BA2121">'Second'</span>, <span style="color: #BA2121">'Third'</span>, <span style="color: #BA2121">'Fourth'</span>, <span style="color: #BA2121">'Fifth'</span>]
|
||||
df<span style="color: #666666">.</span>index <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">10</span>)
|
||||
|
||||
display(df)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df[<span style="color: #BA2121">'Second'</span>]<span style="color: #666666">.</span>mean() )
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>info())
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>describe())
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
||||
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
||||
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
||||
|
||||
df<span style="color: #666666">.</span>cumsum()<span style="color: #666666">.</span>plot(lw<span style="color: #666666">=2.0</span>, figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
df<span style="color: #666666">.</span>plot<span style="color: #666666">.</span>bar(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>), rot<span style="color: #666666">=15</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
We can produce a \( 4\times 4 \) matrix
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">16</span>)<span style="color: #666666">.</span>reshape((<span style="color: #666666">4</span>,<span style="color: #666666">4</span>))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(b)
|
||||
df1 <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(b)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(df1)
|
||||
</pre></div>
|
||||
<p>
|
||||
and many other operations.
|
||||
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -264,6 +205,8 @@ and many other operations.
|
||||
<li><a href="._DimRed-bs025.html">26</a></li>
|
||||
<li><a href="._DimRed-bs026.html">27</a></li>
|
||||
<li class="active"><a href="._DimRed-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs028.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
|
||||
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -221,7 +223,7 @@ 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-bs027.html">28</a></li>
|
||||
<li><a href="._DimRed-bs028.html">29</a></li>
|
||||
<li><a href="._DimRed-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -936,21 +936,70 @@ $$
|
||||
<p>
|
||||
In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
|
||||
\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
|
||||
|
||||
<p>
|
||||
The eigenvalues tell us then how much we need to stretch the
|
||||
corresponding eigenvectors. Dimensions with large eigenvalues have
|
||||
thus large variations (large variance) and define therefore useful
|
||||
dimensions. The data points are more spread out in the direction of
|
||||
these eigenvectors. Smaller eigenvalues mean on the other hand that
|
||||
the corresponding eigenvectors are shrunk accordingly and the data
|
||||
points are tightly bunched together and there is not much variation in
|
||||
these specific directions. Hopefully then we could leave it out
|
||||
dimensions where the eigenvalues are very small. If \( p \) is very large,
|
||||
we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
|
||||
features/predictors.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec16">Classical PCA Theorem </h2>
|
||||
<h2 id="___sec16">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
|
||||
<ul>
|
||||
<p><li> Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.</li>
|
||||
</ul>
|
||||
<p> <br>
|
||||
$$
|
||||
\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
|
||||
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
|
||||
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
|
||||
\dots & \dots & \dots & \dots \dots & \dots \\
|
||||
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
|
||||
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
|
||||
<ul>
|
||||
<p><li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
|
||||
<p><li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
|
||||
<p><li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
|
||||
<p><li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
|
||||
<p><li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
|
||||
</ul>
|
||||
<p>
|
||||
|
||||
After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec17">Prof of the PCA Theorem </h2>
|
||||
<h2 id="___sec17">Classical PCA Theorem </h2>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec18">Getting started with PCA </h2>
|
||||
<h2 id="___sec18">Prof of the PCA Theorem </h2>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec19">Getting started with PCA </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -966,7 +1015,7 @@ X_pca = pca.transform(X_train_scaled)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec19">Principal Component Analysis </h2>
|
||||
<h2 id="___sec20">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1003,7 +1052,7 @@ X2D = X_centered.dot(W2)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec20">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec21">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -1034,7 +1083,7 @@ More material to come here.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec21">More on the PCA </h2>
|
||||
<h2 id="___sec22">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1065,7 +1114,7 @@ X_reduced = pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec22">Incremental PCA </h2>
|
||||
<h2 id="___sec23">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1077,7 +1126,7 @@ instances arrive).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec23">Randomized PCA </h2>
|
||||
<h2 id="___sec24">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1091,7 +1140,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec24">Kernel PCA </h2>
|
||||
<h2 id="___sec25">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1117,7 +1166,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec25">LLE </h2>
|
||||
<h2 id="___sec26">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1129,7 +1178,7 @@ these local relationships are best preserved (more details shortly).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec26">Other techniques </h2>
|
||||
<h2 id="___sec27">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -108,17 +108,18 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -903,19 +904,66 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord
|
||||
\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">Classical PCA Theorem </h2>
|
||||
The eigenvalues tell us then how much we need to stretch the
|
||||
corresponding eigenvectors. Dimensions with large eigenvalues have
|
||||
thus large variations (large variance) and define therefore useful
|
||||
dimensions. The data points are more spread out in the direction of
|
||||
these eigenvectors. Smaller eigenvalues mean on the other hand that
|
||||
the corresponding eigenvectors are shrunk accordingly and the data
|
||||
points are tightly bunched together and there is not much variation in
|
||||
these specific directions. Hopefully then we could leave it out
|
||||
dimensions where the eigenvalues are very small. If \( p \) is very large,
|
||||
we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
|
||||
features/predictors.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">Prof of the PCA Theorem </h2>
|
||||
<h2 id="___sec16">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
|
||||
<ul>
|
||||
<li> Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.</li>
|
||||
</ul>
|
||||
|
||||
$$
|
||||
\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
|
||||
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
|
||||
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
|
||||
\dots & \dots & \dots & \dots \dots & \dots \\
|
||||
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
|
||||
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
|
||||
<ul>
|
||||
<li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
|
||||
<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
|
||||
<li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
|
||||
<li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
|
||||
<li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
|
||||
</ul>
|
||||
|
||||
After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Getting started with PCA </h2>
|
||||
<h2 id="___sec17">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Prof of the PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Getting started with PCA </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -930,7 +978,7 @@ X_pca = pca.transform(X_train_scaled)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Principal Component Analysis </h2>
|
||||
<h2 id="___sec20">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -966,7 +1014,7 @@ X2D = X_centered.dot(W2)
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec20">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec21">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -997,7 +1045,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">More on the PCA </h2>
|
||||
<h2 id="___sec22">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1027,7 +1075,7 @@ X_reduced = pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec22">Incremental PCA </h2>
|
||||
<h2 id="___sec23">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1039,7 +1087,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec23">Randomized PCA </h2>
|
||||
<h2 id="___sec24">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1054,7 +1102,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">Kernel PCA </h2>
|
||||
<h2 id="___sec25">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1083,7 +1131,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec25">LLE </h2>
|
||||
<h2 id="___sec26">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1095,7 +1143,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Other techniques </h2>
|
||||
<h2 id="___sec27">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -113,17 +113,18 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
None,
|
||||
'___sec14'),
|
||||
('Towards the PCA theorem', 2, None, '___sec15'),
|
||||
('Classical PCA Theorem', 2, None, '___sec16'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec17'),
|
||||
('Getting started with PCA', 2, None, '___sec18'),
|
||||
('Principal Component Analysis', 2, None, '___sec19'),
|
||||
('PCA and scikit-learn', 2, None, '___sec20'),
|
||||
('More on the PCA', 2, None, '___sec21'),
|
||||
('Incremental PCA', 2, None, '___sec22'),
|
||||
('Randomized PCA', 2, None, '___sec23'),
|
||||
('Kernel PCA', 2, None, '___sec24'),
|
||||
('LLE', 2, None, '___sec25'),
|
||||
('Other techniques', 2, None, '___sec26')]}
|
||||
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
||||
('Classical PCA Theorem', 2, None, '___sec17'),
|
||||
('Prof of the PCA Theorem', 2, None, '___sec18'),
|
||||
('Getting started with PCA', 2, None, '___sec19'),
|
||||
('Principal Component Analysis', 2, None, '___sec20'),
|
||||
('PCA and scikit-learn', 2, None, '___sec21'),
|
||||
('More on the PCA', 2, None, '___sec22'),
|
||||
('Incremental PCA', 2, None, '___sec23'),
|
||||
('Randomized PCA', 2, None, '___sec24'),
|
||||
('Kernel PCA', 2, None, '___sec25'),
|
||||
('LLE', 2, None, '___sec26'),
|
||||
('Other techniques', 2, None, '___sec27')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -908,19 +909,66 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord
|
||||
\( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">Classical PCA Theorem </h2>
|
||||
The eigenvalues tell us then how much we need to stretch the
|
||||
corresponding eigenvectors. Dimensions with large eigenvalues have
|
||||
thus large variations (large variance) and define therefore useful
|
||||
dimensions. The data points are more spread out in the direction of
|
||||
these eigenvectors. Smaller eigenvalues mean on the other hand that
|
||||
the corresponding eigenvectors are shrunk accordingly and the data
|
||||
points are tightly bunched together and there is not much variation in
|
||||
these specific directions. Hopefully then we could leave it out
|
||||
dimensions where the eigenvalues are very small. If \( p \) is very large,
|
||||
we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
|
||||
features/predictors.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">Prof of the PCA Theorem </h2>
|
||||
<h2 id="___sec16">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
|
||||
<ul>
|
||||
<li> Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.</li>
|
||||
</ul>
|
||||
|
||||
$$
|
||||
\boldsymbol{X}=\begin{bmatrix}
|
||||
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
|
||||
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
|
||||
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
|
||||
\dots & \dots & \dots & \dots \dots & \dots \\
|
||||
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
|
||||
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
|
||||
<ul>
|
||||
<li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
|
||||
<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
|
||||
<li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
|
||||
<li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
|
||||
<li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
|
||||
</ul>
|
||||
|
||||
After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Getting started with PCA </h2>
|
||||
<h2 id="___sec17">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec18">Prof of the PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Getting started with PCA </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -935,7 +983,7 @@ X_pca <span style="color: #666666">=</span> pca<span style="color: #666666">.</s
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Principal Component Analysis </h2>
|
||||
<h2 id="___sec20">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -971,7 +1019,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec20">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec21">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -1002,7 +1050,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec21">More on the PCA </h2>
|
||||
<h2 id="___sec22">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -1032,7 +1080,7 @@ 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="___sec22">Incremental PCA </h2>
|
||||
<h2 id="___sec23">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -1044,7 +1092,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec23">Randomized PCA </h2>
|
||||
<h2 id="___sec24">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -1059,7 +1107,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec24">Kernel PCA </h2>
|
||||
<h2 id="___sec25">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -1088,7 +1136,7 @@ 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="___sec25">LLE </h2>
|
||||
<h2 id="___sec26">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -1100,7 +1148,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec26">Other techniques </h2>
|
||||
<h2 id="___sec27">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -1000,6 +1000,57 @@
|
||||
"In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is\n",
|
||||
"$\\lambda_0 > \\lambda_1 > \\dots > \\lambda_{p-1}$. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"The eigenvalues tell us then how much we need to stretch the\n",
|
||||
"corresponding eigenvectors. Dimensions with large eigenvalues have\n",
|
||||
"thus large variations (large variance) and define therefore useful\n",
|
||||
"dimensions. The data points are more spread out in the direction of\n",
|
||||
"these eigenvectors. Smaller eigenvalues mean on the other hand that\n",
|
||||
"the corresponding eigenvectors are shrunk accordingly and the data\n",
|
||||
"points are tightly bunched together and there is not much variation in\n",
|
||||
"these specific directions. Hopefully then we could leave it out\n",
|
||||
"dimensions where the eigenvalues are very small. If $p$ is very large,\n",
|
||||
"we could then aim at reducing $p$ to $l << p$ and handle only $l$\n",
|
||||
"features/predictors.\n",
|
||||
"\n",
|
||||
"## The Algorithm before theorem\n",
|
||||
"\n",
|
||||
"Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. \n",
|
||||
"* Set up the datapoints for the design/feature matrix $\\boldsymbol{X}$ with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ referring to the column numbers and the entries $n$ being the row elements."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\boldsymbol{X}=\\begin{bmatrix}\n",
|
||||
"x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n",
|
||||
"x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n",
|
||||
"x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n",
|
||||
"\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n",
|
||||
"x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n",
|
||||
"x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n",
|
||||
"\\end{bmatrix},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"* Center the data by subtracting the mean value for each column. This leads to a new matrix $\\boldsymbol{X}\\rightarrow \\overline{\\boldsymbol{X}}$.\n",
|
||||
"\n",
|
||||
"* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}\\overline{\\boldsymbol{X}}^T].\n",
|
||||
"\n",
|
||||
"* Find the eigenpairs of $\\boldsymbol{C}$ with eigenvalues $[\\lambda_0,\\lambda_1,\\dots,\\lambda_{p-1}]$ and eigenvectors $[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$.\n",
|
||||
"\n",
|
||||
"* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.\n",
|
||||
"\n",
|
||||
"* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.\n",
|
||||
"\n",
|
||||
"After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.\n",
|
||||
"\n",
|
||||
"## Classical PCA Theorem\n",
|
||||
"\n",
|
||||
"\n",
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -703,6 +703,44 @@ and since $\bm{C}[\bm{y}]$ is diagonal we have for a given eigenvalue $i$ of the
|
||||
In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
|
||||
$\lambda_0 > \lambda_1 > \dots > \lambda_{p-1}$.
|
||||
|
||||
|
||||
The eigenvalues tell us then how much we need to stretch the
|
||||
corresponding eigenvectors. Dimensions with large eigenvalues have
|
||||
thus large variations (large variance) and define therefore useful
|
||||
dimensions. The data points are more spread out in the direction of
|
||||
these eigenvectors. Smaller eigenvalues mean on the other hand that
|
||||
the corresponding eigenvectors are shrunk accordingly and the data
|
||||
points are tightly bunched together and there is not much variation in
|
||||
these specific directions. Hopefully then we could leave it out
|
||||
dimensions where the eigenvalues are very small. If $p$ is very large,
|
||||
we could then aim at reducing $p$ to $l << p$ and handle only $l$
|
||||
features/predictors.
|
||||
|
||||
!split
|
||||
===== The Algorithm before the Theorem =====
|
||||
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
* Set up the datapoints for the design/feature matrix $\bm{X}$ with $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$ referring to the column numbers and the entries $n$ being the row elements.
|
||||
!bt
|
||||
\[
|
||||
\bm{X}=\begin{bmatrix}
|
||||
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
|
||||
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
|
||||
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
|
||||
\dots & \dots & \dots & \dots \dots & \dots \\
|
||||
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
|
||||
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
|
||||
\end{bmatrix},
|
||||
\]
|
||||
!et
|
||||
* Center the data by subtracting the mean value for each column. This leads to a new matrix $\bm{X}\rightarrow \overline{\bm{X}}$.
|
||||
* Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\bm{X}}\overline{\bm{X}}^T].
|
||||
* Find the eigenpairs of $\bm{C}$ with eigenvalues $[\lambda_0,\lambda_1,\dots,\lambda_{p-1}]$ and eigenvectors $[\bm{s}_0,\bm{s}_1,\dots,\bm{s}_{p-1}]$.
|
||||
* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
|
||||
* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.
|
||||
|
||||
After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
|
||||
|
||||
!split
|
||||
===== Classical PCA Theorem =====
|
||||
|
||||
|
||||
@@ -0,0 +1,68 @@
|
||||
# Common imports
|
||||
import numpy as np
|
||||
from sklearn.neural_network import MLPRegressor
|
||||
from sklearn.metrics import accuracy_score
|
||||
import seaborn as sns
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
def FrankeFunction(x,y):
|
||||
term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
|
||||
term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
|
||||
term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
|
||||
term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
|
||||
return term1 + term2 + term3 + term4
|
||||
|
||||
|
||||
def create_X(x, y, n ):
|
||||
if len(x.shape) > 1:
|
||||
x = np.ravel(x)
|
||||
y = np.ravel(y)
|
||||
|
||||
N = len(x)
|
||||
l = int((n+1)*(n+2)/2) # Number of elements in beta
|
||||
X = np.ones((N,l))
|
||||
|
||||
for i in range(1,n+1):
|
||||
q = int((i)*(i+1)/2)
|
||||
for k in range(i+1):
|
||||
X[:,q+k] = (x**(i-k))*(y**k)
|
||||
|
||||
return X
|
||||
|
||||
|
||||
# Making meshgrid of datapoints and compute Franke's function
|
||||
n = 4
|
||||
N = 100
|
||||
x = np.sort(np.random.uniform(0, 1, N))
|
||||
y = np.sort(np.random.uniform(0, 1, N))
|
||||
z = FrankeFunction(x, y)
|
||||
X = create_X(x, y, n=n)
|
||||
|
||||
# only training data, no advanced splitting
|
||||
X_train = X
|
||||
Y_train = z
|
||||
# only one simple layer with 100 neurons
|
||||
n_hidden_neurons = 100
|
||||
epochs = 100
|
||||
# store models for later use
|
||||
eta_vals = np.logspace(-5, 1, 7)
|
||||
lmbd_vals = np.logspace(-5, 1, 7)
|
||||
# store the models for later use
|
||||
DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
||||
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
||||
sns.set()
|
||||
for i, eta in enumerate(eta_vals):
|
||||
for j, lmbd in enumerate(lmbd_vals):
|
||||
dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
|
||||
alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
|
||||
dnn.fit(X_train, Y_train)
|
||||
DNN_scikit[i][j] = dnn
|
||||
train_accuracy[i][j] = dnn.score(X_train, Y_train)
|
||||
|
||||
fig, ax = plt.subplots(figsize = (10, 10))
|
||||
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
||||
ax.set_title("Training Accuracy")
|
||||
ax.set_ylabel("$\eta$")
|
||||
ax.set_xlabel("$\lambda$")
|
||||
plt.show()
|
||||
|
||||
Reference in New Issue
Block a user