Dim red update
This commit is contained in:
mhjensen
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
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None,
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'___sec0'),
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('Preprocessing our data', 2, None, '___sec1'),
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('Principal Component Analysis', 2, None, '___sec2'),
|
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('PCA and scikit-learn', 2, None, '___sec3'),
|
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('More on the PCA', 2, None, '___sec4'),
|
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('Incremental PCA', 2, None, '___sec5'),
|
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('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
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('LLE', 2, None, '___sec8'),
|
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('Other techniques', 2, None, '___sec9')]}
|
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('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
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('PCA and scikit-learn', 2, None, '___sec4'),
|
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('More on the PCA', 2, None, '___sec5'),
|
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('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
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('Kernel PCA', 2, None, '___sec8'),
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('LLE', 2, None, '___sec9'),
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('Other techniques', 2, None, '___sec10')]}
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end of tocinfo -->
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<body>
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@@ -93,14 +94,15 @@ MathJax.Hub.Config({
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
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</ul>
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</li>
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@@ -135,7 +137,7 @@ MathJax.Hub.Config({
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Oct 10, 2019</h4></center> <!-- date -->
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<center><h4>Oct 12, 2019</h4></center> <!-- date -->
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<br>
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<p>
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@@ -159,7 +161,7 @@ MathJax.Hub.Config({
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<li><a href="._DimRed-bs008.html">9</a></li>
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<li><a href="._DimRed-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._DimRed-bs010.html">11</a></li>
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<li><a href="._DimRed-bs011.html">12</a></li>
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<li><a href="._DimRed-bs001.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
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'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
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('Principal Component Analysis', 2, None, '___sec2'),
|
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('PCA and scikit-learn', 2, None, '___sec3'),
|
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('More on the PCA', 2, None, '___sec4'),
|
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('Incremental PCA', 2, None, '___sec5'),
|
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('Randomized PCA', 2, None, '___sec6'),
|
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('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
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('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
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('Kernel PCA', 2, None, '___sec8'),
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('LLE', 2, None, '___sec9'),
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('Other techniques', 2, None, '___sec10')]}
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end of tocinfo -->
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<body>
|
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@@ -93,14 +94,15 @@ MathJax.Hub.Config({
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
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</ul>
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</li>
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@@ -153,6 +155,8 @@ reduction techniques: the principal component analysis PCA, Kernel PCA, and Loca
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<li><a href="._DimRed-bs008.html">9</a></li>
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<li><a href="._DimRed-bs009.html">10</a></li>
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||||
<li><a href="._DimRed-bs010.html">11</a></li>
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<li><a href="">...</a></li>
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<li><a href="._DimRed-bs011.html">12</a></li>
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<li><a href="._DimRed-bs002.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
|
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@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
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||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
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||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
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||||
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||||
</ul>
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||||
</li>
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@@ -123,15 +125,15 @@ MathJax.Hub.Config({
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<p>
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Before we proceed however, we will discuss how to preprocess our
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data. Till now and in connection with project 1 not met so many cases
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data. Till now and in connection with our previous examples we have not met so many cases
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where we are too sensitive to the scaling of our data. Normally the
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data may need a rescaling and/or may be sensitive to extreme
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values. Scaling the data renders our inputs much more suitable for the
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algorithms we want to emply.
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algorithms we want to employ.
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<p>
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<b>Scikit-Learn</b> has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b> ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).
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This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scale each column of the design matrix so that
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This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scales each column of the design matrix by its Euclidean norm.
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<p>
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</div>
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@@ -154,6 +156,7 @@ This scaling has the drawback that it does not ensure that we have a particular
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<li><a href="._DimRed-bs008.html">9</a></li>
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<li><a href="._DimRed-bs009.html">10</a></li>
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<li><a href="._DimRed-bs010.html">11</a></li>
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<li><a href="._DimRed-bs011.html">12</a></li>
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<li><a href="._DimRed-bs003.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
|
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@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
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('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
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||||
</li>
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@@ -116,38 +118,54 @@ MathJax.Hub.Config({
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<a name="part0003"></a>
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<!-- !split -->
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<h2 id="___sec2" class="anchor">Principal Component Analysis </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
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First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
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<h2 id="___sec2" class="anchor">Simple preprocessing examples </h2>
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<p>
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The following Python code uses NumPy’s <b>svd()</b> function to obtain all the principal components of the
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training set, then extracts the first two principal components
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We show here how we can use a simple regression case (our nuclear binding energies discussed earlier).
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Rescaling our data with different
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<p>
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<!-- 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: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
cancer <span style="color: #666666">=</span> load_breast_cancer()
|
||||
|
||||
<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>
|
||||
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
|
||||
|
||||
<!-- 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)
|
||||
svm <span style="color: #666666">=</span> SVC(C<span style="color: #666666">=100</span>)
|
||||
svm<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test,y_test)))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> MinMaxScaler, StandardScaler
|
||||
|
||||
scaler <span style="color: #666666">=</span> MinMaxScaler()
|
||||
scaler<span style="color: #666666">.</span>fit(X_train)
|
||||
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
||||
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
|
||||
|
||||
|
||||
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy scaled data: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
||||
|
||||
scaler <span style="color: #666666">=</span> StandardScaler()
|
||||
scaler<span style="color: #666666">.</span>fit(X_train)
|
||||
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
||||
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
||||
|
||||
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy scaled data: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
@@ -165,6 +183,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -114,36 +116,41 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0004"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec3" class="anchor">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec3" 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 -->
|
||||
@@ -160,6 +167,7 @@ More material to come here.
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -114,33 +116,36 @@ MathJax.Hub.Config({
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0005"></a>
|
||||
<!-- !split -->
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec4" class="anchor">More on the PCA </h2>
|
||||
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:
|
||||
<h2 id="___sec4" class="anchor">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
|
||||
that it automatically takes care of centering the data):
|
||||
<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 -->
|
||||
@@ -157,6 +162,7 @@ X_reduced <span style="color: #666666">=</span> pca<span style="color: #666666">
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
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|
||||
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||||
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|
||||
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|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
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|
||||
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|
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|
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|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
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end of tocinfo -->
|
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|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -116,13 +118,31 @@ MathJax.Hub.Config({
|
||||
<a name="part0006"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec5" class="anchor">Incremental PCA </h2>
|
||||
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).
|
||||
<h2 id="___sec5" class="anchor">More on the PCA </h2>
|
||||
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 -->
|
||||
@@ -139,6 +159,7 @@ instances arrive).
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -116,18 +118,12 @@ MathJax.Hub.Config({
|
||||
<a name="part0007"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec6" class="anchor">Randomized 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>
|
||||
|
||||
<h2 id="___sec6" class="anchor">Incremental PCA </h2>
|
||||
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>
|
||||
@@ -145,6 +141,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
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|
||||
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|
||||
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|
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|
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|
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||||
('LLE', 2, None, '___sec8'),
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('Other techniques', 2, None, '___sec10')]}
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end of tocinfo -->
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|
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<body>
|
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@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -116,28 +118,14 @@ MathJax.Hub.Config({
|
||||
<a name="part0008"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec7" 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="___sec7" 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>
|
||||
@@ -159,6 +147,7 @@ X_reduced <span style="color: #666666">=</span> rbf_pca<span style="color: #6666
|
||||
<li class="active"><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -116,14 +118,32 @@ MathJax.Hub.Config({
|
||||
<a name="part0009"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec8" class="anchor">LLE </h2>
|
||||
<h2 id="___sec8" 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>
|
||||
@@ -141,6 +161,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<li><a href="._DimRed-bs008.html">9</a></li>
|
||||
<li class="active"><a href="._DimRed-bs009.html">10</a></li>
|
||||
<li><a href="._DimRed-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -46,14 +46,15 @@ Automatically generated HTML file from DocOnce source
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -93,14 +94,15 @@ MathJax.Hub.Config({
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Other techniques</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">Simple preprocessing examples</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Principal Component Analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">PCA and scikit-learn</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on the PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Incremental PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Randomized PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Kernel PCA</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">LLE</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Other techniques</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -135,7 +137,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 10, 2019</h4></center> <!-- date -->
|
||||
<center><h4>Oct 12, 2019</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -159,7 +161,7 @@ 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-bs010.html">11</a></li>
|
||||
<li><a href="._DimRed-bs011.html">12</a></li>
|
||||
<li><a href="._DimRed-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p> <br>
|
||||
<center><h4>Oct 10, 2019</h4></center> <!-- date -->
|
||||
<center><h4>Oct 12, 2019</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -184,15 +184,15 @@ reduction techniques: the principal component analysis PCA, Kernel PCA, and Loca
|
||||
<b></b>
|
||||
<p>
|
||||
Before we proceed however, we will discuss how to preprocess our
|
||||
data. Till now and in connection with project 1 not met so many cases
|
||||
data. Till now and in connection with our previous examples we have not met so many cases
|
||||
where we are too sensitive to the scaling of our data. Normally the
|
||||
data may need a rescaling and/or may be sensitive to extreme
|
||||
values. Scaling the data renders our inputs much more suitable for the
|
||||
algorithms we want to emply.
|
||||
algorithms we want to employ.
|
||||
|
||||
<p>
|
||||
<b>Scikit-Learn</b> has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b> ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scale each column of the design matrix so that
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scales each column of the design matrix by its Euclidean norm.
|
||||
|
||||
|
||||
</div>
|
||||
@@ -200,7 +200,60 @@ This scaling has the drawback that it does not ensure that we have a particular
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec2">Principal Component Analysis </h2>
|
||||
<h2 id="___sec2">Simple preprocessing examples </h2>
|
||||
|
||||
<p>
|
||||
We show here how we can use a simple regression case (our nuclear binding energies discussed earlier).
|
||||
Rescaling our data with different
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> SVC
|
||||
cancer = load_breast_cancer()
|
||||
|
||||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=<span style="color: #B452CD">0</span>)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_train.shape)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_test.shape)
|
||||
|
||||
svm = SVC(C=<span style="color: #B452CD">100</span>)
|
||||
svm.fit(X_train, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy: {:.2f}"</span>.format(svm.score(X_test,y_test)))
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> MinMaxScaler, StandardScaler
|
||||
|
||||
scaler = MinMaxScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature min values before scaling:\n {}"</span>.format(X_train.min(axis=<span style="color: #B452CD">0</span>)))
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature max values before scaling:\n {}"</span>.format(X_train.max(axis=<span style="color: #B452CD">0</span>)))
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature min values before scaling:\n {}"</span>.format(X_train_scaled.min(axis=<span style="color: #B452CD">0</span>)))
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature max values before scaling:\n {}"</span>.format(X_train_scaled.max(axis=<span style="color: #B452CD">0</span>)))
|
||||
|
||||
|
||||
svm.fit(X_train_scaled, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy scaled data: {:.2f}"</span>.format(svm.score(X_test_scaled,y_test)))
|
||||
|
||||
scaler = StandardScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
|
||||
svm.fit(X_train_scaled, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy scaled data: {:.2f}"</span>.format(svm.score(X_test_scaled,y_test)))
|
||||
</pre></div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec3">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -237,7 +290,7 @@ X2D = X_centered.dot(W2)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec3">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec4">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -268,7 +321,7 @@ More material to come here.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec4">More on the PCA </h2>
|
||||
<h2 id="___sec5">More on the PCA </h2>
|
||||
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
|
||||
@@ -297,7 +350,7 @@ X_reduced = pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec5">Incremental PCA </h2>
|
||||
<h2 id="___sec6">Incremental PCA </h2>
|
||||
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
|
||||
@@ -307,7 +360,7 @@ instances arrive).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec6">Randomized PCA </h2>
|
||||
<h2 id="___sec7">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -321,7 +374,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec7">Kernel PCA </h2>
|
||||
<h2 id="___sec8">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -347,7 +400,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec8">LLE </h2>
|
||||
<h2 id="___sec9">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -359,7 +412,7 @@ these local relationships are best preserved (more details shortly).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec9">Other techniques </h2>
|
||||
<h2 id="___sec10">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -66,14 +66,15 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -115,7 +116,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 10, 2019</h4></center> <!-- date -->
|
||||
<center><h4>Oct 12, 2019</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -150,15 +151,15 @@ reduction techniques: the principal component analysis PCA, Kernel PCA, and Loca
|
||||
|
||||
<p>
|
||||
Before we proceed however, we will discuss how to preprocess our
|
||||
data. Till now and in connection with project 1 not met so many cases
|
||||
data. Till now and in connection with our previous examples we have not met so many cases
|
||||
where we are too sensitive to the scaling of our data. Normally the
|
||||
data may need a rescaling and/or may be sensitive to extreme
|
||||
values. Scaling the data renders our inputs much more suitable for the
|
||||
algorithms we want to emply.
|
||||
algorithms we want to employ.
|
||||
|
||||
<p>
|
||||
<b>Scikit-Learn</b> has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b> ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scale each column of the design matrix so that
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scales each column of the design matrix by its Euclidean norm.
|
||||
|
||||
|
||||
</div>
|
||||
@@ -167,7 +168,59 @@ This scaling has the drawback that it does not ensure that we have a particular
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec2">Principal Component Analysis </h2>
|
||||
<h2 id="___sec2">Simple preprocessing examples </h2>
|
||||
|
||||
<p>
|
||||
We show here how we can use a simple regression case (our nuclear binding energies discussed earlier).
|
||||
Rescaling our data with different
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> SVC
|
||||
cancer = load_breast_cancer()
|
||||
|
||||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=<span style="color: #B452CD">0</span>)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_train.shape)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(X_test.shape)
|
||||
|
||||
svm = SVC(C=<span style="color: #B452CD">100</span>)
|
||||
svm.fit(X_train, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy: {:.2f}"</span>.format(svm.score(X_test,y_test)))
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> MinMaxScaler, StandardScaler
|
||||
|
||||
scaler = MinMaxScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature min values before scaling:\n {}"</span>.format(X_train.min(axis=<span style="color: #B452CD">0</span>)))
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature max values before scaling:\n {}"</span>.format(X_train.max(axis=<span style="color: #B452CD">0</span>)))
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature min values before scaling:\n {}"</span>.format(X_train_scaled.min(axis=<span style="color: #B452CD">0</span>)))
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Feature max values before scaling:\n {}"</span>.format(X_train_scaled.max(axis=<span style="color: #B452CD">0</span>)))
|
||||
|
||||
|
||||
svm.fit(X_train_scaled, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy scaled data: {:.2f}"</span>.format(svm.score(X_test_scaled,y_test)))
|
||||
|
||||
scaler = StandardScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
|
||||
svm.fit(X_train_scaled, y_train)
|
||||
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">"Test set accuracy scaled data: {:.2f}"</span>.format(svm.score(X_test_scaled,y_test)))
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec3">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -203,7 +256,7 @@ X2D = X_centered.dot(W2)
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec3">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec4">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -234,7 +287,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec4">More on the PCA </h2>
|
||||
<h2 id="___sec5">More on the PCA </h2>
|
||||
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
|
||||
@@ -262,7 +315,7 @@ X_reduced = pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec5">Incremental PCA </h2>
|
||||
<h2 id="___sec6">Incremental PCA </h2>
|
||||
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
|
||||
@@ -272,7 +325,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec6">Randomized PCA </h2>
|
||||
<h2 id="___sec7">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -287,7 +340,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec7">Kernel PCA </h2>
|
||||
<h2 id="___sec8">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -316,7 +369,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec8">LLE </h2>
|
||||
<h2 id="___sec9">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -328,7 +381,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec9">Other techniques </h2>
|
||||
<h2 id="___sec10">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -71,14 +71,15 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
None,
|
||||
'___sec0'),
|
||||
('Preprocessing our data', 2, None, '___sec1'),
|
||||
('Principal Component Analysis', 2, None, '___sec2'),
|
||||
('PCA and scikit-learn', 2, None, '___sec3'),
|
||||
('More on the PCA', 2, None, '___sec4'),
|
||||
('Incremental PCA', 2, None, '___sec5'),
|
||||
('Randomized PCA', 2, None, '___sec6'),
|
||||
('Kernel PCA', 2, None, '___sec7'),
|
||||
('LLE', 2, None, '___sec8'),
|
||||
('Other techniques', 2, None, '___sec9')]}
|
||||
('Simple preprocessing examples', 2, None, '___sec2'),
|
||||
('Principal Component Analysis', 2, None, '___sec3'),
|
||||
('PCA and scikit-learn', 2, None, '___sec4'),
|
||||
('More on the PCA', 2, None, '___sec5'),
|
||||
('Incremental PCA', 2, None, '___sec6'),
|
||||
('Randomized PCA', 2, None, '___sec7'),
|
||||
('Kernel PCA', 2, None, '___sec8'),
|
||||
('LLE', 2, None, '___sec9'),
|
||||
('Other techniques', 2, None, '___sec10')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -120,7 +121,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 10, 2019</h4></center> <!-- date -->
|
||||
<center><h4>Oct 12, 2019</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -155,15 +156,15 @@ reduction techniques: the principal component analysis PCA, Kernel PCA, and Loca
|
||||
|
||||
<p>
|
||||
Before we proceed however, we will discuss how to preprocess our
|
||||
data. Till now and in connection with project 1 not met so many cases
|
||||
data. Till now and in connection with our previous examples we have not met so many cases
|
||||
where we are too sensitive to the scaling of our data. Normally the
|
||||
data may need a rescaling and/or may be sensitive to extreme
|
||||
values. Scaling the data renders our inputs much more suitable for the
|
||||
algorithms we want to emply.
|
||||
algorithms we want to employ.
|
||||
|
||||
<p>
|
||||
<b>Scikit-Learn</b> has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b> ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scale each column of the design matrix so that
|
||||
This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which ensures that all features are exactly between \( 0 \) and \( 1 \). The <b>Normalizer</b> function scales each column of the design matrix by its Euclidean norm.
|
||||
|
||||
|
||||
</div>
|
||||
@@ -172,7 +173,59 @@ This scaling has the drawback that it does not ensure that we have a particular
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec2">Principal Component Analysis </h2>
|
||||
<h2 id="___sec2">Simple preprocessing examples </h2>
|
||||
|
||||
<p>
|
||||
We show here how we can use a simple regression case (our nuclear binding energies discussed earlier).
|
||||
Rescaling our data with different
|
||||
|
||||
<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">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
cancer <span style="color: #666666">=</span> load_breast_cancer()
|
||||
|
||||
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
|
||||
|
||||
svm <span style="color: #666666">=</span> SVC(C<span style="color: #666666">=100</span>)
|
||||
svm<span style="color: #666666">.</span>fit(X_train, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test,y_test)))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> MinMaxScaler, StandardScaler
|
||||
|
||||
scaler <span style="color: #666666">=</span> MinMaxScaler()
|
||||
scaler<span style="color: #666666">.</span>fit(X_train)
|
||||
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
||||
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}"</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
|
||||
|
||||
|
||||
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy scaled data: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
||||
|
||||
scaler <span style="color: #666666">=</span> StandardScaler()
|
||||
scaler<span style="color: #666666">.</span>fit(X_train)
|
||||
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
||||
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
||||
|
||||
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
||||
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Test set accuracy scaled data: {:.2f}"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec3">Principal Component Analysis </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -208,7 +261,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec3">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec4">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -239,7 +292,7 @@ More material to come here.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec4">More on the PCA </h2>
|
||||
<h2 id="___sec5">More on the PCA </h2>
|
||||
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
|
||||
@@ -267,7 +320,7 @@ 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="___sec5">Incremental PCA </h2>
|
||||
<h2 id="___sec6">Incremental PCA </h2>
|
||||
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
|
||||
@@ -277,7 +330,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec6">Randomized PCA </h2>
|
||||
<h2 id="___sec7">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -292,7 +345,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec7">Kernel PCA </h2>
|
||||
<h2 id="___sec8">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -321,7 +374,7 @@ 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="___sec8">LLE </h2>
|
||||
<h2 id="___sec9">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -333,7 +386,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec9">Other techniques </h2>
|
||||
<h2 id="___sec10">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -10,7 +10,7 @@
|
||||
"<!-- Author: --> \n",
|
||||
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
|
||||
"\n",
|
||||
"Date: **Oct 10, 2019**\n",
|
||||
"Date: **Oct 12, 2019**\n",
|
||||
"\n",
|
||||
"Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
|
||||
"\n",
|
||||
@@ -34,19 +34,79 @@
|
||||
"## Preprocessing our data\n",
|
||||
"\n",
|
||||
"Before we proceed however, we will discuss how to preprocess our\n",
|
||||
"data. Till now and in connection with project 1 not met so many cases\n",
|
||||
"data. Till now and in connection with our previous examples we have not met so many cases\n",
|
||||
"where we are too sensitive to the scaling of our data. Normally the\n",
|
||||
"data may need a rescaling and/or may be sensitive to extreme\n",
|
||||
"values. Scaling the data renders our inputs much more suitable for the\n",
|
||||
"algorithms we want to emply.\n",
|
||||
"algorithms we want to employ.\n",
|
||||
"\n",
|
||||
"**Scikit-Learn** has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The **StandardScaler** function in **Scikit-Learn** ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).\n",
|
||||
"This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in **Scikit-Learn** is the **MinMaxScaler** which ensures that all features are exactly between $0$ and $1$. The **Normalizer** function scale each column of the design matrix so that\n",
|
||||
"This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in **Scikit-Learn** is the **MinMaxScaler** which ensures that all features are exactly between $0$ and $1$. The **Normalizer** function scales each column of the design matrix by its Euclidean norm.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Simple preprocessing examples\n",
|
||||
"\n",
|
||||
"We show here how we can use a simple regression case (our nuclear binding energies discussed earlier).\n",
|
||||
"Rescaling our data with different"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"%matplotlib inline\n",
|
||||
"\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"import numpy as np\n",
|
||||
"from sklearn.model_selection import train_test_split \n",
|
||||
"from sklearn.datasets import load_breast_cancer\n",
|
||||
"from sklearn.svm import SVC\n",
|
||||
"cancer = load_breast_cancer()\n",
|
||||
"\n",
|
||||
"X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n",
|
||||
"print(X_train.shape)\n",
|
||||
"print(X_test.shape)\n",
|
||||
"\n",
|
||||
"svm = SVC(C=100)\n",
|
||||
"svm.fit(X_train, y_train)\n",
|
||||
"print(\"Test set accuracy: {:.2f}\".format(svm.score(X_test,y_test)))\n",
|
||||
"\n",
|
||||
"from sklearn.preprocessing import MinMaxScaler, StandardScaler\n",
|
||||
"\n",
|
||||
"scaler = MinMaxScaler()\n",
|
||||
"scaler.fit(X_train)\n",
|
||||
"X_train_scaled = scaler.transform(X_train)\n",
|
||||
"X_test_scaled = scaler.transform(X_test)\n",
|
||||
"\n",
|
||||
"print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n",
|
||||
"print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n",
|
||||
"\n",
|
||||
"print(\"Feature min values before scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n",
|
||||
"print(\"Feature max values before scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"svm.fit(X_train_scaled, y_train)\n",
|
||||
"print(\"Test set accuracy scaled data: {:.2f}\".format(svm.score(X_test_scaled,y_test)))\n",
|
||||
"\n",
|
||||
"scaler = StandardScaler()\n",
|
||||
"scaler.fit(X_train)\n",
|
||||
"X_train_scaled = scaler.transform(X_train)\n",
|
||||
"X_test_scaled = scaler.transform(X_test)\n",
|
||||
"\n",
|
||||
"svm.fit(X_train_scaled, y_train)\n",
|
||||
"print(\"Test set accuracy scaled data: {:.2f}\".format(svm.score(X_test_scaled,y_test)))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Principal Component Analysis\n",
|
||||
"Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n",
|
||||
"First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n",
|
||||
@@ -57,7 +117,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -84,7 +144,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -108,7 +168,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -130,7 +190,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -159,7 +219,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -182,7 +242,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -228,7 +288,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"execution_count": 8,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
|
||||
Binary file not shown.
Binary file not shown.
Reference in New Issue
Block a user