test
This commit is contained in:
mhjensen
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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<body>
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
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||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -346,7 +347,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week43-bs008.html">9</a></li>
|
||||
<li><a href="._week43-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
||||
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|
||||
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|
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||||
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||||
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||||
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<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -308,7 +309,7 @@ MathJax.Hub.Config({
|
||||
<li> Friday: Principal Component Analysis and Dimensionality Reduction</li>
|
||||
</ul>
|
||||
|
||||
Reading suggestions for both days: "Aurelien Geron's chapters 8
|
||||
We will also study the usage of <a href="https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola" target="_self">Autograd</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html" target="_self">week 40</a> and the <a href="https://github.com/HIPS/autograd" target="_self">Autograd doucmentation</a>.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -327,7 +328,7 @@ Reading suggestions for both days: "Aurelien Geron's chapters 8
|
||||
<li><a href="._week43-bs009.html">10</a></li>
|
||||
<li><a href="._week43-bs010.html">11</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs002.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
|
||||
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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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|
||||
@@ -80,100 +80,100 @@ Automatically generated HTML file from DocOnce source
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
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||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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||||
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|
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|
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|
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|
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|
||||
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|
||||
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|
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|
||||
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|
||||
'___sec50'),
|
||||
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|
||||
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|
||||
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|
||||
'___sec50'),
|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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|
||||
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|
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|
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<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -305,6 +306,10 @@ MathJax.Hub.Config({
|
||||
|
||||
<h2 id="___sec0" class="anchor">Recurrent Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini" target="_self">Overview video</a>.
|
||||
See also lecture on Thursday October 22 and examples from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html" target="_self">week 42</a>.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -323,7 +328,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week43-bs010.html">11</a></li>
|
||||
<li><a href="._week43-bs011.html">12</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs003.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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<body>
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -329,7 +330,7 @@ and output layer to any given precision.
|
||||
<li><a href="._week43-bs011.html">12</a></li>
|
||||
<li><a href="._week43-bs012.html">13</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
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<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs004.html">»</a></li>
|
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</ul>
|
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@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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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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|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +348,7 @@ for the solution to be unique.
|
||||
<li><a href="._week43-bs012.html">13</a></li>
|
||||
<li><a href="._week43-bs013.html">14</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
('The network with one input layer, specified number of hidden '
|
||||
@@ -80,100 +80,100 @@ Automatically generated HTML file from DocOnce source
|
||||
('The trial solution', 2, None, '___sec25'),
|
||||
('The program using Autograd', 2, None, '___sec26'),
|
||||
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|
||||
('Example: Solving the one dimensional Poisson equation using '
|
||||
'Autograd and TensorFlow',
|
||||
('Example: Solving the one dimensional Poisson equation',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
('The specific equation to solve for', 2, None, '___sec29'),
|
||||
('Solving the equation using Autograd', 2, None, '___sec30'),
|
||||
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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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|
||||
('Setting up the network using Autograd', 2, None, '___sec38'),
|
||||
('Setting up the code', 2, None, '___sec32'),
|
||||
('Partial Differential Equations', 2, None, '___sec33'),
|
||||
('Type of problem', 2, None, '___sec34'),
|
||||
('Network requirements', 2, None, '___sec35'),
|
||||
('More details', 2, None, '___sec36'),
|
||||
('Example: The diffusion equation', 2, None, '___sec37'),
|
||||
('Defining the problem', 2, None, '___sec38'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec39'),
|
||||
('Setting up the network using Autograd; The trial solution',
|
||||
2,
|
||||
None,
|
||||
'___sec39'),
|
||||
('Why the jacobian?', 2, None, '___sec40'),
|
||||
'___sec40'),
|
||||
('Why the jacobian?', 2, None, '___sec41'),
|
||||
('Setting up the network using Autograd; The full program',
|
||||
2,
|
||||
None,
|
||||
'___sec41'),
|
||||
'___sec42'),
|
||||
('Example: Solving the wave equation with Neural Networks',
|
||||
2,
|
||||
None,
|
||||
'___sec42'),
|
||||
('The problem to solve for', 2, None, '___sec43'),
|
||||
('The trial solution', 2, None, '___sec44'),
|
||||
('The analytical solution', 2, None, '___sec45'),
|
||||
'___sec43'),
|
||||
('The problem to solve for', 2, None, '___sec44'),
|
||||
('The trial solution', 2, None, '___sec45'),
|
||||
('The analytical solution', 2, None, '___sec46'),
|
||||
('Solving the wave equation - the full program using Autograd',
|
||||
2,
|
||||
None,
|
||||
'___sec46'),
|
||||
'___sec47'),
|
||||
('Resources on differential equations and deep learning',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec48'),
|
||||
'___sec48'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec49'),
|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -352,7 +353,7 @@ As described previously, an optimization method could be used to minimize the pa
|
||||
<li><a href="._week43-bs013.html">14</a></li>
|
||||
<li><a href="._week43-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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|
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -354,7 +355,7 @@ The neural net should then find the parameters \( P \) that minimizes the cost f
|
||||
<li><a href="._week43-bs014.html">15</a></li>
|
||||
<li><a href="._week43-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -336,7 +337,7 @@ Automatic differentiation is a method of finding the derivatives numerically wit
|
||||
<li><a href="._week43-bs015.html">16</a></li>
|
||||
<li><a href="._week43-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -354,7 +355,7 @@ Having an analytical solution at hand, it is possible to use it to compare how w
|
||||
<li><a href="._week43-bs016.html">17</a></li>
|
||||
<li><a href="._week43-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -345,7 +346,7 @@ In this example, \( \gamma = 2 \) and \( g_0 = 10 \).
|
||||
<li><a href="._week43-bs017.html">18</a></li>
|
||||
<li><a href="._week43-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
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<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs010.html">»</a></li>
|
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</ul>
|
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<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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|
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|
||||
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|
||||
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||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -339,7 +340,7 @@ with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and \( h_
|
||||
<li><a href="._week43-bs018.html">19</a></li>
|
||||
<li><a href="._week43-bs019.html">20</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs011.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
('The network with one input layer, specified number of hidden '
|
||||
@@ -80,100 +80,100 @@ Automatically generated HTML file from DocOnce source
|
||||
('The trial solution', 2, None, '___sec25'),
|
||||
('The program using Autograd', 2, None, '___sec26'),
|
||||
('Using forward Euler to solve the ODE', 2, None, '___sec27'),
|
||||
('Example: Solving the one dimensional Poisson equation using '
|
||||
'Autograd and TensorFlow',
|
||||
('Example: Solving the one dimensional Poisson equation',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
('The specific equation to solve for', 2, None, '___sec29'),
|
||||
('Solving the equation using Autograd', 2, None, '___sec30'),
|
||||
('Comparing with a numerical scheme', 2, None, '___sec31'),
|
||||
('Partial Differential Equations', 2, None, '___sec32'),
|
||||
('Type of problem', 2, None, '___sec33'),
|
||||
('Network requirements', 2, None, '___sec34'),
|
||||
('More details', 2, None, '___sec35'),
|
||||
('Example: The diffusion equation', 2, None, '___sec36'),
|
||||
('Defining the problem', 2, None, '___sec37'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec38'),
|
||||
('Setting up the code', 2, None, '___sec32'),
|
||||
('Partial Differential Equations', 2, None, '___sec33'),
|
||||
('Type of problem', 2, None, '___sec34'),
|
||||
('Network requirements', 2, None, '___sec35'),
|
||||
('More details', 2, None, '___sec36'),
|
||||
('Example: The diffusion equation', 2, None, '___sec37'),
|
||||
('Defining the problem', 2, None, '___sec38'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec39'),
|
||||
('Setting up the network using Autograd; The trial solution',
|
||||
2,
|
||||
None,
|
||||
'___sec39'),
|
||||
('Why the jacobian?', 2, None, '___sec40'),
|
||||
'___sec40'),
|
||||
('Why the jacobian?', 2, None, '___sec41'),
|
||||
('Setting up the network using Autograd; The full program',
|
||||
2,
|
||||
None,
|
||||
'___sec41'),
|
||||
'___sec42'),
|
||||
('Example: Solving the wave equation with Neural Networks',
|
||||
2,
|
||||
None,
|
||||
'___sec42'),
|
||||
('The problem to solve for', 2, None, '___sec43'),
|
||||
('The trial solution', 2, None, '___sec44'),
|
||||
('The analytical solution', 2, None, '___sec45'),
|
||||
'___sec43'),
|
||||
('The problem to solve for', 2, None, '___sec44'),
|
||||
('The trial solution', 2, None, '___sec45'),
|
||||
('The analytical solution', 2, None, '___sec46'),
|
||||
('Solving the wave equation - the full program using Autograd',
|
||||
2,
|
||||
None,
|
||||
'___sec46'),
|
||||
'___sec47'),
|
||||
('Resources on differential equations and deep learning',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec48'),
|
||||
'___sec48'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec49'),
|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -351,7 +352,7 @@ $$
|
||||
<li><a href="._week43-bs019.html">20</a></li>
|
||||
<li><a href="._week43-bs020.html">21</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs012.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -360,7 +361,7 @@ is fulfilled as <em>best as possible</em>.
|
||||
<li><a href="._week43-bs020.html">21</a></li>
|
||||
<li><a href="._week43-bs021.html">22</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs013.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -356,7 +357,7 @@ for an input value \( x \).
|
||||
<li><a href="._week43-bs021.html">22</a></li>
|
||||
<li><a href="._week43-bs022.html">23</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs014.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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<body>
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||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -354,7 +355,7 @@ $$
|
||||
<li><a href="._week43-bs022.html">23</a></li>
|
||||
<li><a href="._week43-bs023.html">24</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
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<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs015.html">»</a></li>
|
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</ul>
|
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<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -339,7 +340,7 @@ The input layer will consist of \( N_{\text{input} } \) neurons, passing its ele
|
||||
<li><a href="._week43-bs023.html">24</a></li>
|
||||
<li><a href="._week43-bs024.html">25</a></li>
|
||||
<li><a href="">...</a></li>
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||||
<li><a href="._week43-bs075.html">76</a></li>
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<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
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||||
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -348,7 +349,7 @@ $$
|
||||
<li><a href="._week43-bs024.html">25</a></li>
|
||||
<li><a href="._week43-bs025.html">26</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs017.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -349,7 +350,7 @@ $$
|
||||
<li><a href="._week43-bs025.html">26</a></li>
|
||||
<li><a href="._week43-bs026.html">27</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs018.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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|
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||||
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||||
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||||
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<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -322,7 +323,7 @@ $$
|
||||
It is possible to use other activations functions for the hidden layer also.
|
||||
|
||||
<p>
|
||||
The output $\boldsymbol{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is:
|
||||
The output \( \boldsymbol{x}_i^{\text{hidden}} \) from each \( i \)-th hidden neuron is:
|
||||
|
||||
$$
|
||||
\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big)
|
||||
@@ -332,7 +333,11 @@ $$
|
||||
The outputs \( \boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.
|
||||
|
||||
<p>
|
||||
The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\text{output}}$ and biases \( b_i^{\text{output}} \). In this case, it is assumes that the number of neurons in the output layer is one.
|
||||
The output layer consists of one neuron in this case, and combines the
|
||||
output from each of the neurons in the hidden layers. The output layer
|
||||
combines the results from the hidden layer using some weights \( w_i^{\text{output}} \)
|
||||
and biases \( b_i^{\text{output}} \). In this case,
|
||||
it is assumes that the number of neurons in the output layer is one.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -360,7 +365,7 @@ The output layer consists of one neuron in this case, and combines the output fr
|
||||
<li><a href="._week43-bs026.html">27</a></li>
|
||||
<li><a href="._week43-bs027.html">28</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
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|
||||
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|
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|
||||
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|
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|
||||
'___sec48'),
|
||||
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|
||||
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|
||||
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|
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||||
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|
||||
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|
||||
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|
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||||
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|
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||||
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||||
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||||
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|
||||
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||||
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||||
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|
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
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|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +348,7 @@ $$
|
||||
<li><a href="._week43-bs027.html">28</a></li>
|
||||
<li><a href="._week43-bs028.html">29</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs020.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -348,7 +349,7 @@ In this case we seek a continuous range of values since we are approximating a f
|
||||
<li><a href="._week43-bs028.html">29</a></li>
|
||||
<li><a href="._week43-bs029.html">30</a></li>
|
||||
<li><a href="">...</a></li>
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||||
<li><a href="._week43-bs075.html">76</a></li>
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<li><a href="._week43-bs076.html">77</a></li>
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||||
<li><a href="._week43-bs021.html">»</a></li>
|
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</ul>
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<!-- ------------------- end of main content --------------- -->
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||||
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||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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||||
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|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -303,7 +304,7 @@ MathJax.Hub.Config({
|
||||
<a name="part0021"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec19" class="anchor">Backpropagation </h2>
|
||||
<h2 id="___sec19" class="anchor">Back propagation </h2>
|
||||
|
||||
<p>
|
||||
The next step is to decide how the parameters should be changed such that they minimize the cost function.
|
||||
@@ -347,7 +348,7 @@ Here, gradient descent with a constant step size has been chosen.
|
||||
<li><a href="._week43-bs029.html">30</a></li>
|
||||
<li><a href="._week43-bs030.html">31</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
('The network with one input layer, specified number of hidden '
|
||||
@@ -80,100 +80,100 @@ Automatically generated HTML file from DocOnce source
|
||||
('The trial solution', 2, None, '___sec25'),
|
||||
('The program using Autograd', 2, None, '___sec26'),
|
||||
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|
||||
('Example: Solving the one dimensional Poisson equation using '
|
||||
'Autograd and TensorFlow',
|
||||
('Example: Solving the one dimensional Poisson equation',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
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|
||||
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|
||||
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|
||||
('Partial Differential Equations', 2, None, '___sec32'),
|
||||
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|
||||
('Network requirements', 2, None, '___sec34'),
|
||||
('More details', 2, None, '___sec35'),
|
||||
('Example: The diffusion equation', 2, None, '___sec36'),
|
||||
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|
||||
('Setting up the network using Autograd', 2, None, '___sec38'),
|
||||
('Setting up the code', 2, None, '___sec32'),
|
||||
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|
||||
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|
||||
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|
||||
('More details', 2, None, '___sec36'),
|
||||
('Example: The diffusion equation', 2, None, '___sec37'),
|
||||
('Defining the problem', 2, None, '___sec38'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec39'),
|
||||
('Setting up the network using Autograd; The trial solution',
|
||||
2,
|
||||
None,
|
||||
'___sec39'),
|
||||
('Why the jacobian?', 2, None, '___sec40'),
|
||||
'___sec40'),
|
||||
('Why the jacobian?', 2, None, '___sec41'),
|
||||
('Setting up the network using Autograd; The full program',
|
||||
2,
|
||||
None,
|
||||
'___sec41'),
|
||||
'___sec42'),
|
||||
('Example: Solving the wave equation with Neural Networks',
|
||||
2,
|
||||
None,
|
||||
'___sec42'),
|
||||
('The problem to solve for', 2, None, '___sec43'),
|
||||
('The trial solution', 2, None, '___sec44'),
|
||||
('The analytical solution', 2, None, '___sec45'),
|
||||
'___sec43'),
|
||||
('The problem to solve for', 2, None, '___sec44'),
|
||||
('The trial solution', 2, None, '___sec45'),
|
||||
('The analytical solution', 2, None, '___sec46'),
|
||||
('Solving the wave equation - the full program using Autograd',
|
||||
2,
|
||||
None,
|
||||
'___sec46'),
|
||||
'___sec47'),
|
||||
('Resources on differential equations and deep learning',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec48'),
|
||||
'___sec48'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec49'),
|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -325,7 +326,7 @@ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new}
|
||||
The value of \( \lambda \) decides how large steps the algorithm must take
|
||||
in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$.
|
||||
The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect
|
||||
to the elements in \( \boldsymbol{\omega \).
|
||||
to the elements in \( \boldsymbol{\omega} \).
|
||||
|
||||
<p>
|
||||
In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with
|
||||
@@ -369,7 +370,7 @@ $$
|
||||
<li><a href="._week43-bs030.html">31</a></li>
|
||||
<li><a href="._week43-bs031.html">32</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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|
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|
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|
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|
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|
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|
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|
||||
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|
||||
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|
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|
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|
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||||
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||||
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||||
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||||
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|
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|
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|
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|
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|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -479,7 +480,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week43-bs031.html">32</a></li>
|
||||
<li><a href="._week43-bs032.html">33</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs024.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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|
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<body>
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||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -499,7 +500,7 @@ The number of neurons within each hidden layer are given as a list of integers i
|
||||
<li><a href="._week43-bs032.html">33</a></li>
|
||||
<li><a href="._week43-bs033.html">34</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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<body>
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@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -322,8 +323,8 @@ Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0
|
||||
<p>
|
||||
In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability
|
||||
and high execution time (this might be more apparent in the examples solving PDEs),
|
||||
a network has been constructed using TensorFlow also.
|
||||
For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.
|
||||
using a library like TensorFlow is recommended.
|
||||
Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -351,7 +352,7 @@ For comparison, the forward Euler method has been implemented in order to see ho
|
||||
<li><a href="._week43-bs033.html">34</a></li>
|
||||
<li><a href="._week43-bs034.html">35</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs026.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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||||
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|
||||
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|
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|
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|
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|
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|
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|
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|
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|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +348,7 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
|
||||
<li><a href="._week43-bs034.html">35</a></li>
|
||||
<li><a href="._week43-bs035.html">36</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -67,7 +67,7 @@ Automatically generated HTML file from DocOnce source
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||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -230,7 +230,7 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec16" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec17" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec18" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Backpropagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec19" style="font-size: 80%;"><b>Back propagation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec20" style="font-size: 80%;"><b>Gradient descent</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec21" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec22" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
|
||||
@@ -239,55 +239,56 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec25" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec26" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec27" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec28" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec29" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec30" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec31" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec61" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec62" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec32" style="font-size: 80%;"><b>Setting up the code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec33" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec34" style="font-size: 80%;"><b>Type of problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec35" style="font-size: 80%;"><b>Network requirements</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec36" style="font-size: 80%;"><b>More details</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec37" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec38" style="font-size: 80%;"><b>Defining the problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec39" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec41" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec42" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec43" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec44" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec45" style="font-size: 80%;"><b>The trial solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec46" style="font-size: 80%;"><b>The analytical solution</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs049.html#___sec47" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec48" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec49" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec50" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec51" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec52" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec53" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec54" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec57" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec58" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec59" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec60" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec64" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec65" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec66" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec67" style="font-size: 80%;"><b>The final step</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec68" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec69" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec70" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec71" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec72" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec73" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec74" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec75" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec76" style="font-size: 80%;"><b>LLE</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec77" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -346,7 +347,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week43-bs008.html">9</a></li>
|
||||
<li><a href="._week43-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week43-bs075.html">76</a></li>
|
||||
<li><a href="._week43-bs076.html">77</a></li>
|
||||
<li><a href="._week43-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -166,12 +166,16 @@ MathJax.Hub.Config({
|
||||
</ul>
|
||||
<p>
|
||||
|
||||
Reading suggestions for both days: "Aurelien Geron's chapters 8
|
||||
We will also study the usage of <a href="https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola" target="_blank">Autograd</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html" target="_blank">week 40</a> and the <a href="https://github.com/HIPS/autograd" target="_blank">Autograd doucmentation</a>.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec0">Recurrent Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini" target="_blank">Overview video</a>.
|
||||
See also lecture on Thursday October 22 and examples from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html" target="_blank">week 42</a>.
|
||||
</section>
|
||||
|
||||
|
||||
@@ -584,7 +588,7 @@ $$
|
||||
It is possible to use other activations functions for the hidden layer also.
|
||||
|
||||
<p>
|
||||
The output $\boldsymbol{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is:
|
||||
The output \( \boldsymbol{x}_i^{\text{hidden}} \) from each \( i \)-th hidden neuron is:
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
@@ -596,7 +600,11 @@ $$
|
||||
The outputs \( \boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.
|
||||
|
||||
<p>
|
||||
The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\text{output}}$ and biases \( b_i^{\text{output}} \). In this case, it is assumes that the number of neurons in the output layer is one.
|
||||
The output layer consists of one neuron in this case, and combines the
|
||||
output from each of the neurons in the hidden layers. The output layer
|
||||
combines the results from the hidden layer using some weights \( w_i^{\text{output}} \)
|
||||
and biases \( b_i^{\text{output}} \). In this case,
|
||||
it is assumes that the number of neurons in the output layer is one.
|
||||
</section>
|
||||
|
||||
|
||||
@@ -648,7 +656,7 @@ In this case we seek a continuous range of values since we are approximating a f
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec19">Backpropagation </h2>
|
||||
<h2 id="___sec19">Back propagation </h2>
|
||||
|
||||
<p>
|
||||
The next step is to decide how the parameters should be changed such that they minimize the cost function.
|
||||
@@ -695,7 +703,7 @@ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new}
|
||||
The value of \( \lambda \) decides how large steps the algorithm must take
|
||||
in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$.
|
||||
The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect
|
||||
to the elements in \( \boldsymbol{\omega \).
|
||||
to the elements in \( \boldsymbol{\omega} \).
|
||||
|
||||
<p>
|
||||
In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with
|
||||
@@ -1067,8 +1075,8 @@ Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0
|
||||
<p>
|
||||
In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability
|
||||
and high execution time (this might be more apparent in the examples solving PDEs),
|
||||
a network has been constructed using TensorFlow also.
|
||||
For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.
|
||||
using a library like TensorFlow is recommended.
|
||||
Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.
|
||||
</section>
|
||||
|
||||
|
||||
@@ -1305,7 +1313,7 @@ The network will be the similar as for the exponential decay example, but with s
|
||||
<h2 id="___sec27">Using forward Euler to solve the ODE </h2>
|
||||
|
||||
<p>
|
||||
A straight-forward way of solving an ODE numerically, is to use Euler's method.
|
||||
A straightforward way of solving an ODE numerically, is to use Euler's method.
|
||||
|
||||
<p>
|
||||
Euler's method uses Taylor series to approximate the value at a function \( f \) at a step \( \Delta x \) from \( x \):
|
||||
@@ -1436,13 +1444,11 @@ extending the program that uses the network using Autograd:
|
||||
|
||||
plt.show()
|
||||
</pre></div>
|
||||
<p>
|
||||
Running the program gives
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow </h2>
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation </h2>
|
||||
|
||||
<p>
|
||||
The Poisson equation for \( g(x) \) in one dimension is
|
||||
@@ -1782,13 +1788,18 @@ f(x_2) \\
|
||||
f(x_{N_x - 3}) \\
|
||||
f(x_{N_x - 2})
|
||||
\end{pmatrix} \\
|
||||
A\boldsymbol{g} &= \boldsymbol{f}
|
||||
\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f},
|
||||
\end{aligned}
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p>
|
||||
which makes it possible to solve for the vector \( \boldsymbol{g} \).
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec32">Setting up the code </h2>
|
||||
|
||||
<p>
|
||||
We can then compare the result from this numerical scheme with the output from our network using Autograd:
|
||||
@@ -1987,23 +1998,17 @@ We can then compare the result from this numerical scheme with the output from o
|
||||
plt.legend([<span style="color: #CD5555">'numerical scheme'</span>,<span style="color: #CD5555">'analytical'</span>,<span style="color: #CD5555">'dnn'</span>])
|
||||
plt.show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The program prints out:
|
||||
<p>
|
||||
|
||||
<!-- code=text typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>The max absolute difference between the analytical solution and DNN Autograd: 0.000464088
|
||||
The max absolute difference between the analytical solution and numerical scheme: 0.00266858
|
||||
</pre></div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec32">Partial Differential Equations </h2>
|
||||
<h2 id="___sec33">Partial Differential Equations </h2>
|
||||
|
||||
<p>
|
||||
A partial differential equation (PDE) has a solution here the function is defined by multiple variables.
|
||||
The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.
|
||||
A partial differential equation (PDE) has a solution here the function
|
||||
is defined by multiple variables. The equation may involve all kinds
|
||||
of combinations of which variables the function is differentiated with
|
||||
respect to.
|
||||
|
||||
<p>
|
||||
In general, a partial differential equation for a function \( g(x_1,\dots,x_N) \) with \( N \) variables may be expressed as
|
||||
@@ -2022,7 +2027,7 @@ where \( f \) is an expression involving all kinds of possible mixed derivatives
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec33">Type of problem </h2>
|
||||
<h2 id="___sec34">Type of problem </h2>
|
||||
|
||||
<p>
|
||||
The problem our network must solve for, is similar to the ODE case.
|
||||
@@ -2047,7 +2052,7 @@ The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensu
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec34">Network requirements </h2>
|
||||
<h2 id="___sec35">Network requirements </h2>
|
||||
|
||||
<p>
|
||||
The network tries then the minimize the cost function following the
|
||||
@@ -2064,7 +2069,7 @@ minimize is
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
@@ -2072,15 +2077,13 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec35">More details </h2>
|
||||
<h2 id="___sec36">More details </h2>
|
||||
|
||||
<p>
|
||||
If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
If we let \( \boldsymbol{x} = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x) }{\partial x_N}, \frac{\partial g(\boldsymbol{x) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
@@ -2089,7 +2092,7 @@ If we also have \( M \) different sets of values for \( x_1, \dots, x_N \), that
|
||||
<p> <br>
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2.
|
||||
\end{equation*}
|
||||
$$
|
||||
<p> <br>
|
||||
@@ -2097,7 +2100,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec36">Example: The diffusion equation </h2>
|
||||
<h2 id="___sec37">Example: The diffusion equation </h2>
|
||||
|
||||
<p>
|
||||
In one spatial dimension, the equation reads
|
||||
@@ -2126,7 +2129,7 @@ with \( u(x) \) being some given function.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec37">Defining the problem </h2>
|
||||
<h2 id="___sec38">Defining the problem </h2>
|
||||
|
||||
<p>
|
||||
For this case, we want to find \( g(x,t) \) such that
|
||||
@@ -2162,7 +2165,7 @@ First, we will look into how Autograd could be used in a network tailored to sol
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec38">Setting up the network using Autograd </h2>
|
||||
<h2 id="___sec39">Setting up the network using Autograd </h2>
|
||||
|
||||
<p>
|
||||
The only change to do here, is to extend our network such that
|
||||
@@ -2225,7 +2228,7 @@ network at each possible pair \( (x,t) \), given an array for the desired
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec39">Setting up the network using Autograd; The trial solution </h2>
|
||||
<h2 id="___sec40">Setting up the network using Autograd; The trial solution </h2>
|
||||
|
||||
<p>
|
||||
The cost function must then iterate through the given arrays
|
||||
@@ -2258,7 +2261,7 @@ since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec40">Why the jacobian? </h2>
|
||||
<h2 id="___sec41">Why the jacobian? </h2>
|
||||
|
||||
<p>
|
||||
The Jacobian is used because the program must find the derivative of
|
||||
@@ -2326,7 +2329,7 @@ mixed derivatives of \( g(x,t) \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec41">Setting up the network using Autograd; The full program </h2>
|
||||
<h2 id="___sec42">Setting up the network using Autograd; The full program </h2>
|
||||
|
||||
<p>
|
||||
Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.
|
||||
@@ -2580,7 +2583,7 @@ Using TensorFlow results in a much better execution time. Try it!
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec42">Example: Solving the wave equation with Neural Networks </h2>
|
||||
<h2 id="___sec43">Example: Solving the wave equation with Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
The wave equation is
|
||||
@@ -2613,7 +2616,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec43">The problem to solve for </h2>
|
||||
<h2 id="___sec44">The problem to solve for </h2>
|
||||
|
||||
<p>
|
||||
The wave equation to solve for, is
|
||||
@@ -2647,7 +2650,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec44">The trial solution </h2>
|
||||
<h2 id="___sec45">The trial solution </h2>
|
||||
Setting up the network is done in similar matter as for the example of solving the diffusion equation.
|
||||
The only things we have to change, is the trial solution such that it satisfies the conditions from <a href="#mjx-eqn-20">(20)</a> and the cost function.
|
||||
|
||||
@@ -2675,7 +2678,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec45">The analytical solution </h2>
|
||||
<h2 id="___sec46">The analytical solution </h2>
|
||||
|
||||
<p>
|
||||
The analytical solution for our specific problem, is
|
||||
@@ -2689,7 +2692,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec46">Solving the wave equation - the full program using Autograd </h2>
|
||||
<h2 id="___sec47">Solving the wave equation - the full program using Autograd </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -2919,7 +2922,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec47">Resources on differential equations and deep learning </h2>
|
||||
<h2 id="___sec48">Resources on differential equations and deep learning </h2>
|
||||
|
||||
<ol>
|
||||
<p><li> <a href="https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf" target="_blank">Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al</a></li>
|
||||
@@ -2931,12 +2934,15 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec48">Friday, Principal Component Analysis </h2>
|
||||
<h2 id="___sec49">Friday, Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=fkf4IBRSeEc&ab_channel=SteveBrunton" target="_blank">Overview video</a>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec49">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
<h2 id="___sec50">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
|
||||
<p>
|
||||
The principal component analysis deals with the problem of fitting a
|
||||
@@ -2959,7 +2965,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec50">Introducing the Covariance and Correlation functions </h2>
|
||||
<h2 id="___sec51">Introducing the Covariance and Correlation functions </h2>
|
||||
|
||||
<p>
|
||||
Before we discuss the PCA theorem, we need to remind ourselves about
|
||||
@@ -3032,7 +3038,7 @@ In the above example this is the function we constructed using <b>pandas</b>.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec51">Correlation Function and Design/Feature Matrix </h2>
|
||||
<h2 id="___sec52">Correlation Function and Design/Feature Matrix </h2>
|
||||
|
||||
<p>
|
||||
In our derivation of the various regression algorithms like <b>Ordinary Least Squares</b> or <b>Ridge regression</b>
|
||||
@@ -3103,7 +3109,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec52">Covariance Matrix Examples </h2>
|
||||
<h2 id="___sec53">Covariance Matrix Examples </h2>
|
||||
|
||||
<p>
|
||||
The Numpy function <b>np.cov</b> calculates the covariance elements using
|
||||
@@ -3149,7 +3155,7 @@ C = np.cov(W)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec53">Correlation Matrix </h2>
|
||||
<h2 id="___sec54">Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
The previous example can be converted into the correlation matrix by
|
||||
@@ -3194,7 +3200,7 @@ The above procedure with <b>numpy</b> can be made more compact if we use <b>pand
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec54">Correlation Matrix with Pandas </h2>
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas </h2>
|
||||
|
||||
<p>
|
||||
We whow here how we can set up the correlation matrix using <b>pandas</b>, as done in this simple code
|
||||
@@ -3221,7 +3227,7 @@ We expand this model to the Franke function discussed above.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
<h2 id="___sec56">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -3284,7 +3290,7 @@ matrix without these elements.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec56">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
<h2 id="___sec57">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
|
||||
@@ -3337,7 +3343,7 @@ It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\t
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec57">Towards the PCA theorem </h2>
|
||||
<h2 id="___sec58">Towards the PCA theorem </h2>
|
||||
|
||||
<p>
|
||||
We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
|
||||
@@ -3396,7 +3402,7 @@ features/predictors.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec58">The Algorithm before theorem </h2>
|
||||
<h2 id="___sec59">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
@@ -3429,7 +3435,7 @@ $$
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec59">Writing our own PCA code </h2>
|
||||
<h2 id="___sec60">Writing our own PCA code </h2>
|
||||
|
||||
<p>
|
||||
We will use a simple example first with two-dimensional data
|
||||
@@ -3464,7 +3470,7 @@ X = np.random.multivariate_normal(mean, cov, n)
|
||||
<p>
|
||||
Now we are going to implement the PCA algorithm. We will break it down into various substeps.
|
||||
|
||||
<h3 id="___sec60">Compute the sample mean and center the data </h3>
|
||||
<h3 id="___sec61">Compute the sample mean and center the data </h3>
|
||||
|
||||
<p>
|
||||
The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
|
||||
@@ -3505,7 +3511,7 @@ variance. The diagonal covariance matrix elements will then be one,
|
||||
while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
|
||||
specific case.
|
||||
|
||||
<h3 id="___sec61">Compute the sample covariance </h3>
|
||||
<h3 id="___sec62">Compute the sample covariance </h3>
|
||||
|
||||
<p>
|
||||
Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
|
||||
@@ -3549,7 +3555,7 @@ plt.show()
|
||||
Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
|
||||
The plot shows how the data are clustered around a line with slope close to one. Is this expected?
|
||||
|
||||
<h3 id="___sec62">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
<h3 id="___sec63">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
|
||||
<p>
|
||||
Now we are ready to solve for the principal components! To do so we
|
||||
@@ -3614,7 +3620,7 @@ This code does not contain all the above elements, but it shows how we can use <
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec63">Classical PCA Theorem </h2>
|
||||
<h2 id="___sec64">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
|
||||
@@ -3649,7 +3655,7 @@ The proof which follows will be updated by mid January 2020.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec64">Proof of the PCA Theorem </h2>
|
||||
<h2 id="___sec65">Proof of the PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
|
||||
@@ -3678,7 +3684,7 @@ where the vectors on the rhs are known.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec65">PCA Proof continued </h2>
|
||||
<h2 id="___sec66">PCA Proof continued </h2>
|
||||
|
||||
<p>
|
||||
We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
|
||||
@@ -3729,7 +3735,7 @@ of the projected data.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec66">The final step </h2>
|
||||
<h2 id="___sec67">The final step </h2>
|
||||
|
||||
<p>
|
||||
We could trivially maximize the variance of the projection (and
|
||||
@@ -3791,7 +3797,7 @@ Additional part of the proof for the other eigenvectors will be added by mid Jan
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec67">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
<h2 id="___sec68">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
|
||||
<p>
|
||||
This material will be added by mid January 2020.
|
||||
@@ -3799,7 +3805,7 @@ This material will be added by mid January 2020.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec68">Principal Component Analysis </h2>
|
||||
<h2 id="___sec69">Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
|
||||
@@ -3855,7 +3861,7 @@ X2D = X_centered.dot(W2)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec69">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec70">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -3887,7 +3893,7 @@ variance that lies along the axis of each principal component.
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec70">Back to the Cancer Data </h2>
|
||||
<h2 id="___sec71">Back to the Cancer Data </h2>
|
||||
We can now repeat the above but applied to real data, in this case our breast cancer data.
|
||||
Here we compute performance scores on the training data using logistic regression.
|
||||
<p>
|
||||
@@ -3928,7 +3934,7 @@ We see that our training data after the PCA decomposition has a performance simi
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec71">More on the PCA </h2>
|
||||
<h2 id="___sec72">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -3959,7 +3965,7 @@ X_reduced = pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec72">Incremental PCA </h2>
|
||||
<h2 id="___sec73">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -3971,7 +3977,7 @@ instances arrive).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec73">Randomized PCA </h2>
|
||||
<h2 id="___sec74">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -3982,7 +3988,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec74">Kernel PCA </h2>
|
||||
<h2 id="___sec75">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -4008,7 +4014,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec75">LLE </h2>
|
||||
<h2 id="___sec76">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -4020,7 +4026,7 @@ these local relationships are best preserved (more details shortly).
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec76">Other techniques </h2>
|
||||
<h2 id="___sec77">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
@@ -87,7 +87,7 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
('The network with one input layer, specified number of hidden '
|
||||
@@ -100,100 +100,100 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('The trial solution', 2, None, '___sec25'),
|
||||
('The program using Autograd', 2, None, '___sec26'),
|
||||
('Using forward Euler to solve the ODE', 2, None, '___sec27'),
|
||||
('Example: Solving the one dimensional Poisson equation using '
|
||||
'Autograd and TensorFlow',
|
||||
('Example: Solving the one dimensional Poisson equation',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
('The specific equation to solve for', 2, None, '___sec29'),
|
||||
('Solving the equation using Autograd', 2, None, '___sec30'),
|
||||
('Comparing with a numerical scheme', 2, None, '___sec31'),
|
||||
('Partial Differential Equations', 2, None, '___sec32'),
|
||||
('Type of problem', 2, None, '___sec33'),
|
||||
('Network requirements', 2, None, '___sec34'),
|
||||
('More details', 2, None, '___sec35'),
|
||||
('Example: The diffusion equation', 2, None, '___sec36'),
|
||||
('Defining the problem', 2, None, '___sec37'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec38'),
|
||||
('Setting up the code', 2, None, '___sec32'),
|
||||
('Partial Differential Equations', 2, None, '___sec33'),
|
||||
('Type of problem', 2, None, '___sec34'),
|
||||
('Network requirements', 2, None, '___sec35'),
|
||||
('More details', 2, None, '___sec36'),
|
||||
('Example: The diffusion equation', 2, None, '___sec37'),
|
||||
('Defining the problem', 2, None, '___sec38'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec39'),
|
||||
('Setting up the network using Autograd; The trial solution',
|
||||
2,
|
||||
None,
|
||||
'___sec39'),
|
||||
('Why the jacobian?', 2, None, '___sec40'),
|
||||
'___sec40'),
|
||||
('Why the jacobian?', 2, None, '___sec41'),
|
||||
('Setting up the network using Autograd; The full program',
|
||||
2,
|
||||
None,
|
||||
'___sec41'),
|
||||
'___sec42'),
|
||||
('Example: Solving the wave equation with Neural Networks',
|
||||
2,
|
||||
None,
|
||||
'___sec42'),
|
||||
('The problem to solve for', 2, None, '___sec43'),
|
||||
('The trial solution', 2, None, '___sec44'),
|
||||
('The analytical solution', 2, None, '___sec45'),
|
||||
'___sec43'),
|
||||
('The problem to solve for', 2, None, '___sec44'),
|
||||
('The trial solution', 2, None, '___sec45'),
|
||||
('The analytical solution', 2, None, '___sec46'),
|
||||
('Solving the wave equation - the full program using Autograd',
|
||||
2,
|
||||
None,
|
||||
'___sec46'),
|
||||
'___sec47'),
|
||||
('Resources on differential equations and deep learning',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec48'),
|
||||
'___sec48'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec49'),
|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -245,13 +245,17 @@ MathJax.Hub.Config({
|
||||
<li> Friday: Principal Component Analysis and Dimensionality Reduction</li>
|
||||
</ul>
|
||||
|
||||
Reading suggestions for both days: "Aurelien Geron's chapters 8
|
||||
We will also study the usage of <a href="https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola" target="_blank">Autograd</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html" target="_blank">week 40</a> and the <a href="https://github.com/HIPS/autograd" target="_blank">Autograd doucmentation</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec0">Recurrent Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini" target="_blank">Overview video</a>.
|
||||
See also lecture on Thursday October 22 and examples from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html" target="_blank">week 42</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
@@ -625,7 +629,7 @@ $$
|
||||
It is possible to use other activations functions for the hidden layer also.
|
||||
|
||||
<p>
|
||||
The output $\boldsymbol{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is:
|
||||
The output \( \boldsymbol{x}_i^{\text{hidden}} \) from each \( i \)-th hidden neuron is:
|
||||
|
||||
$$
|
||||
\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big)
|
||||
@@ -635,7 +639,11 @@ $$
|
||||
The outputs \( \boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.
|
||||
|
||||
<p>
|
||||
The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\text{output}}$ and biases \( b_i^{\text{output}} \). In this case, it is assumes that the number of neurons in the output layer is one.
|
||||
The output layer consists of one neuron in this case, and combines the
|
||||
output from each of the neurons in the hidden layers. The output layer
|
||||
combines the results from the hidden layer using some weights \( w_i^{\text{output}} \)
|
||||
and biases \( b_i^{\text{output}} \). In this case,
|
||||
it is assumes that the number of neurons in the output layer is one.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -683,7 +691,7 @@ In this case we seek a continuous range of values since we are approximating a f
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Backpropagation </h2>
|
||||
<h2 id="___sec19">Back propagation </h2>
|
||||
|
||||
<p>
|
||||
The next step is to decide how the parameters should be changed such that they minimize the cost function.
|
||||
@@ -726,7 +734,7 @@ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new}
|
||||
The value of \( \lambda \) decides how large steps the algorithm must take
|
||||
in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$.
|
||||
The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect
|
||||
to the elements in \( \boldsymbol{\omega \).
|
||||
to the elements in \( \boldsymbol{\omega} \).
|
||||
|
||||
<p>
|
||||
In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with
|
||||
@@ -1092,8 +1100,8 @@ Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0
|
||||
<p>
|
||||
In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability
|
||||
and high execution time (this might be more apparent in the examples solving PDEs),
|
||||
a network has been constructed using TensorFlow also.
|
||||
For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.
|
||||
using a library like TensorFlow is recommended.
|
||||
Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1323,7 +1331,7 @@ The network will be the similar as for the exponential decay example, but with s
|
||||
<h2 id="___sec27">Using forward Euler to solve the ODE </h2>
|
||||
|
||||
<p>
|
||||
A straight-forward way of solving an ODE numerically, is to use Euler's method.
|
||||
A straightforward way of solving an ODE numerically, is to use Euler's method.
|
||||
|
||||
<p>
|
||||
Euler's method uses Taylor series to approximate the value at a function \( f \) at a step \( \Delta x \) from \( x \):
|
||||
@@ -1446,13 +1454,10 @@ extending the program that uses the network using Autograd:
|
||||
|
||||
plt.show()
|
||||
</pre></div>
|
||||
<p>
|
||||
Running the program gives
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow </h2>
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation </h2>
|
||||
|
||||
<p>
|
||||
The Poisson equation for \( g(x) \) in one dimension is
|
||||
@@ -1768,13 +1773,18 @@ f(x_2) \\
|
||||
f(x_{N_x - 3}) \\
|
||||
f(x_{N_x - 2})
|
||||
\end{pmatrix} \\
|
||||
A\boldsymbol{g} &= \boldsymbol{f}
|
||||
\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f},
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
<p>
|
||||
which makes it possible to solve for the vector \( \boldsymbol{g} \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Setting up the code </h2>
|
||||
|
||||
<p>
|
||||
We can then compare the result from this numerical scheme with the output from our network using Autograd:
|
||||
|
||||
@@ -1973,21 +1983,15 @@ We can then compare the result from this numerical scheme with the output from o
|
||||
plt.show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The program prints out:
|
||||
<p>
|
||||
|
||||
<!-- code=text typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>The max absolute difference between the analytical solution and DNN Autograd: 0.000464088
|
||||
The max absolute difference between the analytical solution and numerical scheme: 0.00266858
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Partial Differential Equations </h2>
|
||||
<h2 id="___sec33">Partial Differential Equations </h2>
|
||||
|
||||
<p>
|
||||
A partial differential equation (PDE) has a solution here the function is defined by multiple variables.
|
||||
The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.
|
||||
A partial differential equation (PDE) has a solution here the function
|
||||
is defined by multiple variables. The equation may involve all kinds
|
||||
of combinations of which variables the function is differentiated with
|
||||
respect to.
|
||||
|
||||
<p>
|
||||
In general, a partial differential equation for a function \( g(x_1,\dots,x_N) \) with \( N \) variables may be expressed as
|
||||
@@ -2004,7 +2008,7 @@ where \( f \) is an expression involving all kinds of possible mixed derivatives
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec33">Type of problem </h2>
|
||||
<h2 id="___sec34">Type of problem </h2>
|
||||
|
||||
<p>
|
||||
The problem our network must solve for, is similar to the ODE case.
|
||||
@@ -2027,7 +2031,7 @@ The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensu
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec34">Network requirements </h2>
|
||||
<h2 id="___sec35">Network requirements </h2>
|
||||
|
||||
<p>
|
||||
The network tries then the minimize the cost function following the
|
||||
@@ -2043,35 +2047,33 @@ minimize is
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec35">More details </h2>
|
||||
<h2 id="___sec36">More details </h2>
|
||||
|
||||
<p>
|
||||
If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
If we let \( \boldsymbol{x} = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x) }{\partial x_N}, \frac{\partial g(\boldsymbol{x) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2
|
||||
$$
|
||||
|
||||
<p>
|
||||
If we also have \( M \) different sets of values for \( x_1, \dots, x_N \), that is \( \boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big) \) for \( i = 1,\dots,M \) being the rows in matrix \( X \), the cost function can be generalized into
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec36">Example: The diffusion equation </h2>
|
||||
<h2 id="___sec37">Example: The diffusion equation </h2>
|
||||
|
||||
<p>
|
||||
In one spatial dimension, the equation reads
|
||||
@@ -2096,7 +2098,7 @@ with \( u(x) \) being some given function.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec37">Defining the problem </h2>
|
||||
<h2 id="___sec38">Defining the problem </h2>
|
||||
|
||||
<p>
|
||||
For this case, we want to find \( g(x,t) \) such that
|
||||
@@ -2128,7 +2130,7 @@ First, we will look into how Autograd could be used in a network tailored to sol
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec38">Setting up the network using Autograd </h2>
|
||||
<h2 id="___sec39">Setting up the network using Autograd </h2>
|
||||
|
||||
<p>
|
||||
The only change to do here, is to extend our network such that
|
||||
@@ -2190,7 +2192,7 @@ network at each possible pair \( (x,t) \), given an array for the desired
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec39">Setting up the network using Autograd; The trial solution </h2>
|
||||
<h2 id="___sec40">Setting up the network using Autograd; The trial solution </h2>
|
||||
|
||||
<p>
|
||||
The cost function must then iterate through the given arrays
|
||||
@@ -2219,7 +2221,7 @@ since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec40">Why the jacobian? </h2>
|
||||
<h2 id="___sec41">Why the jacobian? </h2>
|
||||
|
||||
<p>
|
||||
The Jacobian is used because the program must find the derivative of
|
||||
@@ -2286,7 +2288,7 @@ mixed derivatives of \( g(x,t) \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec41">Setting up the network using Autograd; The full program </h2>
|
||||
<h2 id="___sec42">Setting up the network using Autograd; The full program </h2>
|
||||
|
||||
<p>
|
||||
Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.
|
||||
@@ -2537,7 +2539,7 @@ Using TensorFlow results in a much better execution time. Try it!
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec42">Example: Solving the wave equation with Neural Networks </h2>
|
||||
<h2 id="___sec43">Example: Solving the wave equation with Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
The wave equation is
|
||||
@@ -2566,7 +2568,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec43">The problem to solve for </h2>
|
||||
<h2 id="___sec44">The problem to solve for </h2>
|
||||
|
||||
<p>
|
||||
The wave equation to solve for, is
|
||||
@@ -2596,7 +2598,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec44">The trial solution </h2>
|
||||
<h2 id="___sec45">The trial solution </h2>
|
||||
Setting up the network is done in similar matter as for the example of solving the diffusion equation.
|
||||
The only things we have to change, is the trial solution such that it satisfies the conditions from \eqref{condwave} and the cost function.
|
||||
|
||||
@@ -2620,7 +2622,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec45">The analytical solution </h2>
|
||||
<h2 id="___sec46">The analytical solution </h2>
|
||||
|
||||
<p>
|
||||
The analytical solution for our specific problem, is
|
||||
@@ -2632,7 +2634,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec46">Solving the wave equation - the full program using Autograd </h2>
|
||||
<h2 id="___sec47">Solving the wave equation - the full program using Autograd </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -2861,7 +2863,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec47">Resources on differential equations and deep learning </h2>
|
||||
<h2 id="___sec48">Resources on differential equations and deep learning </h2>
|
||||
|
||||
<ol>
|
||||
<li> <a href="https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf" target="_blank">Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al</a></li>
|
||||
@@ -2872,12 +2874,15 @@ $$
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec48">Friday, Principal Component Analysis </h2>
|
||||
<h2 id="___sec49">Friday, Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=fkf4IBRSeEc&ab_channel=SteveBrunton" target="_blank">Overview video</a>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec49">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
<h2 id="___sec50">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
|
||||
<p>
|
||||
The principal component analysis deals with the problem of fitting a
|
||||
@@ -2899,7 +2904,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec50">Introducing the Covariance and Correlation functions </h2>
|
||||
<h2 id="___sec51">Introducing the Covariance and Correlation functions </h2>
|
||||
|
||||
<p>
|
||||
Before we discuss the PCA theorem, we need to remind ourselves about
|
||||
@@ -2960,7 +2965,7 @@ In the above example this is the function we constructed using <b>pandas</b>.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec51">Correlation Function and Design/Feature Matrix </h2>
|
||||
<h2 id="___sec52">Correlation Function and Design/Feature Matrix </h2>
|
||||
|
||||
<p>
|
||||
In our derivation of the various regression algorithms like <b>Ordinary Least Squares</b> or <b>Ridge regression</b>
|
||||
@@ -3021,7 +3026,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec52">Covariance Matrix Examples </h2>
|
||||
<h2 id="___sec53">Covariance Matrix Examples </h2>
|
||||
|
||||
<p>
|
||||
The Numpy function <b>np.cov</b> calculates the covariance elements using
|
||||
@@ -3064,7 +3069,7 @@ C = np.cov(W)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec53">Correlation Matrix </h2>
|
||||
<h2 id="___sec54">Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
The previous example can be converted into the correlation matrix by
|
||||
@@ -3109,7 +3114,7 @@ The above procedure with <b>numpy</b> can be made more compact if we use <b>pand
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec54">Correlation Matrix with Pandas </h2>
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas </h2>
|
||||
|
||||
<p>
|
||||
We whow here how we can set up the correlation matrix using <b>pandas</b>, as done in this simple code
|
||||
@@ -3136,7 +3141,7 @@ We expand this model to the Franke function discussed above.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
<h2 id="___sec56">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -3199,7 +3204,7 @@ matrix without these elements.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec56">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
<h2 id="___sec57">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
|
||||
@@ -3242,7 +3247,7 @@ It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\t
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec57">Towards the PCA theorem </h2>
|
||||
<h2 id="___sec58">Towards the PCA theorem </h2>
|
||||
|
||||
<p>
|
||||
We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
|
||||
@@ -3293,7 +3298,7 @@ features/predictors.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec58">The Algorithm before theorem </h2>
|
||||
<h2 id="___sec59">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
@@ -3324,7 +3329,7 @@ $$
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec59">Writing our own PCA code </h2>
|
||||
<h2 id="___sec60">Writing our own PCA code </h2>
|
||||
|
||||
<p>
|
||||
We will use a simple example first with two-dimensional data
|
||||
@@ -3357,7 +3362,7 @@ X = np.random.multivariate_normal(mean, cov, n)
|
||||
<p>
|
||||
Now we are going to implement the PCA algorithm. We will break it down into various substeps.
|
||||
|
||||
<h3 id="___sec60">Compute the sample mean and center the data </h3>
|
||||
<h3 id="___sec61">Compute the sample mean and center the data </h3>
|
||||
|
||||
<p>
|
||||
The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
|
||||
@@ -3394,7 +3399,7 @@ variance. The diagonal covariance matrix elements will then be one,
|
||||
while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
|
||||
specific case.
|
||||
|
||||
<h3 id="___sec61">Compute the sample covariance </h3>
|
||||
<h3 id="___sec62">Compute the sample covariance </h3>
|
||||
|
||||
<p>
|
||||
Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
|
||||
@@ -3436,7 +3441,7 @@ plt.show()
|
||||
Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
|
||||
The plot shows how the data are clustered around a line with slope close to one. Is this expected?
|
||||
|
||||
<h3 id="___sec62">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
<h3 id="___sec63">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
|
||||
<p>
|
||||
Now we are ready to solve for the principal components! To do so we
|
||||
@@ -3500,7 +3505,7 @@ This code does not contain all the above elements, but it shows how we can use <
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec63">Classical PCA Theorem </h2>
|
||||
<h2 id="___sec64">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
|
||||
@@ -3533,7 +3538,7 @@ The proof which follows will be updated by mid January 2020.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec64">Proof of the PCA Theorem </h2>
|
||||
<h2 id="___sec65">Proof of the PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
|
||||
@@ -3556,7 +3561,7 @@ where the vectors on the rhs are known.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec65">PCA Proof continued </h2>
|
||||
<h2 id="___sec66">PCA Proof continued </h2>
|
||||
|
||||
<p>
|
||||
We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
|
||||
@@ -3597,7 +3602,7 @@ of the projected data.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec66">The final step </h2>
|
||||
<h2 id="___sec67">The final step </h2>
|
||||
|
||||
<p>
|
||||
We could trivially maximize the variance of the projection (and
|
||||
@@ -3651,7 +3656,7 @@ Additional part of the proof for the other eigenvectors will be added by mid Jan
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec67">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
<h2 id="___sec68">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
|
||||
<p>
|
||||
This material will be added by mid January 2020.
|
||||
@@ -3659,7 +3664,7 @@ This material will be added by mid January 2020.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec68">Principal Component Analysis </h2>
|
||||
<h2 id="___sec69">Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
|
||||
@@ -3714,7 +3719,7 @@ X2D = X_centered.dot(W2)
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec69">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec70">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -3746,7 +3751,7 @@ variance that lies along the axis of each principal component.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec70">Back to the Cancer Data </h2>
|
||||
<h2 id="___sec71">Back to the Cancer Data </h2>
|
||||
We can now repeat the above but applied to real data, in this case our breast cancer data.
|
||||
Here we compute performance scores on the training data using logistic regression.
|
||||
<p>
|
||||
@@ -3787,7 +3792,7 @@ We see that our training data after the PCA decomposition has a performance simi
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec71">More on the PCA </h2>
|
||||
<h2 id="___sec72">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -3817,7 +3822,7 @@ X_reduced = pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec72">Incremental PCA </h2>
|
||||
<h2 id="___sec73">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -3829,7 +3834,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec73">Randomized PCA </h2>
|
||||
<h2 id="___sec74">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -3840,7 +3845,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec74">Kernel PCA </h2>
|
||||
<h2 id="___sec75">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -3869,7 +3874,7 @@ X_reduced = rbf_pca.fit_transform(X)
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec75">LLE </h2>
|
||||
<h2 id="___sec76">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -3881,7 +3886,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec76">Other techniques </h2>
|
||||
<h2 id="___sec77">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
+127
-122
@@ -92,7 +92,7 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('Final technicalities II', 2, None, '___sec16'),
|
||||
('Final technicalities III', 2, None, '___sec17'),
|
||||
('Final technicalities IV', 2, None, '___sec18'),
|
||||
('Backpropagation', 2, None, '___sec19'),
|
||||
('Back propagation', 2, None, '___sec19'),
|
||||
('Gradient descent', 2, None, '___sec20'),
|
||||
('The code for solving the ODE', 2, None, '___sec21'),
|
||||
('The network with one input layer, specified number of hidden '
|
||||
@@ -105,100 +105,100 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
('The trial solution', 2, None, '___sec25'),
|
||||
('The program using Autograd', 2, None, '___sec26'),
|
||||
('Using forward Euler to solve the ODE', 2, None, '___sec27'),
|
||||
('Example: Solving the one dimensional Poisson equation using '
|
||||
'Autograd and TensorFlow',
|
||||
('Example: Solving the one dimensional Poisson equation',
|
||||
2,
|
||||
None,
|
||||
'___sec28'),
|
||||
('The specific equation to solve for', 2, None, '___sec29'),
|
||||
('Solving the equation using Autograd', 2, None, '___sec30'),
|
||||
('Comparing with a numerical scheme', 2, None, '___sec31'),
|
||||
('Partial Differential Equations', 2, None, '___sec32'),
|
||||
('Type of problem', 2, None, '___sec33'),
|
||||
('Network requirements', 2, None, '___sec34'),
|
||||
('More details', 2, None, '___sec35'),
|
||||
('Example: The diffusion equation', 2, None, '___sec36'),
|
||||
('Defining the problem', 2, None, '___sec37'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec38'),
|
||||
('Setting up the code', 2, None, '___sec32'),
|
||||
('Partial Differential Equations', 2, None, '___sec33'),
|
||||
('Type of problem', 2, None, '___sec34'),
|
||||
('Network requirements', 2, None, '___sec35'),
|
||||
('More details', 2, None, '___sec36'),
|
||||
('Example: The diffusion equation', 2, None, '___sec37'),
|
||||
('Defining the problem', 2, None, '___sec38'),
|
||||
('Setting up the network using Autograd', 2, None, '___sec39'),
|
||||
('Setting up the network using Autograd; The trial solution',
|
||||
2,
|
||||
None,
|
||||
'___sec39'),
|
||||
('Why the jacobian?', 2, None, '___sec40'),
|
||||
'___sec40'),
|
||||
('Why the jacobian?', 2, None, '___sec41'),
|
||||
('Setting up the network using Autograd; The full program',
|
||||
2,
|
||||
None,
|
||||
'___sec41'),
|
||||
'___sec42'),
|
||||
('Example: Solving the wave equation with Neural Networks',
|
||||
2,
|
||||
None,
|
||||
'___sec42'),
|
||||
('The problem to solve for', 2, None, '___sec43'),
|
||||
('The trial solution', 2, None, '___sec44'),
|
||||
('The analytical solution', 2, None, '___sec45'),
|
||||
'___sec43'),
|
||||
('The problem to solve for', 2, None, '___sec44'),
|
||||
('The trial solution', 2, None, '___sec45'),
|
||||
('The analytical solution', 2, None, '___sec46'),
|
||||
('Solving the wave equation - the full program using Autograd',
|
||||
2,
|
||||
None,
|
||||
'___sec46'),
|
||||
'___sec47'),
|
||||
('Resources on differential equations and deep learning',
|
||||
2,
|
||||
None,
|
||||
'___sec47'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec48'),
|
||||
'___sec48'),
|
||||
('Friday, Principal Component Analysis', 2, None, '___sec49'),
|
||||
('Basic ideas of the Principal Component Analysis (PCA)',
|
||||
2,
|
||||
None,
|
||||
'___sec49'),
|
||||
'___sec50'),
|
||||
('Introducing the Covariance and Correlation functions',
|
||||
2,
|
||||
None,
|
||||
'___sec50'),
|
||||
'___sec51'),
|
||||
('Correlation Function and Design/Feature Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec51'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec52'),
|
||||
('Correlation Matrix', 2, None, '___sec53'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec54'),
|
||||
'___sec52'),
|
||||
('Covariance Matrix Examples', 2, None, '___sec53'),
|
||||
('Correlation Matrix', 2, None, '___sec54'),
|
||||
('Correlation Matrix with Pandas', 2, None, '___sec55'),
|
||||
('Correlation Matrix with Pandas and the Franke function',
|
||||
2,
|
||||
None,
|
||||
'___sec55'),
|
||||
'___sec56'),
|
||||
('Rewriting the Covariance and/or Correlation Matrix',
|
||||
2,
|
||||
None,
|
||||
'___sec56'),
|
||||
('Towards the PCA theorem', 2, None, '___sec57'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec58'),
|
||||
('Writing our own PCA code', 2, None, '___sec59'),
|
||||
'___sec57'),
|
||||
('Towards the PCA theorem', 2, None, '___sec58'),
|
||||
('The Algorithm before the Theorem', 2, None, '___sec59'),
|
||||
('Writing our own PCA code', 2, None, '___sec60'),
|
||||
('Compute the sample mean and center the data',
|
||||
3,
|
||||
None,
|
||||
'___sec60'),
|
||||
('Compute the sample covariance', 3, None, '___sec61'),
|
||||
'___sec61'),
|
||||
('Compute the sample covariance', 3, None, '___sec62'),
|
||||
('Diagonalize the sample covariance matrix to obtain the '
|
||||
'principal components',
|
||||
3,
|
||||
None,
|
||||
'___sec62'),
|
||||
('Classical PCA Theorem', 2, None, '___sec63'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec64'),
|
||||
('PCA Proof continued', 2, None, '___sec65'),
|
||||
('The final step', 2, None, '___sec66'),
|
||||
'___sec63'),
|
||||
('Classical PCA Theorem', 2, None, '___sec64'),
|
||||
('Proof of the PCA Theorem', 2, None, '___sec65'),
|
||||
('PCA Proof continued', 2, None, '___sec66'),
|
||||
('The final step', 2, None, '___sec67'),
|
||||
('Geometric Interpretation and link with Singular Value '
|
||||
'Decomposition',
|
||||
2,
|
||||
None,
|
||||
'___sec67'),
|
||||
('Principal Component Analysis', 2, None, '___sec68'),
|
||||
('PCA and scikit-learn', 2, None, '___sec69'),
|
||||
('Back to the Cancer Data', 2, None, '___sec70'),
|
||||
('More on the PCA', 2, None, '___sec71'),
|
||||
('Incremental PCA', 2, None, '___sec72'),
|
||||
('Randomized PCA', 2, None, '___sec73'),
|
||||
('Kernel PCA', 2, None, '___sec74'),
|
||||
('LLE', 2, None, '___sec75'),
|
||||
('Other techniques', 2, None, '___sec76')]}
|
||||
'___sec68'),
|
||||
('Principal Component Analysis', 2, None, '___sec69'),
|
||||
('PCA and scikit-learn', 2, None, '___sec70'),
|
||||
('Back to the Cancer Data', 2, None, '___sec71'),
|
||||
('More on the PCA', 2, None, '___sec72'),
|
||||
('Incremental PCA', 2, None, '___sec73'),
|
||||
('Randomized PCA', 2, None, '___sec74'),
|
||||
('Kernel PCA', 2, None, '___sec75'),
|
||||
('LLE', 2, None, '___sec76'),
|
||||
('Other techniques', 2, None, '___sec77')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -250,13 +250,17 @@ MathJax.Hub.Config({
|
||||
<li> Friday: Principal Component Analysis and Dimensionality Reduction</li>
|
||||
</ul>
|
||||
|
||||
Reading suggestions for both days: "Aurelien Geron's chapters 8
|
||||
We will also study the usage of <a href="https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola" target="_blank">Autograd</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html" target="_blank">week 40</a> and the <a href="https://github.com/HIPS/autograd" target="_blank">Autograd doucmentation</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec0">Recurrent Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini" target="_blank">Overview video</a>.
|
||||
See also lecture on Thursday October 22 and examples from <a href="https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html" target="_blank">week 42</a>.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
@@ -630,7 +634,7 @@ $$
|
||||
It is possible to use other activations functions for the hidden layer also.
|
||||
|
||||
<p>
|
||||
The output $\boldsymbol{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is:
|
||||
The output \( \boldsymbol{x}_i^{\text{hidden}} \) from each \( i \)-th hidden neuron is:
|
||||
|
||||
$$
|
||||
\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big)
|
||||
@@ -640,7 +644,11 @@ $$
|
||||
The outputs \( \boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.
|
||||
|
||||
<p>
|
||||
The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\text{output}}$ and biases \( b_i^{\text{output}} \). In this case, it is assumes that the number of neurons in the output layer is one.
|
||||
The output layer consists of one neuron in this case, and combines the
|
||||
output from each of the neurons in the hidden layers. The output layer
|
||||
combines the results from the hidden layer using some weights \( w_i^{\text{output}} \)
|
||||
and biases \( b_i^{\text{output}} \). In this case,
|
||||
it is assumes that the number of neurons in the output layer is one.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -688,7 +696,7 @@ In this case we seek a continuous range of values since we are approximating a f
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec19">Backpropagation </h2>
|
||||
<h2 id="___sec19">Back propagation </h2>
|
||||
|
||||
<p>
|
||||
The next step is to decide how the parameters should be changed such that they minimize the cost function.
|
||||
@@ -731,7 +739,7 @@ for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new}
|
||||
The value of \( \lambda \) decides how large steps the algorithm must take
|
||||
in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$.
|
||||
The notation \( \nabla_{\boldsymbol{\omega}} \) express the gradient with respect
|
||||
to the elements in \( \boldsymbol{\omega \).
|
||||
to the elements in \( \boldsymbol{\omega} \).
|
||||
|
||||
<p>
|
||||
In our case, we have to minimize the cost function \( C(\boldsymbol{x}, P) \) with
|
||||
@@ -1097,8 +1105,8 @@ Also, at \( t = 0 \) the population has the size \( g(0) = g_0 \), where \( g_0
|
||||
<p>
|
||||
In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability
|
||||
and high execution time (this might be more apparent in the examples solving PDEs),
|
||||
a network has been constructed using TensorFlow also.
|
||||
For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.
|
||||
using a library like TensorFlow is recommended.
|
||||
Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1328,7 +1336,7 @@ The network will be the similar as for the exponential decay example, but with s
|
||||
<h2 id="___sec27">Using forward Euler to solve the ODE </h2>
|
||||
|
||||
<p>
|
||||
A straight-forward way of solving an ODE numerically, is to use Euler's method.
|
||||
A straightforward way of solving an ODE numerically, is to use Euler's method.
|
||||
|
||||
<p>
|
||||
Euler's method uses Taylor series to approximate the value at a function \( f \) at a step \( \Delta x \) from \( x \):
|
||||
@@ -1451,13 +1459,10 @@ extending the program that uses the network using Autograd:
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
Running the program gives
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow </h2>
|
||||
<h2 id="___sec28">Example: Solving the one dimensional Poisson equation </h2>
|
||||
|
||||
<p>
|
||||
The Poisson equation for \( g(x) \) in one dimension is
|
||||
@@ -1773,13 +1778,18 @@ f(x_2) \\
|
||||
f(x_{N_x - 3}) \\
|
||||
f(x_{N_x - 2})
|
||||
\end{pmatrix} \\
|
||||
A\boldsymbol{g} &= \boldsymbol{f}
|
||||
\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f},
|
||||
\end{aligned}
|
||||
$$
|
||||
|
||||
<p>
|
||||
which makes it possible to solve for the vector \( \boldsymbol{g} \).
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Setting up the code </h2>
|
||||
|
||||
<p>
|
||||
We can then compare the result from this numerical scheme with the output from our network using Autograd:
|
||||
|
||||
@@ -1978,21 +1988,15 @@ We can then compare the result from this numerical scheme with the output from o
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The program prints out:
|
||||
<p>
|
||||
|
||||
<!-- code=text typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>The max absolute difference between the analytical solution and DNN Autograd: 0.000464088
|
||||
The max absolute difference between the analytical solution and numerical scheme: 0.00266858
|
||||
</pre></div>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec32">Partial Differential Equations </h2>
|
||||
<h2 id="___sec33">Partial Differential Equations </h2>
|
||||
|
||||
<p>
|
||||
A partial differential equation (PDE) has a solution here the function is defined by multiple variables.
|
||||
The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.
|
||||
A partial differential equation (PDE) has a solution here the function
|
||||
is defined by multiple variables. The equation may involve all kinds
|
||||
of combinations of which variables the function is differentiated with
|
||||
respect to.
|
||||
|
||||
<p>
|
||||
In general, a partial differential equation for a function \( g(x_1,\dots,x_N) \) with \( N \) variables may be expressed as
|
||||
@@ -2009,7 +2013,7 @@ where \( f \) is an expression involving all kinds of possible mixed derivatives
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec33">Type of problem </h2>
|
||||
<h2 id="___sec34">Type of problem </h2>
|
||||
|
||||
<p>
|
||||
The problem our network must solve for, is similar to the ODE case.
|
||||
@@ -2032,7 +2036,7 @@ The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensu
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec34">Network requirements </h2>
|
||||
<h2 id="___sec35">Network requirements </h2>
|
||||
|
||||
<p>
|
||||
The network tries then the minimize the cost function following the
|
||||
@@ -2048,35 +2052,33 @@ minimize is
|
||||
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec35">More details </h2>
|
||||
<h2 id="___sec36">More details </h2>
|
||||
|
||||
<p>
|
||||
If we let \( \boldsymbol{x = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
If we let \( \boldsymbol{x} = \big( x_1, \dots, x_N \big) \) be an array containing the values for \( x_1, \dots, x_N \) respectively, the cost function can be reformulated into the following:
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x) }{\partial x_N}, \frac{\partial g(\boldsymbol{x) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x) }{\partial x_N^n} \right) \right)^2
|
||||
\end{equation*}
|
||||
C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2
|
||||
$$
|
||||
|
||||
<p>
|
||||
If we also have \( M \) different sets of values for \( x_1, \dots, x_N \), that is \( \boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big) \) for \( i = 1,\dots,M \) being the rows in matrix \( X \), the cost function can be generalized into
|
||||
$$
|
||||
\begin{equation*}
|
||||
c\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2
|
||||
C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2.
|
||||
\end{equation*}
|
||||
$$
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec36">Example: The diffusion equation </h2>
|
||||
<h2 id="___sec37">Example: The diffusion equation </h2>
|
||||
|
||||
<p>
|
||||
In one spatial dimension, the equation reads
|
||||
@@ -2101,7 +2103,7 @@ with \( u(x) \) being some given function.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec37">Defining the problem </h2>
|
||||
<h2 id="___sec38">Defining the problem </h2>
|
||||
|
||||
<p>
|
||||
For this case, we want to find \( g(x,t) \) such that
|
||||
@@ -2133,7 +2135,7 @@ First, we will look into how Autograd could be used in a network tailored to sol
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec38">Setting up the network using Autograd </h2>
|
||||
<h2 id="___sec39">Setting up the network using Autograd </h2>
|
||||
|
||||
<p>
|
||||
The only change to do here, is to extend our network such that
|
||||
@@ -2195,7 +2197,7 @@ network at each possible pair \( (x,t) \), given an array for the desired
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec39">Setting up the network using Autograd; The trial solution </h2>
|
||||
<h2 id="___sec40">Setting up the network using Autograd; The trial solution </h2>
|
||||
|
||||
<p>
|
||||
The cost function must then iterate through the given arrays
|
||||
@@ -2224,7 +2226,7 @@ since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec40">Why the jacobian? </h2>
|
||||
<h2 id="___sec41">Why the jacobian? </h2>
|
||||
|
||||
<p>
|
||||
The Jacobian is used because the program must find the derivative of
|
||||
@@ -2291,7 +2293,7 @@ mixed derivatives of \( g(x,t) \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec41">Setting up the network using Autograd; The full program </h2>
|
||||
<h2 id="___sec42">Setting up the network using Autograd; The full program </h2>
|
||||
|
||||
<p>
|
||||
Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.
|
||||
@@ -2542,7 +2544,7 @@ Using TensorFlow results in a much better execution time. Try it!
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec42">Example: Solving the wave equation with Neural Networks </h2>
|
||||
<h2 id="___sec43">Example: Solving the wave equation with Neural Networks </h2>
|
||||
|
||||
<p>
|
||||
The wave equation is
|
||||
@@ -2571,7 +2573,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec43">The problem to solve for </h2>
|
||||
<h2 id="___sec44">The problem to solve for </h2>
|
||||
|
||||
<p>
|
||||
The wave equation to solve for, is
|
||||
@@ -2601,7 +2603,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec44">The trial solution </h2>
|
||||
<h2 id="___sec45">The trial solution </h2>
|
||||
Setting up the network is done in similar matter as for the example of solving the diffusion equation.
|
||||
The only things we have to change, is the trial solution such that it satisfies the conditions from \eqref{condwave} and the cost function.
|
||||
|
||||
@@ -2625,7 +2627,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec45">The analytical solution </h2>
|
||||
<h2 id="___sec46">The analytical solution </h2>
|
||||
|
||||
<p>
|
||||
The analytical solution for our specific problem, is
|
||||
@@ -2637,7 +2639,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec46">Solving the wave equation - the full program using Autograd </h2>
|
||||
<h2 id="___sec47">Solving the wave equation - the full program using Autograd </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -2866,7 +2868,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec47">Resources on differential equations and deep learning </h2>
|
||||
<h2 id="___sec48">Resources on differential equations and deep learning </h2>
|
||||
|
||||
<ol>
|
||||
<li> <a href="https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf" target="_blank">Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al</a></li>
|
||||
@@ -2877,12 +2879,15 @@ $$
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec48">Friday, Principal Component Analysis </h2>
|
||||
<h2 id="___sec49">Friday, Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.youtube.com/watch?v=fkf4IBRSeEc&ab_channel=SteveBrunton" target="_blank">Overview video</a>
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec49">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
<h2 id="___sec50">Basic ideas of the Principal Component Analysis (PCA) </h2>
|
||||
|
||||
<p>
|
||||
The principal component analysis deals with the problem of fitting a
|
||||
@@ -2904,7 +2909,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec50">Introducing the Covariance and Correlation functions </h2>
|
||||
<h2 id="___sec51">Introducing the Covariance and Correlation functions </h2>
|
||||
|
||||
<p>
|
||||
Before we discuss the PCA theorem, we need to remind ourselves about
|
||||
@@ -2965,7 +2970,7 @@ In the above example this is the function we constructed using <b>pandas</b>.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec51">Correlation Function and Design/Feature Matrix </h2>
|
||||
<h2 id="___sec52">Correlation Function and Design/Feature Matrix </h2>
|
||||
|
||||
<p>
|
||||
In our derivation of the various regression algorithms like <b>Ordinary Least Squares</b> or <b>Ridge regression</b>
|
||||
@@ -3026,7 +3031,7 @@ $$
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec52">Covariance Matrix Examples </h2>
|
||||
<h2 id="___sec53">Covariance Matrix Examples </h2>
|
||||
|
||||
<p>
|
||||
The Numpy function <b>np.cov</b> calculates the covariance elements using
|
||||
@@ -3069,7 +3074,7 @@ C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec53">Correlation Matrix </h2>
|
||||
<h2 id="___sec54">Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
The previous example can be converted into the correlation matrix by
|
||||
@@ -3114,7 +3119,7 @@ The above procedure with <b>numpy</b> can be made more compact if we use <b>pand
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec54">Correlation Matrix with Pandas </h2>
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas </h2>
|
||||
|
||||
<p>
|
||||
We whow here how we can set up the correlation matrix using <b>pandas</b>, as done in this simple code
|
||||
@@ -3141,7 +3146,7 @@ We expand this model to the Franke function discussed above.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec55">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
<h2 id="___sec56">Correlation Matrix with Pandas and the Franke function </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -3204,7 +3209,7 @@ matrix without these elements.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec56">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
<h2 id="___sec57">Rewriting the Covariance and/or Correlation Matrix </h2>
|
||||
|
||||
<p>
|
||||
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
|
||||
@@ -3247,7 +3252,7 @@ It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\t
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec57">Towards the PCA theorem </h2>
|
||||
<h2 id="___sec58">Towards the PCA theorem </h2>
|
||||
|
||||
<p>
|
||||
We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
|
||||
@@ -3298,7 +3303,7 @@ features/predictors.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec58">The Algorithm before theorem </h2>
|
||||
<h2 id="___sec59">The Algorithm before theorem </h2>
|
||||
|
||||
<p>
|
||||
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
|
||||
@@ -3329,7 +3334,7 @@ $$
|
||||
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec59">Writing our own PCA code </h2>
|
||||
<h2 id="___sec60">Writing our own PCA code </h2>
|
||||
|
||||
<p>
|
||||
We will use a simple example first with two-dimensional data
|
||||
@@ -3362,7 +3367,7 @@ X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>r
|
||||
<p>
|
||||
Now we are going to implement the PCA algorithm. We will break it down into various substeps.
|
||||
|
||||
<h3 id="___sec60">Compute the sample mean and center the data </h3>
|
||||
<h3 id="___sec61">Compute the sample mean and center the data </h3>
|
||||
|
||||
<p>
|
||||
The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
|
||||
@@ -3399,7 +3404,7 @@ variance. The diagonal covariance matrix elements will then be one,
|
||||
while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our
|
||||
specific case.
|
||||
|
||||
<h3 id="___sec61">Compute the sample covariance </h3>
|
||||
<h3 id="___sec62">Compute the sample covariance </h3>
|
||||
|
||||
<p>
|
||||
Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
|
||||
@@ -3441,7 +3446,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
|
||||
The plot shows how the data are clustered around a line with slope close to one. Is this expected?
|
||||
|
||||
<h3 id="___sec62">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
<h3 id="___sec63">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
||||
|
||||
<p>
|
||||
Now we are ready to solve for the principal components! To do so we
|
||||
@@ -3505,7 +3510,7 @@ This code does not contain all the above elements, but it shows how we can use <
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec63">Classical PCA Theorem </h2>
|
||||
<h2 id="___sec64">Classical PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
We assume now that we have a design matrix \( \boldsymbol{X} \) which has been
|
||||
@@ -3538,7 +3543,7 @@ The proof which follows will be updated by mid January 2020.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec64">Proof of the PCA Theorem </h2>
|
||||
<h2 id="___sec65">Proof of the PCA Theorem </h2>
|
||||
|
||||
<p>
|
||||
To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
|
||||
@@ -3561,7 +3566,7 @@ where the vectors on the rhs are known.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec65">PCA Proof continued </h2>
|
||||
<h2 id="___sec66">PCA Proof continued </h2>
|
||||
|
||||
<p>
|
||||
We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
|
||||
@@ -3602,7 +3607,7 @@ of the projected data.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec66">The final step </h2>
|
||||
<h2 id="___sec67">The final step </h2>
|
||||
|
||||
<p>
|
||||
We could trivially maximize the variance of the projection (and
|
||||
@@ -3656,7 +3661,7 @@ Additional part of the proof for the other eigenvectors will be added by mid Jan
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec67">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
<h2 id="___sec68">Geometric Interpretation and link with Singular Value Decomposition </h2>
|
||||
|
||||
<p>
|
||||
This material will be added by mid January 2020.
|
||||
@@ -3664,7 +3669,7 @@ This material will be added by mid January 2020.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec68">Principal Component Analysis </h2>
|
||||
<h2 id="___sec69">Principal Component Analysis </h2>
|
||||
|
||||
<p>
|
||||
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
|
||||
@@ -3719,7 +3724,7 @@ X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666"
|
||||
<p>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec69">PCA and scikit-learn </h2>
|
||||
<h2 id="___sec70">PCA and scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
|
||||
@@ -3751,7 +3756,7 @@ variance that lies along the axis of each principal component.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec70">Back to the Cancer Data </h2>
|
||||
<h2 id="___sec71">Back to the Cancer Data </h2>
|
||||
We can now repeat the above but applied to real data, in this case our breast cancer data.
|
||||
Here we compute performance scores on the training data using logistic regression.
|
||||
<p>
|
||||
@@ -3792,7 +3797,7 @@ We see that our training data after the PCA decomposition has a performance simi
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec71">More on the PCA </h2>
|
||||
<h2 id="___sec72">More on the PCA </h2>
|
||||
|
||||
<p>
|
||||
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
|
||||
@@ -3822,7 +3827,7 @@ 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="___sec72">Incremental PCA </h2>
|
||||
<h2 id="___sec73">Incremental PCA </h2>
|
||||
|
||||
<p>
|
||||
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
|
||||
@@ -3834,7 +3839,7 @@ instances arrive).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec73">Randomized PCA </h2>
|
||||
<h2 id="___sec74">Randomized PCA </h2>
|
||||
|
||||
<p>
|
||||
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
|
||||
@@ -3845,7 +3850,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec74">Kernel PCA </h2>
|
||||
<h2 id="___sec75">Kernel PCA </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
@@ -3874,7 +3879,7 @@ 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="___sec75">LLE </h2>
|
||||
<h2 id="___sec76">LLE </h2>
|
||||
|
||||
<p>
|
||||
Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
|
||||
@@ -3886,7 +3891,7 @@ these local relationships are best preserved (more details shortly).
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec76">Other techniques </h2>
|
||||
<h2 id="___sec77">Other techniques </h2>
|
||||
|
||||
<p>
|
||||
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
|
||||
|
||||
Binary file not shown.
@@ -21,10 +21,12 @@
|
||||
"\n",
|
||||
"* Friday: Principal Component Analysis and Dimensionality Reduction\n",
|
||||
"\n",
|
||||
"Reading suggestions for both days: \"Aurelien Geron's chapters 8 \n",
|
||||
"We will also study the usage of [Autograd](https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola) in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from [week 40](https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html) and the [Autograd doucmentation](https://github.com/HIPS/autograd).\n",
|
||||
"\n",
|
||||
"## Recurrent Neural Networks\n",
|
||||
"\n",
|
||||
"[Overview video](https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini).\n",
|
||||
"See also lecture on Thursday October 22 and examples from [week 42](https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html).\n",
|
||||
"\n",
|
||||
"## Solving ODEs with Deep Learning\n",
|
||||
"\n",
|
||||
@@ -513,7 +515,7 @@
|
||||
"source": [
|
||||
"It is possible to use other activations functions for the hidden layer also.\n",
|
||||
"\n",
|
||||
"The output $\\boldsymbol{x}_i^{\\text{hidden} }$from each $i$-th hidden neuron is:\n",
|
||||
"The output $\\boldsymbol{x}_i^{\\text{hidden}}$ from each $i$-th hidden neuron is:\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\boldsymbol{x}_i^{\\text{hidden} } = f\\big( \\boldsymbol{z}_{i}^{\\text{hidden}} \\big)\n",
|
||||
@@ -521,7 +523,11 @@
|
||||
"\n",
|
||||
"The outputs $\\boldsymbol{x}_i^{\\text{hidden} } $ are then sent to the output layer.\n",
|
||||
"\n",
|
||||
"The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\\text{output}}$ and biases $b_i^{\\text{output}}$. In this case, it is assumes that the number of neurons in the output layer is one.\n",
|
||||
"The output layer consists of one neuron in this case, and combines the\n",
|
||||
"output from each of the neurons in the hidden layers. The output layer\n",
|
||||
"combines the results from the hidden layer using some weights $w_i^{\\text{output}}$\n",
|
||||
"and biases $b_i^{\\text{output}}$. In this case,\n",
|
||||
"it is assumes that the number of neurons in the output layer is one.\n",
|
||||
"\n",
|
||||
"## Final technicalities III\n",
|
||||
"\n",
|
||||
@@ -578,7 +584,7 @@
|
||||
"source": [
|
||||
"In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n",
|
||||
"\n",
|
||||
"## Backpropagation\n",
|
||||
"## Back propagation\n",
|
||||
"\n",
|
||||
"The next step is to decide how the parameters should be changed such that they minimize the cost function.\n",
|
||||
"\n",
|
||||
@@ -630,7 +636,7 @@
|
||||
"The value of $\\lambda$ decides how large steps the algorithm must take\n",
|
||||
"in the direction of $ \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})$.\n",
|
||||
"The notation $\\nabla_{\\boldsymbol{\\omega}}$ express the gradient with respect\n",
|
||||
"to the elements in $\\boldsymbol{\\omega$.\n",
|
||||
"to the elements in $\\boldsymbol{\\omega}$.\n",
|
||||
"\n",
|
||||
"In our case, we have to minimize the cost function $C(\\boldsymbol{x}, P)$ with\n",
|
||||
"respect to the two sets of weights and biases, that is for the hidden\n",
|
||||
@@ -1027,8 +1033,8 @@
|
||||
"\n",
|
||||
"In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability\n",
|
||||
"and high execution time (this might be more apparent in the examples solving PDEs),\n",
|
||||
"a network has been constructed using TensorFlow also.\n",
|
||||
"For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.\n",
|
||||
"using a library like TensorFlow is recommended.\n",
|
||||
"Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n",
|
||||
"\n",
|
||||
"## Setting up the problem\n",
|
||||
"\n",
|
||||
@@ -1261,7 +1267,7 @@
|
||||
"source": [
|
||||
"## Using forward Euler to solve the ODE\n",
|
||||
"\n",
|
||||
"A straight-forward way of solving an ODE numerically, is to use Euler's method.\n",
|
||||
"A straightforward way of solving an ODE numerically, is to use Euler's method.\n",
|
||||
"\n",
|
||||
"Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\\Delta x$ from $x$:\n",
|
||||
"\n",
|
||||
@@ -1423,11 +1429,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Running the program gives\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow\n",
|
||||
"## Example: Solving the one dimensional Poisson equation\n",
|
||||
"\n",
|
||||
"The Poisson equation for $g(x)$ in one dimension is"
|
||||
]
|
||||
@@ -1845,7 +1847,7 @@
|
||||
"f(x_{N_x - 3}) \\\\\n",
|
||||
"f(x_{N_x - 2})\n",
|
||||
"\\end{pmatrix} \\\\\n",
|
||||
"A\\boldsymbol{g} &= \\boldsymbol{f}\n",
|
||||
"\\boldsymbol{A}\\boldsymbol{g} &= \\boldsymbol{f},\n",
|
||||
"\\end{aligned}\n",
|
||||
"$$"
|
||||
]
|
||||
@@ -1856,6 +1858,8 @@
|
||||
"source": [
|
||||
"which makes it possible to solve for the vector $\\boldsymbol{g}$.\n",
|
||||
"\n",
|
||||
"## Setting up the code\n",
|
||||
"\n",
|
||||
"We can then compare the result from this numerical scheme with the output from our network using Autograd:"
|
||||
]
|
||||
},
|
||||
@@ -2059,29 +2063,16 @@
|
||||
" plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The program prints out:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
" The max absolute difference between the analytical solution and DNN Autograd: 0.000464088\n",
|
||||
" The max absolute difference between the analytical solution and numerical scheme: 0.00266858\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Partial Differential Equations\n",
|
||||
"\n",
|
||||
"A partial differential equation (PDE) has a solution here the function is defined by multiple variables.\n",
|
||||
"The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.\n",
|
||||
"A partial differential equation (PDE) has a solution here the function\n",
|
||||
"is defined by multiple variables. The equation may involve all kinds\n",
|
||||
"of combinations of which variables the function is differentiated with\n",
|
||||
"respect to.\n",
|
||||
"\n",
|
||||
"In general, a partial differential equation for a function $g(x_1,\\dots,x_N)$ with $N$ variables may be expressed as"
|
||||
]
|
||||
@@ -2153,7 +2144,7 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"c\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n",
|
||||
"C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
@@ -2163,7 +2154,7 @@
|
||||
"source": [
|
||||
"## More details\n",
|
||||
"\n",
|
||||
"If we let $\\boldsymbol{x = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:"
|
||||
"If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -2171,7 +2162,7 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"c\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x) }{\\partial x_N^n} \\right) \\right)^2\n",
|
||||
"C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
@@ -2187,7 +2178,7 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"c\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2\n",
|
||||
"C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
@@ -3073,6 +3064,7 @@
|
||||
"\n",
|
||||
"## Friday, Principal Component Analysis\n",
|
||||
"\n",
|
||||
"[Overview video](https://www.youtube.com/watch?v=fkf4IBRSeEc&ab_channel=SteveBrunton)\n",
|
||||
"\n",
|
||||
"## Basic ideas of the Principal Component Analysis (PCA)\n",
|
||||
"\n",
|
||||
|
||||
Reference in New Issue
Block a user