adding gates
This commit is contained in:
Morten Hjorth-Jensen
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
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('Examples of XOR, OR and AND gates',
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2,
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None,
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'examples-of-xor-or-and-and-gates'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
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</ul>
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</li>
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@@ -326,7 +331,7 @@ MathJax.Hub.Config({
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<center>[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b></center>
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<br>
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<p>
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<center><h4>Oct 7, 2021</h4></center> <!-- date -->
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<center><h4>Oct 8, 2021</h4></center> <!-- date -->
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<br>
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<p>
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@@ -350,7 +355,7 @@ MathJax.Hub.Config({
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<li><a href="._week40-bs008.html">9</a></li>
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<li><a href="._week40-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs001.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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||||
2,
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None,
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'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
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('Examples of XOR, OR and AND gates',
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2,
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None,
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'examples-of-xor-or-and-and-gates'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
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</ul>
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</li>
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@@ -340,7 +345,7 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
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<li><a href="._week40-bs009.html">10</a></li>
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<li><a href="._week40-bs010.html">11</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs002.html">»</a></li>
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</ul>
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||||
<!-- ------------------- end of main content --------------- -->
|
||||
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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||||
2,
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||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
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None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -330,7 +335,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week40-bs010.html">11</a></li>
|
||||
<li><a href="._week40-bs011.html">12</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
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<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs003.html">»</a></li>
|
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|
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -342,7 +347,7 @@ perform a parameter update.
|
||||
<li><a href="._week40-bs011.html">12</a></li>
|
||||
<li><a href="._week40-bs012.html">13</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs004.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -354,7 +359,7 @@ In our notes with SGD we mean stochastic gradient descent with mini-batches.
|
||||
<li><a href="._week40-bs012.html">13</a></li>
|
||||
<li><a href="._week40-bs013.html">14</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs005.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -343,7 +348,7 @@ $$
|
||||
<li><a href="._week40-bs013.html">14</a></li>
|
||||
<li><a href="._week40-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -346,7 +351,7 @@ minibatches. We denote these minibatches by \( B_k \) where
|
||||
<li><a href="._week40-bs014.html">15</a></li>
|
||||
<li><a href="._week40-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
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||||
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -353,7 +358,7 @@ $$
|
||||
<li><a href="._week40-bs015.html">16</a></li>
|
||||
<li><a href="._week40-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +352,7 @@ the number of minibatches, as exemplified in the code below.
|
||||
<li><a href="._week40-bs016.html">17</a></li>
|
||||
<li><a href="._week40-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -361,7 +366,7 @@ all \( n \) datapoints.
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||||
<li><a href="._week40-bs017.html">18</a></li>
|
||||
<li><a href="._week40-bs018.html">19</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
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<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs010.html">»</a></li>
|
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</ul>
|
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<!-- ------------------- end of main content --------------- -->
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +352,7 @@ gave the lowest value.
|
||||
<li><a href="._week40-bs018.html">19</a></li>
|
||||
<li><a href="._week40-bs019.html">20</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs011.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -381,7 +386,7 @@ We note that we have defined several hyperparameters. These are now the number o
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<li><a href="._week40-bs019.html">20</a></li>
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<li><a href="._week40-bs020.html">21</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
|
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<li><a href="._week40-bs012.html">»</a></li>
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</ul>
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
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||||
None,
|
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'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -405,7 +410,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week40-bs020.html">21</a></li>
|
||||
<li><a href="._week40-bs021.html">22</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs013.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
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||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -367,7 +372,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
|
||||
<li><a href="._week40-bs021.html">22</a></li>
|
||||
<li><a href="._week40-bs022.html">23</a></li>
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||||
<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs014.html">»</a></li>
|
||||
</ul>
|
||||
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||||
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -360,7 +365,7 @@ $$
|
||||
<li><a href="._week40-bs022.html">23</a></li>
|
||||
<li><a href="._week40-bs023.html">24</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs015.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -386,7 +391,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
|
||||
<li><a href="._week40-bs023.html">24</a></li>
|
||||
<li><a href="._week40-bs024.html">25</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs016.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -358,7 +363,7 @@ ADAM.
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||||
<li><a href="._week40-bs024.html">25</a></li>
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<li><a href="._week40-bs025.html">26</a></li>
|
||||
<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs017.html">»</a></li>
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<!-- ------------------- end of main content --------------- -->
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -361,7 +366,7 @@ learning rate for flat directions.
|
||||
<li><a href="._week40-bs025.html">26</a></li>
|
||||
<li><a href="._week40-bs026.html">27</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
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<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs018.html">»</a></li>
|
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</ul>
|
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<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -379,7 +384,7 @@ $$
|
||||
<li><a href="._week40-bs026.html">27</a></li>
|
||||
<li><a href="._week40-bs027.html">28</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs019.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -344,7 +349,7 @@ Geron's text, see chapter 11, has several interesting discussions.
|
||||
<li><a href="._week40-bs027.html">28</a></li>
|
||||
<li><a href="._week40-bs028.html">29</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs020.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -415,7 +420,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week40-bs028.html">29</a></li>
|
||||
<li><a href="._week40-bs029.html">30</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs021.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -363,7 +368,7 @@ grad_analytical <span style="color: #666666">=</span> <span style="color: #66666
|
||||
<li><a href="._week40-bs029.html">30</a></li>
|
||||
<li><a href="._week40-bs030.html">31</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs022.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -380,7 +385,7 @@ Note that the grad function will not produce the true gradient of the function.
|
||||
<li><a href="._week40-bs030.html">31</a></li>
|
||||
<li><a href="._week40-bs031.html">32</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs023.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -364,7 +369,7 @@ could expect form a gradient-evaluting function.
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||||
<li><a href="._week40-bs031.html">32</a></li>
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<li><a href="._week40-bs032.html">33</a></li>
|
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<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
|
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs024.html">»</a></li>
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -356,7 +361,7 @@ f4_grad_analytical <span style="color: #666666">=</span> x<span style="color: #6
|
||||
<li><a href="._week40-bs032.html">33</a></li>
|
||||
<li><a href="._week40-bs033.html">34</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs025.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -353,7 +358,7 @@ x <span style="color: #666666">=</span> <span style="color: #666666">2.7</span>
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<li><a href="._week40-bs033.html">34</a></li>
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<li><a href="._week40-bs034.html">35</a></li>
|
||||
<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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||||
<li><a href="._week40-bs026.html">»</a></li>
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|
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
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||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
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|
||||
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|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -376,7 +381,7 @@ f6_grad_analytical <span style="color: #666666">=</span> <span style="color: #66
|
||||
<li><a href="._week40-bs034.html">35</a></li>
|
||||
<li><a href="._week40-bs035.html">36</a></li>
|
||||
<li><a href="">...</a></li>
|
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<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs027.html">»</a></li>
|
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|
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -368,7 +373,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
|
||||
<li><a href="._week40-bs035.html">36</a></li>
|
||||
<li><a href="._week40-bs036.html">37</a></li>
|
||||
<li><a href="">...</a></li>
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||||
<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs028.html">»</a></li>
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|
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
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('Examples of XOR, OR and AND gates',
|
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2,
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None,
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'examples-of-xor-or-and-and-gates'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -356,7 +361,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
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<li><a href="._week40-bs036.html">37</a></li>
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<li><a href="._week40-bs037.html">38</a></li>
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||||
<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs029.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -372,7 +377,7 @@ x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>a
|
||||
<li><a href="._week40-bs037.html">38</a></li>
|
||||
<li><a href="._week40-bs038.html">39</a></li>
|
||||
<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
|
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<li><a href="._week40-bs030.html">»</a></li>
|
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</ul>
|
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<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -343,7 +348,7 @@ a <span style="color: #666666">/=</span>b
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<li><a href="._week40-bs038.html">39</a></li>
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<li><a href="._week40-bs039.html">40</a></li>
|
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<li><a href="">...</a></li>
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<li><a href="._week40-bs062.html">63</a></li>
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<li><a href="._week40-bs063.html">64</a></li>
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<li><a href="._week40-bs031.html">»</a></li>
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|
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@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
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('Mathematical model', 2, None, 'mathematical-model'),
|
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('Mathematical model', 2, None, 'mathematical-model'),
|
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@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -341,7 +346,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week40-bs039.html">40</a></li>
|
||||
<li><a href="._week40-bs040.html">41</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
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<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs032.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -344,7 +349,7 @@ a weight variable.
|
||||
<li><a href="._week40-bs040.html">41</a></li>
|
||||
<li><a href="._week40-bs041.html">42</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs033.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -393,7 +398,7 @@ humanities to life science and medicine.
|
||||
<li><a href="._week40-bs041.html">42</a></li>
|
||||
<li><a href="._week40-bs042.html">43</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs034.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -357,7 +362,7 @@ methods we discussed earlier.
|
||||
<li><a href="._week40-bs042.html">43</a></li>
|
||||
<li><a href="._week40-bs043.html">44</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs035.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -348,7 +353,7 @@ to <em>all</em> nodes in the subsequent layer, making this a so-called
|
||||
<li><a href="._week40-bs043.html">44</a></li>
|
||||
<li><a href="._week40-bs044.html">45</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs036.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -356,7 +361,7 @@ recognition.
|
||||
<li><a href="._week40-bs044.html">45</a></li>
|
||||
<li><a href="._week40-bs045.html">46</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs037.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +352,7 @@ especially well-suited for handwriting and speech recognition.
|
||||
<li><a href="._week40-bs045.html">46</a></li>
|
||||
<li><a href="._week40-bs046.html">47</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs038.html">»</a></li>
|
||||
</ul>
|
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<!-- ------------------- end of main content --------------- -->
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||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
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2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -347,7 +352,7 @@ type of NN due the unusual activation functions.
|
||||
<li><a href="._week40-bs046.html">47</a></li>
|
||||
<li><a href="._week40-bs047.html">48</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs039.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -344,7 +349,7 @@ Such networks are often called <em>multilayer perceptrons</em> (MLPs).
|
||||
<li><a href="._week40-bs047.html">48</a></li>
|
||||
<li><a href="._week40-bs048.html">49</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs040.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -348,7 +353,7 @@ as to not restrict the range of output values.
|
||||
<li><a href="._week40-bs048.html">49</a></li>
|
||||
<li><a href="._week40-bs049.html">50</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs041.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -342,7 +347,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week40-bs049.html">50</a></li>
|
||||
<li><a href="._week40-bs050.html">51</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs042.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,20 +312,47 @@ MathJax.Hub.Config({
|
||||
<a name="part0042"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
<h2 id="examples-of-xor-or-and-and-gates" class="anchor">Examples of XOR, OR and AND gates </h2>
|
||||
|
||||
<p>
|
||||
The output \( y \) is produced via the activation function \( f \)
|
||||
$$
|
||||
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
|
||||
$$
|
||||
Let us first try to fit various gates using standard linear regression
|
||||
|
||||
This function receives \( x_i \) as inputs.
|
||||
Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \).
|
||||
In an FFNN of such neurons, the <em>inputs</em> \( x_i \) are the <em>outputs</em> of
|
||||
the neurons in the preceding layer. Furthermore, an MLP is
|
||||
fully-connected, which means that each neuron receives a weighted sum
|
||||
of the outputs of <em>all</em> neurons in the previous layer.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">"""</span>
|
||||
<span style="color: #BA2121; font-style: italic">Simple code that tests XOR, OR and AND gates with linear regression</span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #408080; font-style: italic"># Design matrix</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The invers of X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The XOR gate </span>
|
||||
yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
|
||||
ThetaXOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The OR gate </span>
|
||||
yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
|
||||
ThetaOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The OR gate </span>
|
||||
yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
|
||||
ThetaAND <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
What is happening here?
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -348,7 +380,7 @@ of the outputs of <em>all</em> neurons in the previous layer.
|
||||
<li><a href="._week40-bs050.html">51</a></li>
|
||||
<li><a href="._week40-bs051.html">52</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs043.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -310,46 +315,17 @@ MathJax.Hub.Config({
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
|
||||
|
||||
The output \( y \) is produced via the activation function \( f \)
|
||||
$$
|
||||
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
|
||||
\tag{7}
|
||||
\end{equation}
|
||||
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
|
||||
$$
|
||||
|
||||
<p>
|
||||
Here \( b_i \) is the so-called bias which is normally needed in
|
||||
case of zero activation weights or inputs. How to fix the biases and
|
||||
the weights will be discussed below. The value of \( z_i^1 \) is the
|
||||
argument to the activation function \( f_i \) of each node \( i \), The
|
||||
variable \( M \) stands for all possible inputs to a given node \( i \) in the
|
||||
first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
|
||||
\tag{8}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where we assume that all nodes in the same layer have identical
|
||||
activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
|
||||
In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
|
||||
\tag{9}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( N_l \) is the number of nodes in layer \( l \). When the output of
|
||||
all the nodes in the first hidden layer are computed, the values of
|
||||
the subsequent layer can be calculated and so forth until the output
|
||||
is obtained.
|
||||
This function receives \( x_i \) as inputs.
|
||||
Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \).
|
||||
In an FFNN of such neurons, the <em>inputs</em> \( x_i \) are the <em>outputs</em> of
|
||||
the neurons in the preceding layer. Furthermore, an MLP is
|
||||
fully-connected, which means that each neuron receives a weighted sum
|
||||
of the outputs of <em>all</em> neurons in the previous layer.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -377,7 +353,7 @@ is obtained.
|
||||
<li><a href="._week40-bs051.html">52</a></li>
|
||||
<li><a href="._week40-bs052.html">53</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs044.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -310,29 +315,47 @@ MathJax.Hub.Config({
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
The output of neuron \( i \) in layer 2 is thus,
|
||||
First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
|
||||
\tag{10}\\
|
||||
&= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
|
||||
\tag{11}
|
||||
\end{align}
|
||||
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
|
||||
\tag{7}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
|
||||
<p>
|
||||
Here \( b_i \) is the so-called bias which is normally needed in
|
||||
case of zero activation weights or inputs. How to fix the biases and
|
||||
the weights will be discussed below. The value of \( z_i^1 \) is the
|
||||
argument to the activation function \( f_i \) of each node \( i \), The
|
||||
variable \( M \) stands for all possible inputs to a given node \( i \) in the
|
||||
first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
|
||||
\tag{12}\\
|
||||
&= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
|
||||
+ b_1^3\right]
|
||||
\tag{13}
|
||||
\end{align}
|
||||
\begin{equation}
|
||||
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
|
||||
\tag{8}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where we assume that all nodes in the same layer have identical
|
||||
activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
|
||||
In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
|
||||
\tag{9}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( N_l \) is the number of nodes in layer \( l \). When the output of
|
||||
all the nodes in the first hidden layer are computed, the values of
|
||||
the subsequent layer can be calculated and so forth until the output
|
||||
is obtained.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -359,7 +382,7 @@ $$
|
||||
<li><a href="._week40-bs052.html">53</a></li>
|
||||
<li><a href="._week40-bs053.html">54</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs045.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -310,19 +315,28 @@ MathJax.Hub.Config({
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
We can generalize this expression to an MLP with \( l \) hidden
|
||||
layers. The complete functional form is,
|
||||
The output of neuron \( i \) in layer 2 is thus,
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
|
||||
\tag{14}
|
||||
y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
|
||||
\tag{10}\\
|
||||
&= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
|
||||
\tag{11}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
which illustrates a basic property of MLPs: The only independent
|
||||
variables are the input values \( x_n \).
|
||||
where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
|
||||
|
||||
$$
|
||||
\begin{align}
|
||||
y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
|
||||
\tag{12}\\
|
||||
&= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
|
||||
+ b_1^3\right]
|
||||
\tag{13}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -350,7 +364,7 @@ variables are the input values \( x_n \).
|
||||
<li><a href="._week40-bs053.html">54</a></li>
|
||||
<li><a href="._week40-bs054.html">55</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs046.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -310,28 +315,19 @@ MathJax.Hub.Config({
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
This confirms that an MLP, despite its quite convoluted mathematical
|
||||
form, is nothing more than an analytic function, specifically a
|
||||
mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
|
||||
\hat{y} \in \mathbb{R}^m \).
|
||||
|
||||
<p>
|
||||
Furthermore, the flexibility and universality of an MLP can be
|
||||
illustrated by realizing that the expression is essentially a nested
|
||||
sum of scaled activation functions of the form
|
||||
We can generalize this expression to an MLP with \( l \) hidden
|
||||
layers. The complete functional form is,
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
f(x) = c_1 f(c_2 x + c_3) + c_4
|
||||
\tag{15}
|
||||
\end{equation}
|
||||
\begin{align}
|
||||
&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
|
||||
\tag{14}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where the parameters \( c_i \) are weights and biases. By adjusting these
|
||||
parameters, the activation functions can be shifted up and down or
|
||||
left and right, change slope or be rescaled which is the key to the
|
||||
flexibility of a neural network.
|
||||
which illustrates a basic property of MLPs: The only independent
|
||||
variables are the input values \( x_n \).
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -359,7 +355,7 @@ flexibility of a neural network.
|
||||
<li><a href="._week40-bs054.html">55</a></li>
|
||||
<li><a href="._week40-bs055.html">56</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs047.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,42 +312,32 @@ MathJax.Hub.Config({
|
||||
<a name="part0047"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="matrix-vector-notation" class="anchor">Matrix-vector notation </h3>
|
||||
<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
We can introduce a more convenient notation for the activations in an A NN.
|
||||
This confirms that an MLP, despite its quite convoluted mathematical
|
||||
form, is nothing more than an analytic function, specifically a
|
||||
mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
|
||||
\hat{y} \in \mathbb{R}^m \).
|
||||
|
||||
<p>
|
||||
Additionally, we can represent the biases and activations
|
||||
as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
|
||||
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
|
||||
Furthermore, the flexibility and universality of an MLP can be
|
||||
illustrated by realizing that the expression is essentially a nested
|
||||
sum of scaled activation functions of the form
|
||||
|
||||
<p>
|
||||
We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
|
||||
With this notation, the sum becomes a matrix-vector multiplication, and we can write
|
||||
the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
|
||||
$$
|
||||
\begin{equation}
|
||||
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
|
||||
f_2\left(\left[\begin{array}{ccc}
|
||||
w^2_{11} &w^2_{12} &w^2_{13} \\
|
||||
w^2_{21} &w^2_{22} &w^2_{23} \\
|
||||
w^2_{31} &w^2_{32} &w^2_{33} \\
|
||||
\end{array} \right] \cdot
|
||||
\left[\begin{array}{c}
|
||||
y^1_1 \\
|
||||
y^1_2 \\
|
||||
y^1_3 \\
|
||||
\end{array}\right] +
|
||||
\left[\begin{array}{c}
|
||||
b^2_1 \\
|
||||
b^2_2 \\
|
||||
b^2_3 \\
|
||||
\end{array}\right]\right).
|
||||
\tag{16}
|
||||
f(x) = c_1 f(c_2 x + c_3) + c_4
|
||||
\tag{15}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
where the parameters \( c_i \) are weights and biases. By adjusting these
|
||||
parameters, the activation functions can be shifted up and down or
|
||||
left and right, change slope or be rescaled which is the key to the
|
||||
flexibility of a neural network.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -369,7 +364,7 @@ $$
|
||||
<li><a href="._week40-bs055.html">56</a></li>
|
||||
<li><a href="._week40-bs056.html">57</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs048.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,26 +312,42 @@ MathJax.Hub.Config({
|
||||
<a name="part0048"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="matrix-vector-notation-and-activation" class="anchor">Matrix-vector notation and activation </h3>
|
||||
<h3 id="matrix-vector-notation" class="anchor">Matrix-vector notation </h3>
|
||||
|
||||
<p>
|
||||
The activation of node \( i \) in layer 2 is
|
||||
We can introduce a more convenient notation for the activations in an A NN.
|
||||
|
||||
<p>
|
||||
Additionally, we can represent the biases and activations
|
||||
as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
|
||||
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
|
||||
|
||||
<p>
|
||||
We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
|
||||
With this notation, the sum becomes a matrix-vector multiplication, and we can write
|
||||
the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
|
||||
$$
|
||||
\begin{equation}
|
||||
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
|
||||
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
|
||||
\tag{17}
|
||||
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
|
||||
f_2\left(\left[\begin{array}{ccc}
|
||||
w^2_{11} &w^2_{12} &w^2_{13} \\
|
||||
w^2_{21} &w^2_{22} &w^2_{23} \\
|
||||
w^2_{31} &w^2_{32} &w^2_{33} \\
|
||||
\end{array} \right] \cdot
|
||||
\left[\begin{array}{c}
|
||||
y^1_1 \\
|
||||
y^1_2 \\
|
||||
y^1_3 \\
|
||||
\end{array}\right] +
|
||||
\left[\begin{array}{c}
|
||||
b^2_1 \\
|
||||
b^2_2 \\
|
||||
b^2_3 \\
|
||||
\end{array}\right]\right).
|
||||
\tag{16}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
This is not just a convenient and compact notation, but also a useful
|
||||
and intuitive way to think about MLPs: The output is calculated by a
|
||||
series of matrix-vector multiplications and vector additions that are
|
||||
used as input to the activation functions. For each operation
|
||||
\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -353,7 +374,7 @@ used as input to the activation functions. For each operation
|
||||
<li><a href="._week40-bs056.html">57</a></li>
|
||||
<li><a href="._week40-bs057.html">58</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs049.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,21 +312,27 @@ MathJax.Hub.Config({
|
||||
<a name="part0049"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="activation-functions" class="anchor">Activation functions </h3>
|
||||
<h3 id="matrix-vector-notation-and-activation" class="anchor">Matrix-vector notation and activation </h3>
|
||||
|
||||
<p>
|
||||
A property that characterizes a neural network, other than its
|
||||
connectivity, is the choice of activation function(s). As described
|
||||
in, the following restrictions are imposed on an activation function
|
||||
for a FFNN to fulfill the universal approximation theorem
|
||||
The activation of node \( i \) in layer 2 is
|
||||
|
||||
<ul>
|
||||
<li> Non-constant</li>
|
||||
<li> Bounded</li>
|
||||
<li> Monotonically-increasing</li>
|
||||
<li> Continuous</li>
|
||||
</ul>
|
||||
$$
|
||||
\begin{equation}
|
||||
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
|
||||
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
|
||||
\tag{17}
|
||||
\end{equation}
|
||||
$$
|
||||
|
||||
<p>
|
||||
This is not just a convenient and compact notation, but also a useful
|
||||
and intuitive way to think about MLPs: The output is calculated by a
|
||||
series of matrix-vector multiplications and vector additions that are
|
||||
used as input to the activation functions. For each operation
|
||||
\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -347,7 +358,7 @@ for a FFNN to fulfill the universal approximation theorem
|
||||
<li><a href="._week40-bs057.html">58</a></li>
|
||||
<li><a href="._week40-bs058.html">59</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs050.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,29 +312,21 @@ MathJax.Hub.Config({
|
||||
<a name="part0050"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="activation-functions-logistic-and-hyperbolic-ones" class="anchor">Activation functions, Logistic and Hyperbolic ones </h3>
|
||||
<h3 id="activation-functions" class="anchor">Activation functions </h3>
|
||||
|
||||
<p>
|
||||
The second requirement excludes all linear functions. Furthermore, in
|
||||
a MLP with only linear activation functions, each layer simply
|
||||
performs a linear transformation of its inputs.
|
||||
A property that characterizes a neural network, other than its
|
||||
connectivity, is the choice of activation function(s). As described
|
||||
in, the following restrictions are imposed on an activation function
|
||||
for a FFNN to fulfill the universal approximation theorem
|
||||
|
||||
<p>
|
||||
Regardless of the number of layers, the output of the NN will be
|
||||
nothing but a linear function of the inputs. Thus we need to introduce
|
||||
some kind of non-linearity to the NN to be able to fit non-linear
|
||||
functions Typical examples are the logistic <em>Sigmoid</em>
|
||||
<ul>
|
||||
<li> Non-constant</li>
|
||||
<li> Bounded</li>
|
||||
<li> Monotonically-increasing</li>
|
||||
<li> Continuous</li>
|
||||
</ul>
|
||||
|
||||
$$
|
||||
f(x) = \frac{1}{1 + e^{-x}},
|
||||
$$
|
||||
|
||||
and the <em>hyperbolic tangent</em> function
|
||||
$$
|
||||
f(x) = \tanh(x)
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -355,7 +352,7 @@ $$
|
||||
<li><a href="._week40-bs058.html">59</a></li>
|
||||
<li><a href="._week40-bs059.html">60</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs051.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,90 +312,28 @@ MathJax.Hub.Config({
|
||||
<a name="part0051"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="relevance" class="anchor">Relevance </h3>
|
||||
<h3 id="activation-functions-logistic-and-hyperbolic-ones" class="anchor">Activation functions, Logistic and Hyperbolic ones </h3>
|
||||
|
||||
<p>
|
||||
The <em>sigmoid</em> function are more biologically plausible because the
|
||||
output of inactive neurons are zero. Such activation function are
|
||||
called <em>one-sided</em>. However, it has been shown that the hyperbolic
|
||||
tangent performs better than the sigmoid for training MLPs. has
|
||||
become the most popular for <em>deep neural networks</em>
|
||||
The second requirement excludes all linear functions. Furthermore, in
|
||||
a MLP with only linear activation functions, each layer simply
|
||||
performs a linear transformation of its inputs.
|
||||
|
||||
<p>
|
||||
Regardless of the number of layers, the output of the NN will be
|
||||
nothing but a linear function of the inputs. Thus we need to introduce
|
||||
some kind of non-linearity to the NN to be able to fit non-linear
|
||||
functions Typical examples are the logistic <em>Sigmoid</em>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">"""The sigmoid function (or the logistic curve) is a </span>
|
||||
<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
|
||||
<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
|
||||
<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
|
||||
$$
|
||||
f(x) = \frac{1}{1 + e^{-x}},
|
||||
$$
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
|
||||
and the <em>hyperbolic tangent</em> function
|
||||
$$
|
||||
f(x) = \tanh(x)
|
||||
$$
|
||||
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
|
||||
sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
|
||||
sigma <span style="color: #666666">=</span> sigma_fn(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, sigma)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sigmoid function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Step Function"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
|
||||
step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">>=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
|
||||
step <span style="color: #666666">=</span> step_fn(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, step)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'step function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Sine Function"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
|
||||
t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, t)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sine function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Plots a graph of the squashing function used by a rectified linear</span>
|
||||
<span style="color: #BA2121; font-style: italic">unit"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
|
||||
zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
|
||||
y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, y)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'Rectified linear unit'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -417,7 +360,7 @@ plt<span style="color: #666666">.</span>show()
|
||||
<li><a href="._week40-bs059.html">60</a></li>
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs052.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,41 +312,90 @@ MathJax.Hub.Config({
|
||||
<a name="part0052"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="the-multilayer-perceptron-mlp" class="anchor">The multilayer perceptron (MLP) </h2>
|
||||
<h3 id="relevance" class="anchor">Relevance </h3>
|
||||
|
||||
<p>
|
||||
The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
|
||||
|
||||
<ol>
|
||||
<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
|
||||
<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
|
||||
<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
|
||||
</ol>
|
||||
|
||||
As a convention it is normal to call a network with one layer of input units, one layer of hidden
|
||||
units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
|
||||
The <em>sigmoid</em> function are more biologically plausible because the
|
||||
output of inactive neurons are zero. Such activation function are
|
||||
called <em>one-sided</em>. However, it has been shown that the hyperbolic
|
||||
tangent performs better than the sigmoid for training MLPs. has
|
||||
become the most popular for <em>deep neural networks</em>
|
||||
|
||||
<p>
|
||||
For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
|
||||
Hereafter we will call the various entities of a layer for nodes.
|
||||
There are also no connections within a single layer.
|
||||
|
||||
<p>
|
||||
The number of input nodes does not need to equal the number of output
|
||||
nodes. This applies also to the hidden layers. Each layer may have its
|
||||
own number of nodes and activation functions.
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">"""The sigmoid function (or the logistic curve) is a </span>
|
||||
<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
|
||||
<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
|
||||
<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
|
||||
|
||||
<p>
|
||||
The hidden layers have their name from the fact that they are not
|
||||
linked to observables and as we will see below when we define the
|
||||
so-called activation \( \hat{z} \), we can think of this as a basis
|
||||
expansion of the original inputs \( \hat{x} \). The difference however
|
||||
between neural networks and say linear regression is that now these
|
||||
basis functions (which will correspond to the weights in the network)
|
||||
are learned from data. This results in an important difference between
|
||||
neural networks and deep learning approaches on one side and methods
|
||||
like logistic regression or linear regression and their modifications on the other side.
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
|
||||
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
|
||||
sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
|
||||
sigma <span style="color: #666666">=</span> sigma_fn(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, sigma)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sigmoid function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Step Function"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
|
||||
step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">>=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
|
||||
step <span style="color: #666666">=</span> step_fn(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, step)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'step function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Sine Function"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
|
||||
t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, t)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sine function'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #BA2121; font-style: italic">"""Plots a graph of the squashing function used by a rectified linear</span>
|
||||
<span style="color: #BA2121; font-style: italic">unit"""</span>
|
||||
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
|
||||
zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
|
||||
y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
|
||||
|
||||
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
|
||||
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
|
||||
ax<span style="color: #666666">.</span>plot(z, y)
|
||||
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
|
||||
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
|
||||
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'Rectified linear unit'</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -368,7 +422,7 @@ like logistic regression or linear regression and their modifications on the oth
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs053.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,30 +312,40 @@ MathJax.Hub.Config({
|
||||
<a name="part0053"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="from-one-to-many-layers-the-universal-approximation-theorem" class="anchor">From one to many layers, the universal approximation theorem </h2>
|
||||
<h2 id="the-multilayer-perceptron-mlp" class="anchor">The multilayer perceptron (MLP) </h2>
|
||||
|
||||
<p>
|
||||
A neural network with only one layer, what we called the simple
|
||||
perceptron, is best suited if we have a standard binary model with
|
||||
clear (linear) boundaries between the outcomes. As such it could
|
||||
equally well be replaced by standard linear regression or logistic
|
||||
regression. Networks with one or more hidden layers approximate
|
||||
systems with more complex boundaries.
|
||||
The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
|
||||
|
||||
<ol>
|
||||
<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
|
||||
<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
|
||||
<li> The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.</li>
|
||||
</ol>
|
||||
|
||||
As a convention it is normal to call a network with one layer of input units, one layer of hidden
|
||||
units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
|
||||
|
||||
<p>
|
||||
As stated earlier,
|
||||
an important theorem in studies of neural networks, restated without
|
||||
proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_self">universal approximation
|
||||
theorem</a>.
|
||||
For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
|
||||
Hereafter we will call the various entities of a layer for nodes.
|
||||
There are also no connections within a single layer.
|
||||
|
||||
<p>
|
||||
It states that a feed-forward network with a single hidden layer
|
||||
containing a finite number of neurons can approximate continuous
|
||||
functions on compact subsets of real functions. The theorem thus
|
||||
states that simple neural networks can represent a wide variety of
|
||||
interesting functions when given appropriate parameters. It is the
|
||||
multilayer feedforward architecture itself which gives neural networks
|
||||
the potential of being universal approximators.
|
||||
The number of input nodes does not need to equal the number of output
|
||||
nodes. This applies also to the hidden layers. Each layer may have its
|
||||
own number of nodes and activation functions.
|
||||
|
||||
<p>
|
||||
The hidden layers have their name from the fact that they are not
|
||||
linked to observables and as we will see below when we define the
|
||||
so-called activation \( \hat{z} \), we can think of this as a basis
|
||||
expansion of the original inputs \( \hat{x} \). The difference however
|
||||
between neural networks and say linear regression is that now these
|
||||
basis functions (which will correspond to the weights in the network)
|
||||
are learned from data. This results in an important difference between
|
||||
neural networks and deep learning approaches on one side and methods
|
||||
like logistic regression or linear regression and their modifications on the other side.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -357,6 +372,8 @@ the potential of being universal approximators.
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs054.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,33 +312,30 @@ MathJax.Hub.Config({
|
||||
<a name="part0054"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" class="anchor">Deriving the back propagation code for a multilayer perceptron model </h2>
|
||||
<h2 id="from-one-to-many-layers-the-universal-approximation-theorem" class="anchor">From one to many layers, the universal approximation theorem </h2>
|
||||
|
||||
<p>
|
||||
As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
|
||||
The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
|
||||
This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_self">back propagation algorithm</a>.
|
||||
A neural network with only one layer, what we called the simple
|
||||
perceptron, is best suited if we have a standard binary model with
|
||||
clear (linear) boundaries between the outcomes. As such it could
|
||||
equally well be replaced by standard linear regression or logistic
|
||||
regression. Networks with one or more hidden layers approximate
|
||||
systems with more complex boundaries.
|
||||
|
||||
<p>
|
||||
The questions we want to ask are how do changes in the biases and the
|
||||
weights in our network change the cost function and how can we use the
|
||||
final output to modify the weights?
|
||||
As stated earlier,
|
||||
an important theorem in studies of neural networks, restated without
|
||||
proof here, is the <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_self">universal approximation
|
||||
theorem</a>.
|
||||
|
||||
<p>
|
||||
To derive these equations let us start with a plain regression problem
|
||||
and define our cost function as
|
||||
|
||||
$$
|
||||
{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
where the $t_i$s are our \( n \) targets (the values we want to
|
||||
reproduce), while the outputs of the network after having propagated
|
||||
all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
|
||||
how the basic equations arising from the back propagation algorithm
|
||||
can be modified in order to study classification problems with \( K \)
|
||||
classes.
|
||||
It states that a feed-forward network with a single hidden layer
|
||||
containing a finite number of neurons can approximate continuous
|
||||
functions on compact subsets of real functions. The theorem thus
|
||||
states that simple neural networks can represent a wide variety of
|
||||
interesting functions when given appropriate parameters. It is the
|
||||
multilayer feedforward architecture itself which gives neural networks
|
||||
the potential of being universal approximators.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -359,6 +361,7 @@ classes.
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs055.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,40 +312,33 @@ MathJax.Hub.Config({
|
||||
<a name="part0055"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="definitions" class="anchor">Definitions </h2>
|
||||
<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" class="anchor">Deriving the back propagation code for a multilayer perceptron model </h2>
|
||||
|
||||
<p>
|
||||
With our definition of the targets \( \hat{t} \), the outputs of the
|
||||
network \( \hat{y} \) and the inputs \( \hat{x} \) we
|
||||
define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
|
||||
\( l \)-th layer as a function of the bias, the weights which add up from
|
||||
the previous layer \( l-1 \) and the forward passes/outputs
|
||||
\( \hat{a}^{l-1} \) from the previous layer as
|
||||
As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
|
||||
The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
|
||||
This leads us to the famous <a href="https://www.nature.com/articles/323533a0" target="_self">back propagation algorithm</a>.
|
||||
|
||||
<p>
|
||||
The questions we want to ask are how do changes in the biases and the
|
||||
weights in our network change the cost function and how can we use the
|
||||
final output to modify the weights?
|
||||
|
||||
<p>
|
||||
To derive these equations let us start with a plain regression problem
|
||||
and define our cost function as
|
||||
|
||||
$$
|
||||
z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
|
||||
{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
|
||||
represents the total number of nodes/neurons/units of layer \( l-1 \). The
|
||||
figure here illustrates this equation. We can rewrite this in a more
|
||||
compact form as the matrix-vector products we discussed earlier,
|
||||
|
||||
$$
|
||||
\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
|
||||
$$
|
||||
|
||||
<p>
|
||||
With the activation values \( \hat{z}^l \) we can in turn define the
|
||||
output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
|
||||
activation function. In the examples here we will use the sigmoid
|
||||
function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
|
||||
and their nodes. It means we have
|
||||
|
||||
$$
|
||||
a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
|
||||
$$
|
||||
where the $t_i$s are our \( n \) targets (the values we want to
|
||||
reproduce), while the outputs of the network after having propagated
|
||||
all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
|
||||
how the basic equations arising from the back propagation algorithm
|
||||
can be modified in order to study classification problems with \( K \)
|
||||
classes.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
@@ -365,6 +363,7 @@ $$
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs056.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,23 +312,39 @@ MathJax.Hub.Config({
|
||||
<a name="part0056"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="derivatives-and-the-chain-rule" class="anchor">Derivatives and the chain rule </h2>
|
||||
<h2 id="definitions" class="anchor">Definitions </h2>
|
||||
|
||||
<p>
|
||||
From the definition of the activation \( z_j^l \) we have
|
||||
$$
|
||||
\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
|
||||
$$
|
||||
With our definition of the targets \( \hat{t} \), the outputs of the
|
||||
network \( \hat{y} \) and the inputs \( \hat{x} \) we
|
||||
define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
|
||||
\( l \)-th layer as a function of the bias, the weights which add up from
|
||||
the previous layer \( l-1 \) and the forward passes/outputs
|
||||
\( \hat{a}^{l-1} \) from the previous layer as
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
|
||||
z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
|
||||
$$
|
||||
|
||||
<p>
|
||||
With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
|
||||
where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
|
||||
represents the total number of nodes/neurons/units of layer \( l-1 \). The
|
||||
figure here illustrates this equation. We can rewrite this in a more
|
||||
compact form as the matrix-vector products we discussed earlier,
|
||||
|
||||
$$
|
||||
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
|
||||
\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
|
||||
$$
|
||||
|
||||
<p>
|
||||
With the activation values \( \hat{z}^l \) we can in turn define the
|
||||
output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
|
||||
activation function. In the examples here we will use the sigmoid
|
||||
function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
|
||||
and their nodes. It means we have
|
||||
|
||||
$$
|
||||
a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
@@ -348,6 +369,7 @@ $$
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs057.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,26 +312,23 @@ MathJax.Hub.Config({
|
||||
<a name="part0057"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="derivative-of-the-cost-function" class="anchor">Derivative of the cost function </h2>
|
||||
<h2 id="derivatives-and-the-chain-rule" class="anchor">Derivatives and the chain rule </h2>
|
||||
|
||||
<p>
|
||||
With these definitions we can now compute the derivative of the cost function in terms of the weights.
|
||||
From the definition of the activation \( z_j^l \) we have
|
||||
$$
|
||||
\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
|
||||
$$
|
||||
|
||||
<p>
|
||||
Let us specialize to the output layer \( l=L \). Our cost function is
|
||||
With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
|
||||
$$
|
||||
{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
|
||||
$$
|
||||
|
||||
The derivative of this function with respect to the weights is
|
||||
|
||||
$$
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
|
||||
$$
|
||||
|
||||
The last partial derivative can easily be computed and reads (by applying the chain rule)
|
||||
$$
|
||||
\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
|
||||
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
|
||||
$$
|
||||
|
||||
<p>
|
||||
@@ -350,6 +352,7 @@ $$
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs058.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,51 +312,26 @@ MathJax.Hub.Config({
|
||||
<a name="part0058"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="bringing-it-together-first-back-propagation-equation" class="anchor">Bringing it together, first back propagation equation </h2>
|
||||
<h2 id="derivative-of-the-cost-function" class="anchor">Derivative of the cost function </h2>
|
||||
|
||||
<p>
|
||||
We have thus
|
||||
$$
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
|
||||
$$
|
||||
With these definitions we can now compute the derivative of the cost function in terms of the weights.
|
||||
|
||||
<p>
|
||||
Defining
|
||||
Let us specialize to the output layer \( l=L \). Our cost function is
|
||||
$$
|
||||
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
|
||||
{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
|
||||
$$
|
||||
|
||||
and using the Hadamard product of two vectors we can write this as
|
||||
$$
|
||||
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This is an important expression. The second term on the right handside
|
||||
measures how fast the cost function is changing as a function of the $j$th
|
||||
output activation. If, for example, the cost function doesn't depend
|
||||
much on a particular output node \( j \), then \( \delta_j^L \) will be small,
|
||||
which is what we would expect. The first term on the right, measures
|
||||
how fast the activation function \( f \) is changing at a given activation
|
||||
value \( z_j^L \).
|
||||
|
||||
<p>
|
||||
Notice that everything in the above equations is easily computed. In
|
||||
particular, we compute \( z_j^L \) while computing the behaviour of the
|
||||
network, and it is only a small additional overhead to compute
|
||||
\( f'(z^L_j) \). The exact form of the derivative with respect to the
|
||||
output depends on the form of the cost function.
|
||||
However, provided the cost function is known there should be little
|
||||
trouble in calculating
|
||||
The derivative of this function with respect to the weights is
|
||||
|
||||
$$
|
||||
\frac{\partial {\cal C}}{\partial (a_j^L)}
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
|
||||
$$
|
||||
|
||||
<p>
|
||||
With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
|
||||
The last partial derivative can easily be computed and reads (by applying the chain rule)
|
||||
$$
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
|
||||
\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
|
||||
$$
|
||||
|
||||
<p>
|
||||
@@ -374,6 +354,7 @@ $$
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs059.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,21 +312,54 @@ MathJax.Hub.Config({
|
||||
<a name="part0059"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="derivatives-in-terms-of-z_j-l" class="anchor">Derivatives in terms of \( z_j^L \) </h2>
|
||||
<h2 id="bringing-it-together-first-back-propagation-equation" class="anchor">Bringing it together, first back propagation equation </h2>
|
||||
|
||||
<p>
|
||||
It is also easy to see that our previous equation can be written as
|
||||
|
||||
We have thus
|
||||
$$
|
||||
\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
|
||||
$$
|
||||
|
||||
which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
|
||||
<p>
|
||||
Defining
|
||||
$$
|
||||
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
|
||||
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
|
||||
$$
|
||||
|
||||
That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
|
||||
and using the Hadamard product of two vectors we can write this as
|
||||
$$
|
||||
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This is an important expression. The second term on the right handside
|
||||
measures how fast the cost function is changing as a function of the $j$th
|
||||
output activation. If, for example, the cost function doesn't depend
|
||||
much on a particular output node \( j \), then \( \delta_j^L \) will be small,
|
||||
which is what we would expect. The first term on the right, measures
|
||||
how fast the activation function \( f \) is changing at a given activation
|
||||
value \( z_j^L \).
|
||||
|
||||
<p>
|
||||
Notice that everything in the above equations is easily computed. In
|
||||
particular, we compute \( z_j^L \) while computing the behaviour of the
|
||||
network, and it is only a small additional overhead to compute
|
||||
\( f'(z^L_j) \). The exact form of the derivative with respect to the
|
||||
output depends on the form of the cost function.
|
||||
However, provided the cost function is known there should be little
|
||||
trouble in calculating
|
||||
|
||||
$$
|
||||
\frac{\partial {\cal C}}{\partial (a_j^L)}
|
||||
$$
|
||||
|
||||
<p>
|
||||
With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
|
||||
$$
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -340,6 +378,7 @@ That is, the error \( \delta_j^L \) is exactly equal to the rate of change of th
|
||||
<li><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs060.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -307,66 +312,21 @@ MathJax.Hub.Config({
|
||||
<a name="part0060"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="bringing-it-together" class="anchor">Bringing it together </h2>
|
||||
<h2 id="derivatives-in-terms-of-z_j-l" class="anchor">Derivatives in terms of \( z_j^L \) </h2>
|
||||
|
||||
<p>
|
||||
We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
|
||||
|
||||
<p>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
It is also easy to see that our previous equation can be written as
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
|
||||
\tag{18}
|
||||
\end{equation}
|
||||
\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
|
||||
$$
|
||||
|
||||
and
|
||||
which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
|
||||
$$
|
||||
\begin{equation}
|
||||
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
|
||||
\tag{19}
|
||||
\end{equation}
|
||||
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
\begin{equation}
|
||||
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
|
||||
\tag{20}
|
||||
\end{equation}
|
||||
$$
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
An interesting consequence of the above equations is that when the
|
||||
activation \( a_k^{L-1} \) is small, the gradient term, that is the
|
||||
derivative of the cost function with respect to the weights, will also
|
||||
tend to be small. We say then that the weight learns slowly, meaning
|
||||
that it changes slowly when we minimize the weights via say gradient
|
||||
descent. In this case we say the system learns slowly.
|
||||
|
||||
<p>
|
||||
Another interesting feature is that is when the activation function,
|
||||
represented by the sigmoid function here, is rather flat when we move towards
|
||||
its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
|
||||
cases, the derivatives of the activation function will also be close
|
||||
to zero, meaning again that the gradients will be small and the
|
||||
network learns slowly again.
|
||||
|
||||
<p>
|
||||
We need a fourth equation and we are set. We are going to propagate
|
||||
backwards in order to the determine the weights and biases. In order
|
||||
to do so we need to represent the error in the layer before the final
|
||||
one \( L-1 \) in terms of the errors in the final output layer.
|
||||
|
||||
<p>
|
||||
That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
<ul class="pagination">
|
||||
@@ -384,6 +344,7 @@ one \( L-1 \) in terms of the errors in the final output layer.
|
||||
<li class="active"><a href="._week40-bs060.html">61</a></li>
|
||||
<li><a href="._week40-bs061.html">62</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs061.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs039.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs040.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs041.html#illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptropn model and a multi-perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs046.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs042.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs047.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs048.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs049.html#matrix-vector-notation-and-activation" style="font-size: 80%;"> Matrix-vector notation and activation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs050.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs051.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs052.html#relevance" style="font-size: 80%;"> Relevance</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs053.html#the-multilayer-perceptron-mlp" style="font-size: 80%;"><b>The multilayer perceptron (MLP)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs054.html#from-one-to-many-layers-the-universal-approximation-theorem" style="font-size: 80%;"><b>From one to many layers, the universal approximation theorem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs055.html#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" style="font-size: 80%;"><b>Deriving the back propagation code for a multilayer perceptron model</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs056.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs057.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs058.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs059.html#bringing-it-together-first-back-propagation-equation" style="font-size: 80%;"><b>Bringing it together, first back propagation equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs060.html#derivatives-in-terms-of-z_j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs061.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs062.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week40-bs063.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the Back propagation algorithm</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -326,7 +331,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 7, 2021</h4></center> <!-- date -->
|
||||
<center><h4>Oct 8, 2021</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -350,7 +355,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._week40-bs008.html">9</a></li>
|
||||
<li><a href="._week40-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._week40-bs062.html">63</a></li>
|
||||
<li><a href="._week40-bs063.html">64</a></li>
|
||||
<li><a href="._week40-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b></center>
|
||||
<br>
|
||||
<p> <br>
|
||||
<center><h4>Oct 7, 2021</h4></center> <!-- date -->
|
||||
<center><h4>Oct 8, 2021</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -1438,6 +1438,51 @@ as to not restrict the range of output values.
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="examples-of-xor-or-and-and-gates">Examples of XOR, OR and AND gates </h2>
|
||||
|
||||
<p>
|
||||
Let us first try to fit various gates using standard linear regression
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%;"><span></span><span style="color: #CD5555">"""</span>
|
||||
<span style="color: #CD5555">Simple code that tests XOR, OR and AND gates with linear regression</span>
|
||||
<span style="color: #CD5555">"""</span>
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #228B22"># Design matrix</span>
|
||||
X = np.array([ [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>], [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>], [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>],[<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>]],dtype=np.float64)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The X.TX matrix:{</span>X.T @ X<span style="color: #CD5555">}"</span>)
|
||||
Xinv = np.linalg.pinv(X.T @ X)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The invers of X.TX matrix:{</span>Xinv<span style="color: #CD5555">}"</span>)
|
||||
|
||||
<span style="color: #228B22"># The XOR gate </span>
|
||||
yXOR = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span> ,<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>])
|
||||
ThetaXOR = Xinv @ X.T @ yXOR
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the XOR gate:{</span>ThetaXOR<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the XOR gate:{</span>X @ ThetaXOR<span style="color: #CD5555">}"</span>)
|
||||
|
||||
|
||||
<span style="color: #228B22"># The OR gate </span>
|
||||
yOR = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span> ,<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>])
|
||||
ThetaOR = Xinv @ X.T @ yOR
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the OR gate:{</span>ThetaOR<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the OR gate:{</span>X @ ThetaOR<span style="color: #CD5555">}"</span>)
|
||||
|
||||
|
||||
<span style="color: #228B22"># The OR gate </span>
|
||||
yAND = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span> ,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>])
|
||||
ThetaAND = Xinv @ X.T @ yAND
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the AND gate:{</span>ThetaAND<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the AND gate:{</span>X @ ThetaAND<span style="color: #CD5555">}"</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
What is happening here?
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="mathematical-model">Mathematical model </h2>
|
||||
|
||||
|
||||
@@ -158,6 +158,10 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -254,7 +258,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 7, 2021</h4></center> <!-- date -->
|
||||
<center><h4>Oct 8, 2021</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1491,6 +1495,51 @@ as to not restrict the range of output values.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="examples-of-xor-or-and-and-gates">Examples of XOR, OR and AND gates </h2>
|
||||
|
||||
<p>
|
||||
Let us first try to fit various gates using standard linear regression
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
|
||||
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%;"><span></span><span style="color: #CD5555">"""</span>
|
||||
<span style="color: #CD5555">Simple code that tests XOR, OR and AND gates with linear regression</span>
|
||||
<span style="color: #CD5555">"""</span>
|
||||
|
||||
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
|
||||
<span style="color: #228B22"># Design matrix</span>
|
||||
X = np.array([ [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span>], [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>], [<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>],[<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>]],dtype=np.float64)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The X.TX matrix:{</span>X.T @ X<span style="color: #CD5555">}"</span>)
|
||||
Xinv = np.linalg.pinv(X.T @ X)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The invers of X.TX matrix:{</span>Xinv<span style="color: #CD5555">}"</span>)
|
||||
|
||||
<span style="color: #228B22"># The XOR gate </span>
|
||||
yXOR = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span> ,<span style="color: #B452CD">1</span>, <span style="color: #B452CD">0</span>])
|
||||
ThetaXOR = Xinv @ X.T @ yXOR
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the XOR gate:{</span>ThetaXOR<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the XOR gate:{</span>X @ ThetaXOR<span style="color: #CD5555">}"</span>)
|
||||
|
||||
|
||||
<span style="color: #228B22"># The OR gate </span>
|
||||
yOR = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span> ,<span style="color: #B452CD">1</span>, <span style="color: #B452CD">1</span>])
|
||||
ThetaOR = Xinv @ X.T @ yOR
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the OR gate:{</span>ThetaOR<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the OR gate:{</span>X @ ThetaOR<span style="color: #CD5555">}"</span>)
|
||||
|
||||
|
||||
<span style="color: #228B22"># The OR gate </span>
|
||||
yAND = np.array( [ <span style="color: #B452CD">0</span>, <span style="color: #B452CD">0</span> ,<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>])
|
||||
ThetaAND = Xinv @ X.T @ yAND
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The values of theta for the AND gate:{</span>ThetaAND<span style="color: #CD5555">}"</span>)
|
||||
<span style="color: #658b00">print</span>(<span style="color: #CD5555">f"The linear regression prediction for the AND gate:{</span>X @ ThetaAND<span style="color: #CD5555">}"</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
What is happening here?
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="mathematical-model">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
|
||||
@@ -163,6 +163,10 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
|
||||
('Examples of XOR, OR and AND gates',
|
||||
2,
|
||||
None,
|
||||
'examples-of-xor-or-and-and-gates'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
('Mathematical model', 2, None, 'mathematical-model'),
|
||||
@@ -259,7 +263,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 7, 2021</h4></center> <!-- date -->
|
||||
<center><h4>Oct 8, 2021</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -1496,6 +1500,51 @@ as to not restrict the range of output values.
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="examples-of-xor-or-and-and-gates">Examples of XOR, OR and AND gates </h2>
|
||||
|
||||
<p>
|
||||
Let us first try to fit various gates using standard linear regression
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">"""</span>
|
||||
<span style="color: #BA2121; font-style: italic">Simple code that tests XOR, OR and AND gates with linear regression</span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #408080; font-style: italic"># Design matrix</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The invers of X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The XOR gate </span>
|
||||
yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
|
||||
ThetaXOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The OR gate </span>
|
||||
yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
|
||||
ThetaOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The OR gate </span>
|
||||
yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
|
||||
ThetaAND <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">f"The linear regression prediction for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">"</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
What is happening here?
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="mathematical-model">Mathematical model </h2>
|
||||
|
||||
<p>
|
||||
|
||||
Binary file not shown.
@@ -10,7 +10,7 @@
|
||||
"<!-- Author: --> \n",
|
||||
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA\n",
|
||||
"\n",
|
||||
"Date: **Oct 7, 2021**\n",
|
||||
"Date: **Oct 8, 2021**\n",
|
||||
"\n",
|
||||
"Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
|
||||
"\n",
|
||||
@@ -1540,6 +1540,59 @@
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Examples of XOR, OR and AND gates\n",
|
||||
"\n",
|
||||
"Let us first try to fit various gates using standard linear regression"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"\"\"\"\n",
|
||||
"Simple code that tests XOR, OR and AND gates with linear regression\n",
|
||||
"\"\"\"\n",
|
||||
"\n",
|
||||
"import numpy as np\n",
|
||||
"# Design matrix\n",
|
||||
"X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)\n",
|
||||
"print(f\"The X.TX matrix:{X.T @ X}\")\n",
|
||||
"Xinv = np.linalg.pinv(X.T @ X)\n",
|
||||
"print(f\"The invers of X.TX matrix:{Xinv}\")\n",
|
||||
"\n",
|
||||
"# The XOR gate \n",
|
||||
"yXOR = np.array( [ 0, 1 ,1, 0])\n",
|
||||
"ThetaXOR = Xinv @ X.T @ yXOR\n",
|
||||
"print(f\"The values of theta for the XOR gate:{ThetaXOR}\")\n",
|
||||
"print(f\"The linear regression prediction for the XOR gate:{X @ ThetaXOR}\")\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# The OR gate \n",
|
||||
"yOR = np.array( [ 0, 1 ,1, 1])\n",
|
||||
"ThetaOR = Xinv @ X.T @ yOR\n",
|
||||
"print(f\"The values of theta for the OR gate:{ThetaOR}\")\n",
|
||||
"print(f\"The linear regression prediction for the OR gate:{X @ ThetaOR}\")\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# The OR gate \n",
|
||||
"yAND = np.array( [ 0, 0 ,0, 1])\n",
|
||||
"ThetaAND = Xinv @ X.T @ yAND\n",
|
||||
"print(f\"The values of theta for the AND gate:{ThetaAND}\")\n",
|
||||
"print(f\"The linear regression prediction for the AND gate:{X @ ThetaAND}\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"What is happening here?\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Mathematical model\n",
|
||||
"\n",
|
||||
"The output $y$ is produced via the activation function $f$"
|
||||
|
||||
@@ -1103,6 +1103,47 @@ as to not restrict the range of output values.
|
||||
FIGURE: [figures/nns.png, width=600 frac=0.8] In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer.
|
||||
|
||||
|
||||
!split
|
||||
===== Examples of XOR, OR and AND gates =====
|
||||
|
||||
Let us first try to fit various gates using standard linear regression
|
||||
|
||||
!bc pycod
|
||||
"""
|
||||
Simple code that tests XOR, OR and AND gates with linear regression
|
||||
"""
|
||||
|
||||
import numpy as np
|
||||
# Design matrix
|
||||
X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
|
||||
print(f"The X.TX matrix:{X.T @ X}")
|
||||
Xinv = np.linalg.pinv(X.T @ X)
|
||||
print(f"The invers of X.TX matrix:{Xinv}")
|
||||
|
||||
# The XOR gate
|
||||
yXOR = np.array( [ 0, 1 ,1, 0])
|
||||
ThetaXOR = Xinv @ X.T @ yXOR
|
||||
print(f"The values of theta for the XOR gate:{ThetaXOR}")
|
||||
print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
|
||||
|
||||
|
||||
# The OR gate
|
||||
yOR = np.array( [ 0, 1 ,1, 1])
|
||||
ThetaOR = Xinv @ X.T @ yOR
|
||||
print(f"The values of theta for the OR gate:{ThetaOR}")
|
||||
print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
|
||||
|
||||
|
||||
# The OR gate
|
||||
yAND = np.array( [ 0, 0 ,0, 1])
|
||||
ThetaAND = Xinv @ X.T @ yAND
|
||||
print(f"The values of theta for the AND gate:{ThetaAND}")
|
||||
print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
|
||||
!ec
|
||||
|
||||
What is happening here?
|
||||
|
||||
|
||||
!split
|
||||
===== Mathematical model =====
|
||||
|
||||
|
||||
Reference in New Issue
Block a user