mhjensen
199 lines
7.5 KiB
HTML
199 lines
7.5 KiB
HTML
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'sections': [('What is Machine Learning?', 2, None, '___sec0'),
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<a class="navbar-brand" href="NeuralNet-bs.html">Data Analysis and Machine Learning: Elements of machine learning</a>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs001.html#___sec0" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs002.html#___sec1" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs004.html#___sec3" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs008.html#___sec7" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<a name="part0009"></a>
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<h2 id="___sec8" class="anchor">Mathematical model </h2>
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<p>
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First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \),
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$$
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\begin{equation}
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u_i^1 = \sum_{j=1}^2 w_{ij}^1 x_j + b_i^1
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\tag{3}
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\end{equation}
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$$
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This value is the argument to the activation function \( f_1 \) of each neuron \( i \),
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producing the output \( y_i^1 \) of all neurons in layer 1,
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$$
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\begin{equation}
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y_i^1 = f_1(u_i^1) = f_1\left(\sum_{j=1}^2 w_{ij}^1 x_j + b_i^1\right)
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\tag{4}
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\end{equation}
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$$
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where we assume that all nodes in the same layer have identical activation functions, hence the notation \( f_l \)
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$$
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\begin{equation}
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y_i^l = f_l(u_i^l) = f_l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
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\tag{5}
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\end{equation}
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$$
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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,
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the values of the subsequent layer can be calculated and so forth until the output is obtained.
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<p>
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<p>
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<li><a href="._NeuralNet-bs008.html">«</a></li>
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<li><a href="._NeuralNet-bs000.html">1</a></li>
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<li><a href="._NeuralNet-bs001.html">2</a></li>
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<li><a href="._NeuralNet-bs002.html">3</a></li>
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<li><a href="._NeuralNet-bs003.html">4</a></li>
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<li><a href="._NeuralNet-bs004.html">5</a></li>
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<li><a href="._NeuralNet-bs005.html">6</a></li>
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<li><a href="._NeuralNet-bs006.html">7</a></li>
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<li><a href="._NeuralNet-bs007.html">8</a></li>
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<li><a href="._NeuralNet-bs008.html">9</a></li>
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<li><a href="._NeuralNet-bs010.html">11</a></li>
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<li><a href="._NeuralNet-bs011.html">12</a></li>
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<li><a href="._NeuralNet-bs012.html">13</a></li>
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<li><a href="._NeuralNet-bs013.html">14</a></li>
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