Morten Hjorth-Jensen
347 lines
19 KiB
HTML
347 lines
19 KiB
HTML
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<!-- navigation toc: --> <li><a href="._week40-bs001.html#lecture-monday-september-29-2025" style="font-size: 80%;"><b>Lecture Monday September 29, 2025</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs002.html#suggested-readings-and-videos" style="font-size: 80%;"><b>Suggested readings and videos</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs003.html#lab-sessions-tuesday-and-wednesday" style="font-size: 80%;"><b>Lab sessions Tuesday and Wednesday</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs004.html#logistic-regression-from-last-week" style="font-size: 80%;"><b>Logistic Regression, from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs005.html#classification-problems" style="font-size: 80%;"><b>Classification problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs006.html#optimization-and-deep-learning" style="font-size: 80%;"><b>Optimization and Deep learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs007.html#basics" style="font-size: 80%;"><b>Basics</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs008.html#two-parameters" style="font-size: 80%;"><b>Two parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs009.html#maximum-likelihood" style="font-size: 80%;"><b>Maximum likelihood</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs010.html#the-cost-function-rewritten" style="font-size: 80%;"><b>The cost function rewritten</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs011.html#minimizing-the-cross-entropy" style="font-size: 80%;"><b>Minimizing the cross entropy</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs012.html#a-more-compact-expression" style="font-size: 80%;"><b>A more compact expression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs013.html#extending-to-more-predictors" style="font-size: 80%;"><b>Extending to more predictors</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs014.html#including-more-classes" style="font-size: 80%;"><b>Including more classes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs015.html#more-classes" style="font-size: 80%;"><b>More classes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs016.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;"><b>Optimization, the central part of any Machine Learning algortithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs017.html#revisiting-our-logistic-regression-case" style="font-size: 80%;"><b>Revisiting our Logistic Regression case</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs018.html#the-equations-to-solve" style="font-size: 80%;"><b>The equations to solve</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs019.html#solving-using-newton-raphson-s-method" style="font-size: 80%;"><b>Solving using Newton-Raphson's method</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs020.html#example-code-for-logistic-regression" style="font-size: 80%;"><b>Example code for Logistic Regression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs020.html#synthetic-data-generation" style="font-size: 80%;"> Synthetic data generation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs021.html#using-scikit-learn" style="font-size: 80%;"><b>Using <b>Scikit-learn</b></b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs022.html#using-the-correlation-matrix" style="font-size: 80%;"><b>Using the correlation matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs023.html#discussing-the-correlation-data" style="font-size: 80%;"><b>Discussing the correlation data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs024.html#other-measures-in-classification-studies" style="font-size: 80%;"><b>Other measures in classification studies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs025.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs026.html#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs027.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs028.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs029.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs030.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs031.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs032.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs033.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-bs034.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs035.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-bs036.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs037.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs042.html#mathematical-model" style="font-size: 80%;"><b>Mathematical model</b></a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs043.html#matrix-vector-notation" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs044.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-bs045.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week40-bs046.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-bs047.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0039"></a>
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<!-- !split -->
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<h2 id="mathematical-model" class="anchor">Mathematical model </h2>
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<p>First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),</p>
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$$
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\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
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\tag{2}
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\end{equation}
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$$
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<p>Here \( b_i \) is the so-called bias which is normally needed in
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case of zero activation weights or inputs. How to fix the biases and
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the weights will be discussed below. The value of \( z_i^1 \) is the
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argument to the activation function \( f_i \) of each node \( i \), The
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variable \( M \) stands for all possible inputs to a given node \( i \) in the
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first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
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</p>
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$$
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\begin{equation}
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y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
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\tag{3}
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\end{equation}
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$$
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<p>where we assume that all nodes in the same layer have identical
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activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
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In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
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</p>
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$$
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\begin{equation}
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y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
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\tag{4}
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\end{equation}
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$$
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<p>where \( N_l \) is the number of nodes in layer \( l \). When the output of
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all the nodes in the first hidden layer are computed, the values of
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the subsequent layer can be calculated and so forth until the output
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is obtained.
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</p>
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<p>
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<!-- Bootstrap footer
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<footer>
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<a href="https://..."><img width="250" align=right src="https://..."></a>
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</footer>
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-->
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<center style="font-size:80%">
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<!-- copyright only on the titlepage -->
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</center>
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</body>
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</html>
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