mhjensen
194 lines
7.4 KiB
HTML
194 lines
7.4 KiB
HTML
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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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<a name="part0012"></a>
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<h2 id="___sec11" class="anchor">Mathematical model </h2>
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<p>
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This confirms that an MLP,
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despite its quite convoluted mathematical form, is nothing more than an analytic function, specifically a
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mapping of real-valued vectors \( \vec{x} \in \mathbb{R}^n \rightarrow \vec{y} \in \mathbb{R}^m \).
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In our example, \( n=2 \) and \( m=1 \). Consequentially,
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the number of input and output values of the function we want to fit must be equal to the number of inputs and outputs of our MLP.
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<p>
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Furthermore, the flexibility and universality of a MLP can be illustrated by realizing that
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the expression is essentially a nested sum of scaled activation functions of the form
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$$
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\begin{equation}
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h(x) = c_1 f(c_2 x + c_3) + c_4
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\tag{11}
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\end{equation}
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$$
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where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions
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can be shifted up and down or left and right, change slope or be rescaled
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which is the key to the flexibility of a NN.
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$$
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\begin{equation}
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f_o = f(u_o) = u_o
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\tag{12}
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\end{equation}
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$$
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