Radial basis function network: Difference between revisions

Revision as of 15:17, 18 December 2020

Revision as of 06:59, 5 March 2021

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==Network architecture==

[[Image:Radial funktion network.svg|thumb|250px|right|Figure 1: Architecture of a radial basis function network. An input vector <math>x</math> is used as input to all radial basis functions, each with different parameters. The output of the network is a linear combination of the outputs from radial basis functions.]]

Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers <math>\mathbf{x} \in \mathbb{R}^n</math>. The output of the network is then a scalar function of the input vector, <math> \varphi : \mathbb{R}^n \to \mathbb{R} </math>, and is given by

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The parameters <math> a_i </math>, <math> \mathbf{c}_i </math>, and <math> \beta_i </math> are determined in a manner that optimizes the fit between <math> \varphi </math> and the data.

[[Image:Unnormalized radial basis functions.svg|thumb|250px|right|Figure 2: Two unnormalized radial basis functions in one input dimension. The basis function centers are located at <math> c_1=0.75 </math> and <math> c_2=3.25 </math>.

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|width=250

|image1=Normalized radial basis functions.svg

|caption1=Figure 3: Two normalized radial basis functions in one input dimension ([[logistic function|sigmoids]]). The basis function centers are located at <math> c_1=0.75 </math> and <math> c_2=3.25 </math>.

|image2=3 Normalized radial basis functions.svg

|caption2=Figure 4: Three normalized radial basis functions in one input dimension. The additional basis function has center at <math> c_3=2.75 </math>

|image3=4 Normalized radial basis functions.svg

|caption3=Figure 5: Four normalized radial basis functions in one input dimension. The fourth basis function has center at <math> c_4=0 </math>. Note that the first basis function (dark blue) has become localized.

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