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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Radial basis function network ： ウィキペディア英語版 Radial basis function network In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.〔 ==Network architecture== 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 $\backslash mathbf\; \backslash in\; \backslash mathbb^n$. The output of the network is then a scalar function of the input vector, $\backslash varphi\; :\; \backslash mathbb^n\; \backslash to\; \backslash mathbb$, and is given by :$\backslash varphi(\backslash mathbf)\; =\; \backslash sum\_^N\; a\_i\; \backslash rho(||\backslash mathbf-\backslash mathbf\_i||)$ where $N$ is the number of neurons in the hidden layer, $\backslash mathbf\; c\_i$ is the center vector for neuron $i$, and $a\_i$ is the weight of neuron $i$ in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better in general) and the radial basis function is commonly taken to be Gaussian :$\backslash rho\; \backslash big\; (\; \backslash left\; \backslash Vert\; \backslash mathbf\; -\; \backslash mathbf\_i\; \backslash right\; \backslash Vert\; \backslash big\; )\; =\; \backslash exp\; \backslash left(-\backslash beta\; \backslash left\; \backslash Vert\; \backslash mathbf\; -\; \backslash mathbf\_i\; \backslash right\; \backslash Vert\; ^2\; \backslash right)$. The Gaussian basis functions are local to the center vector in the sense that :$\backslash lim\_\backslash rho(\backslash left\; \backslash Vert\; \backslash mathbf\; -\; \backslash mathbf\_i\; \backslash right\; \backslash Vert)\; =\; 0$ i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of $\backslash mathbb^n$. This means that an RBF network with enough hidden neurons can approximate any continuous function with arbitrary precision. The parameters $a\_i$, $\backslash mathbf\_i$, and $\backslash beta\_i$ are determined in a manner that optimizes the fit between $\backslash varphi$ and the data. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Radial basis function network」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.