動径基底関数ネットワークに基づく非線形回帰モデルとその推定

DOI Web Site 被引用文献8件 参考文献15件

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タイトル別名
  • Estimating Nonlinear Regression Models based on Radial Basis Function Networks
  • ドウケイ キテイ カンスウ ネットワーク ニ モトヅク ヒセンケイ カイキ モデル ト ソノ スイテイ

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Neural networks have received considerable attention as useful tools for analyzing data with complex structure. We consider the use of radial basis function networks in constructing nonlinear regression models.<BR>Suppose that we have n observations {(yi, xi);i=1, ..., n}, where yi are independent random response variables and xi are vectors of d explanatory variables. Consider the regression model<BR>yi=u(xi)+εi, i=1, ..., n, <BR>where u(·) is an unknown smooth function and the errors εi are independently distributed n(0, σ2). Our aim is to estimate the function u(·) from the observed data, for which we use the radial basis function (RBF) network<BR>u(xi)= ∑ωkφk(xi)+ω0<BR>where φk(x) is the basis function given by<BR>φk(x)=exp(-||x-ck||2/2vs2k), k=1, ..., M.<BR>We introduce the Gaussian basis function with hyperparameter v that adjusts the amount of overlapping basis functions.<BR>In the first stage the centres ckkand scale factors s2k of the basis functions are determined by using the k-means clustering algorithm based on the input data set {xi;i=1, ..., n}. In the second stage we estimate the weights ωk by the regularization method which maximizes the penalized log-likelihood<BR> $sum;logf(yi|x;ω, σ2)- λ/2 ω'Qψ, <BR>where ω=(ω0, ω1, ..., ωM)', λ is a smoothing parameter and Q is some fixed (M+1)×(M+1) nonnegative-definite matrix.<BR>A crucial issue in the RBF network regression model is the choice of smoothing parameters v, λ and also the number of basis functions that control the smoothness of the fitted function by regularization. We present an information-theoretic criterion for evaluating the nonlinear regression model based on the RBF network. The information criterion proposed is applied to choose the smoothing parameters and the number of basis functions.<BR>We use Monte Carlo experiments and a real data example to examine the performance of the RBF network nonlinear modeling. The simulation results show that our nonlinear modeling performs well in various situations, and that clear improvements are obtained for the use of the hyperparameter in the radial basis functions.

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