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Multilayer perceptron with functional inputs: an inverse regression approach. (English) Zbl 1164.62339

A functional sliced inverse regression (FSIR) technique is considered for fitting the model \[ Y=f(\langle X,a_1\rangle,\dots,\langle X,a_p\rangle,\varepsilon), \] where \(Y\) is the response, \(X\) is a functional regressor (a random function in \(L_2[a,b]\)), \(f\) is an unknown regression function, \(a_1\) are unknown functions, and \(\varepsilon\) is an error term. Consistency of regularized FSIR estimates for \(a_i\) is demonstrated. It is proposed to estimate the function \(f\) by a multilayer perceptron technique. Consistency of the proposed training algorithm for such perceptrons is shown. Applications to real data are considered.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G99 Nonparametric inference
62J12 Generalized linear models (logistic models)

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