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Functional response regression analysis. (English) Zbl 1416.62322
Functional data analysis with functional responses and scalar or vector covariates is studied. The response function \(Y(t)\) can be written as a smooth random function \(Z(t)\) plus a measurement error. Such function \(Z(t)\) has a mean function \(\mu(t)\) and a covariance function \(\operatorname{Cov}(Z(s),Z(t))=G(s,t).\) Hence, \(Z(t)\) has a Karhunen-Loeve decomposition. Then, the rank \(K\) functional principal component analysis justifies considering only a finite number of terms in the decomposition, resulting on \(Z(t)=\mu(t) + \sum^{K}_{k=1} u_k \phi_k(t).\) The atemporal covariates \(X\) enter in the composition of each of the \(u_k\)s, i.e., each of the \(u_k\)s is a linear function of the covariates \(X\) plus a random error. Putting such \(Z(t)\) back into the expression of \(Y(t)\) we have the response function written as function of the \(\mu(t),\) the \(\phi(t)\)s, the covariates and an error which contemplates the two previously mentioned errors. Using B-spline smoothing for both \(\mu(t)\) and \(\phi(t)\)s a supervised least square estimation is performed. The B-spline approximation theory allows to prove that all the estimators used in the modelling are consistent, as well as, that the estimators of the parameters in the linear function of the covariates satisfy a central limit theorem, after some regularity conditions. Simulation experiments are done to access the model performance. Then, the model is applied to an atmospheric particulate matter (PM) data set obtaining sound results.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
62J12 Generalized linear models (logistic models)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
60F05 Central limit and other weak theorems
62P35 Applications of statistics to physics
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