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November 2013 Consistency of the mean and the principal components of spatially distributed functional data
Siegfried Hörmann, Piotr Kokoszka
Bernoulli 19(5A): 1535-1558 (November 2013). DOI: 10.3150/12-BEJ418

Abstract

This paper develops a framework for the estimation of the functional mean and the functional principal components when the functions form a random field. More specifically, the data we study consist of curves $X(\mathbf{s}_{k};t)$, $t\in[0,T]$, observed at spatial points $\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{N}$. We establish conditions for the sample average (in space) of the $X(\mathbf{s}_{k})$ to be a consistent estimator of the population mean function, and for the usual empirical covariance operator to be a consistent estimator of the population covariance operator. These conditions involve an interplay of the assumptions on an appropriately defined dependence between the functions $X(\mathbf{s}_{k})$ and the assumptions on the spatial distribution of the points $\mathbf{s}_{k}$. The rates of convergence may be the same as for i.i.d. functional samples, but generally depend on the strength of dependence and appropriately quantified distances between the points $\mathbf{s}_{k}$. We also formulate conditions for the lack of consistency.

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Siegfried Hörmann. Piotr Kokoszka. "Consistency of the mean and the principal components of spatially distributed functional data." Bernoulli 19 (5A) 1535 - 1558, November 2013. https://doi.org/10.3150/12-BEJ418

Information

Published: November 2013
First available in Project Euclid: 5 November 2013

zbMATH: 06254533
MathSciNet: MR3129024
Digital Object Identifier: 10.3150/12-BEJ418

Keywords: consistency , estimation , functional data , functional principal components , spatial dependence , spatial sampling design

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

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Vol.19 • No. 5A • November 2013
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