New approaches in Functional Data Analysis: inference for incomplete or correlated data and nonlinear methods

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Functional Data Analysis (FDA) is a branch of statistics concerned with the analysis of complex data objects, such as curves and surfaces, that can be viewed as realizations of a random function. Functional data arise naturally in many applications, and are becoming more and more accessible and consequently prevalent, due to the advances in computer technology in the last decades. The two main features of functional data are: (i) they belong to an infinite-dimensional space; (ii) they have some smoothness properties. The classical setup in FDA is to suppose that we have a collection of independent realizations of a random curve X and that each of these realizations is observed at a finite subset of [0,1] and with measurement error. Traditional approaches to analyze such data typically follow a two-step procedure: first a smoothing step; further, a dimension reduction step where each curve is approximated by a linear combination of a finite number of smooth basis functions. However, many interesting features in FDA such as time-localized features and multiscale variations are not well-captured by a linear expansion. The first objective of my research program is to make methodological advances to the nonlinear approach for analyzing functional data. In order to do this, I plan to elaborate an estimation procedure of the covariance function for data containing time-localized features (e.g. peaks) and to explore how to do inference for functional data that lie in non-linear manifolds. Another setup of interest in FDA is the one where each function of a dataset is only observed on a random subinterval of the whole domain of definition [0,1]. These data are called functional fragments and their analysis is quite challenging since we have no statistical information on the unknown covariance function outside a band around the diagonal. My objectives in this area are to develop a method to estimate the missing part of each discretized fragment of a dataset using low-rank matrix completion techniques, and a method to do registration of functional fragments. Traditionally, in FDA we work with samples of independent realizations of random functions. However, in practice, it might very well happen that the observed functions are spatially correlated. My last objective is to explore inference for such data. More precisely, I want to elaborate a testing procedure for spatial multivariate functional data and to build a functional regression model where the responses are count data with an excess of zero counts and the explanatory variables are spatially correlated functional data. To conclude, my research program aims to tackle challenging objectives in three areas of FDA: (a) nonlinear approach in analyzing functional data, (b) analysis of functional fragments, and (c) inference for spatially correlated functional data. The novel methodology of this proposal could open the way to solving many difficult problems in FDA, both theoretical and applied.