# Functional data analysis with increasing number of projections

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## Date

2013-12-16

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## Abstract

Functional principal components (FPC’s) provide the most important and
most extensively used tool for dimension reduction and inference for functional data.
The selection of the number, d, of the FPC’s to be used in a specific procedure has
attracted a fair amount of attention, and a number of reasonably effective approaches
exist. Intuitively, they assume that the functional data can be sufficiently well approximated
by a projection onto a finite–dimensional subspace, and the error resulting from
such an approximation does not impact the conclusions. This has been shown to be a
very effective approach, but it is desirable to understand the behavior of many inferential
procedures by considering the projections on subspaces spanned by an increasing
number of the FPC’s. Such an approach reflects more fully the infinite–dimensional
nature of functional data, and allows to derive procedures which are fairly insensitive
to the selection of d. This is accomplished by considering limits as d → ∞ with the
sample size.
We propose a specific framework in which we let d → ∞ by deriving a normal
approximation for the partial sum process
⌊Xdu⌋
j=1
⌊XNx⌋
i=1
i,j , 0 ≤ u ≤ 1, 0 ≤ x ≤ 1,
where N is the sample size and i,j is the score of the ith function with respect to
the jth FPC. Our approximation can be used to derive statistics that use segments of
observations and segments of the FPC’s. We apply our general results to derive two
inferential procedures for the mean function: a change–point test and a two–sample test.
In addition to the asymptotic theory, the tests are assessed through a small simulation
study and a data example.

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## Keywords

functional data, principal components, normal approximation, increasing dimension, change in mean