Functional data analysis in the Banach space of continuous functions

Dette, Holger; Kokot, Kevin; Aue, Alexander

doi:10.1214/19-aos1842

“…Hall and Van Keilegom (2007) were concerned with the effect of pre-processing discrete data into functions on two-sample testing procedures. Horváth et al (2013) investigated two-sample tests for the equality of means of two functional time series taking values in the Hilbert space of square integrable functions, and Dette et al (2019) introduced multiplier bootstrap-assisted two-sample tests for functional time series taking values in the Banach space of continuous functions. Panaretos et al (2010), Fremdt et al (2013), Pigoli et al (2014), Paparoditis and Sapatinas (2016) and Guo et al (2016) provided procedures for testing the equality of covariance operators in functional samples.…”

Section: Introductionmentioning

confidence: 99%

Two-sample tests for relevant differences in the eigenfunctions of covariance operators

Aue¹,

Dette²,

Rice³

2019

Preprint

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This paper deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eigenfunctions of the respective covariance operators. To this end, a novel testing approach is introduced that connects with, and extends, existing literature in two main ways. First, tests are set up in the relevant testing framework, where interest is not in testing an exact null hypothesis but rather in detecting deviations deemed sufficiently relevant, with relevance determined by the practitioner and perhaps guided by domain experts. Second, the proposed test statistics rely on a self-normalization principle that helps to avoid the notoriously difficult task of estimating the long-run covariance structure of the underlying functional time series. The main theoretical result of this paper is the derivation of the large-sample behavior of the proposed test statistics. Empirical evidence, indicating that the proposed procedures work well in finite samples and compare favorably with competing methods, is provided through a simulation study, and an application to annual temperature data.

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“…In particular, relevant deviations between covariance operators in the sup-norm haveto our best knowledge -not been considered so far and requires a different methodology as the space under consideration is a Banach but not a Hilbert space. There does not exist so much literature on functional data analysis considering Banach spaces and exemplarily we mention the recent work of Dette et al (2020a) who considered relevant hypotheses for the mean function and Liebl and Reimherr (2019), who developed confidence bands for functional parameters.…”

Section: Related Literaturementioning

confidence: 99%

“…We apply the central limit theorem as formulated in Theorem 2.1 in Dette et al (2020a) to the sequence of C(T 2 )-valued random variables ((Z j − µ) ⊗2 ) j∈N = (η ⊗2 j ) j∈N . It can be easily seen that conditions (A1), (A2) and (A4) in this reference are satisfied.…”

Section: Data Examplementioning

confidence: 99%

Detecting relevant differences in the covariance operators of functional time series -- a sup-norm approach

Dette¹,

Kokot²

2020

Preprint

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0

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In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead we propose to formulate the null hypotheses in the form that "the distance between the operators is small", where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.

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“…Hall and Van Keilegom (2007) were concerned with the effect of pre-processing discrete data into functions on two-sample testing procedures. Horváth et al (2013) investigated two-sample tests for the equality of means of two functional time series taking values in the Hilbert space of square integrable functions, and Dette et al (2020a) introduced multiplier bootstrap-assisted two-sample tests for functional time series taking values in the Banach space of continuous functions. Panaretos et al (2010), Fremdt et al (2013), Pigoli et al (2014), Paparoditis and Sapatinas (2016) and Guo et al (2016) provided procedures for testing the equality of covariance operators in functional samples.…”

Section: Introductionmentioning

confidence: 99%

Two-Sample Tests for Relevant Differences in the Eigenfunctions of Covariance Operators

Aue

¹

,

Dette

²

,

Rice

³

2023

Self Cite

View full text Add to dashboard Cite

This paper deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eigenfunctions of the respective covariance operators. To this end, a novel testing approach is introduced that connects with, and extends, existing literature in two main ways. First, tests are set up in the relevant testing framework, where interest is not in testing an exact null hypothesis but rather in detecting deviations deemed sufficiently relevant, with relevance determined by the practitioner and perhaps guided by domain experts. Second, the proposed test statistics rely on a self-normalization principle that helps to avoid the notoriously difficult task of estimating the long-run covariance structure of the underlying functional time series. The main theoretical result of this paper is the derivation of the large-sample behavior of the proposed test statistics. Empirical evidence, indicating that the proposed procedures work well in finite samples and compare favorably with competing methods, is provided through a simulation study, and an application to annual temperature data.

show abstract

Functional data analysis in the Banach space of continuous functions

Cited by 34 publications

References 39 publications

Two-sample tests for relevant differences in the eigenfunctions of covariance operators

Two-sample tests for relevant differences in the eigenfunctions of covariance operators

Detecting relevant differences in the covariance operators of functional time series -- a sup-norm approach

Two-Sample Tests for Relevant Differences in the Eigenfunctions of Covariance Operators

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