Extremal dependence measure for functional data

https://doi.org/10.1016/j.jmva.2021.104887Get rights and content

Abstract

Principal component analysis is one of the most fundamental tools of functional data analysis. It leads to an efficient representation of infinitely dimensional objects, like curves, by means of multivariate vectors of scores. We study the dependence between extremal values of the scores using the extremal dependence measure (EDM). The EDM has been proposed and studied for positive bivariate observations. After extending it to multivariate observations, we focus on its application to the vectors of scores of functional data. Estimated scores form a triangular array of dependent random variables. We derive condition guaranteeing that a suitable estimator of the EDM based on these scores converges to the population EDM and is asymptotically normal. These conditions are completely different from those encountered in the second-order theory of functional data. They are formulated within the framework of functional regular variation. Large sample theory is complemented by an application to intraday return curves for certain stocks and by a simulation study.

Introduction

We first concisely state main contributions of the paper with the caveat that detailed definitions and formulations will be provided in the following. Consider a sample of functions Xi(t),tT, such that each of them has the same distribution as X. The Karhunen–Loéve expansion is X(t)=j=1ξjvj(t). The functions vj are the functional principal components (FPCs) and the random variables ξj are their scores. We want to estimate extremal dependence of ξj and ξj. We define a measure of such a dependence, which we denote by D(ξj,ξj). We then define an estimator of D(ξj,ξj) and formulate conditions under which it is consistent (Theorem 1) and asymptotically normal (Theorem 2). The main difficulty is that the population scores ξij=Xi,vj are not observable.

This paper thus makes a contribution at the nexus of functional data analysis (FDA) and extreme value theory (EVT). We assume that the reader is familiar with mathematical foundations of functional data analysis and central principles of extreme value theory. The FDA background given in Chapters 2 and 3 of [17] is sufficient, and more detailed treatment is provided in [18]. Recent advances in FDA are surveyed in [15], [1], and [5].

Chapters 2 and 6 of [31] provide sufficient background in extreme value theory. Other references are cited when needed. We assume that all functions are elements of the space L2=L2(T), where the measure space T is such that L2(T), with the usual inner product, is a separable Hilbert space. This will be ensured if the measure on T is σ-finite and defined on a countably generated σ-algebra, see e.g. Proposition 3.4.5 in [2]. In particular, T can be taken to be a complete separable metric space (Polish space).

Suppose X1,,Xn are mean zero iid functions in L2 with EXi2<, and denote by X a generic random function with the same distribution as each Xi. A main dimension reduction tool of functional data analysis is to project the infinite dimensional functions Xi onto a finite dimensional subspace spanned by the FPCs. We now recall the required definitions. Consider the population covariance operator of X, defined by C(x)E[X,xX],xL2.The eigenfunctions of C are the FPCs, denoted by vj,j1, i.e., C(vj)=λjvj, where the λj are the eigenvalues of C. The FPCs lead to the commonly used Karhunen–Loéve expansion Xi(t)=j=1ξijvj(t),ξij=Xi,vj,Eξij2=λj.The FPCs vj and the eigenvalues λj are estimated by vˆj and λˆj, which are solutions to the equations Ĉ(vˆj)(t)=λˆjvˆj(t),foralmostalltT,where Ĉ is the sample covariance operator defined by Ĉ(x)(t)=1ni=1nXi,xXi,xL2.Each curve Xi can then be approximated by a linear combination of a finite set of the estimated FPCs vˆj, i.e., Xi(t)j=1pξˆijvˆj(t), where the ξˆij=Xi,vˆj are the sample scores. Each ξˆij quantifies the contribution of the curve vˆj to the shape of the curve Xi. Thus, the vector of the sample scores, [ξˆi1,,ξˆip], encodes the shape of Xi to a good approximation. To illustrate, Fig. 1 displays the first three sample FPCs, vˆ1,vˆ2,vˆ3, for intraday return curves Ri,1i1378, for Walmart stock from July 05, 2006 to Dec 30, 2011. These data are described in detail in Section II of the supplement. The curves Ri show how a return on an investment changes throughout a trading day as two examples are shown in Fig. 2. The curve vˆ1 is a monotonic trend throughout the day. If the score corresponding to it is large, trading in this stock on a given day was dominated by a systematic increase (or decline if the score is negative) in the price of the stock. Notice the gradually decreasing slope of vˆ1, which reflects the well-known fact that the most intense trading takes place after the opening of the trading floor. The second FPC, vˆ2, has a large score, if there is a significant reversal in investor sentiment during a given trading day. These observations are illustrated in Fig. 2.

The main interest in this paper is the estimation of extremal dependence between the scores corresponding to different FPCs. Extremal dependence is a tendency of large values of one component to be coupled with large values of another component of a random vector. In the context of our Walmart stock example, extreme dependence between the first and second scores indicates that an extremely high monotonic trend and a pronounced reversion tend to occur simultaneously. We assess extremal dependence of the scores by means of the extremal dependence measure (EDM), which is constructed based on the theory of heavy-tailed regularly-varying random vectors. There has been considerable research on quantifying the tail dependence between extreme values in a heavy-tailed random vector. [23], [24], [25] defined the coefficient of tail dependence, which was later generalized to the extremogram by [7]. While these approaches are essentially based on the exponent measure of a random vector, the EDM is defined in terms of the spectral measure. The EDM was introduced by [30] and further investigated by [22]. Important related papers are [14] and [3].

In this paper, we quantify extremal dependence of scores using the EDM. To estimate the EDM of population scores, we consider an extension of the estimator proposed by [22]. It is important to emphasize that in our functional setting, the estimator can only be computed using the sample scores ξˆij=Xi,vˆj, not the population scores ξij=Xi,vj because the ξij are unobservable. Establishing large sample properties of any estimator based on sample scores requires taking the effect of the estimation of the scores into account. Since the estimator Ĉ in (3) depends on the whole sample X1,,Xn, the vectors [ξˆi1,,ξˆip] are no longer independent, even if X1,,Xn are i.i.d functions. They form a triangular array of dependent identically distributed vectors of dimension p. We also note that the population scores satisfy Cov(ξij,ξij)=0 if jj and the sample correlation of the sample scores ξˆij and ξˆij is also zero. However, the correlation is a measure of the overall dependence, and there may be strong dependence, e.g. between the positive parts ξij+ and ξij+, in particular there may be extremal dependence in specific quadrants. Another point to keep in mind is that for regularly varying observations, zero covariance does not imply independence.

The remainder of the paper is organized as follows. In Section 2, we introduce preliminaries on multivariate regular variation and the EDM, and extend the concept of the EDM to multivariate data. Our main large sample results are presented in Section 3, which deals with the EDM for scores of functional observations. Section 4 presents a number of preliminary results. These results allow us to streamline the exposition of the proofs of the results of Section 3, which are presented in Section 5.

The paper is accompanied by online Supplementary Material, which contains several sections. Section 1 explains how to normalize tail indexes of components of multivariate vectors. This is a well-researched topic in EVT, but may be less known in the FDA community, so a brief account needed to understand the application in Section 2 of the supplement is provided. Sections 2 Multivariate regular variation and the EDM, 3 The EDM for scores of functional data, present, respectively, an application to functional return data and a simulation study. Section 4 contains additional tables discussed in Section 3.

We hope that this work will be received with some interest by researchers working in two exciting and dynamic fields: functional data analysis and extreme value theory.

Section snippets

Multivariate regular variation and the EDM

We start by introducing multivariate regular variation for random vectors with positive components because the extremal dependence measure (EDM) was defined in such context. Following [31], we denote by Ed=0,d{0} the nonnegative orthant compactified at infinity. We denote by M+(Ed) the space of Radon measures on Ed, and by v the vague convergence in M+(Ed). An Ed-valued random vector Z=[Z1,,Zd] with distribution function F is regularly varying with index α, α>0, if there exist a sequence b

The EDM for scores of functional data

In this section, we consider the estimation of the EDM of scores of functional data. Following the framework introduced in Section 1, recall that X1,,Xn are mean zero iid functions in L2 with EXi2<, and that each Xi admits the Karhunen–Loéve expansion (2). The unknown population scores ξij=Xi,vj in (2) are estimated by the sample scores ξˆij=Xi,vˆj, where the vˆj are estimators of the FPCs vj. We introduce the following random variables: Yd=[ξ1,,ξd],ξj=X,vj,Yid=[ξi1,,ξid],ξij=Xi,vj, Ŷd

Preliminary results

We put together several preliminary results in this section to avoid burdening the proofs in Section 5, so that readers can keep track of the main flow of the argument made in Section 5.

The first lemma follows from Lemma 3.7 of [20] and is needed to prove Lemma 2.

Lemma 1

Suppose random variables Hm(n), m,n1, satisfy 0Hm(n)1 and m1,Hm(n)P0,asn. Then, m=12mHm(n)P0,asn.

In the following lemma, we present a sufficient condition to guarantee the convergence between random measures defined on a

Proofs of the results of Section 3

Proof of Proposition 3

First, note that π(X)>yb(n) and π(X)/π(X) iff (yb(n))1XAπ(). Observe that, for any set S in B(Sd), nPr(π(X)>yb(n),π(X)/π(X)S)=nPrXyb(n)Aπ(S).To prove the regular variation of π(X) in Rd, we will apply Theorem 2.3 of [26]. To do this, we must show that the Aπ(S) are continuity sets of ν, i.e., ν(Aπ(S))=0. The verification uses the same idea described in the proof of Proposition 3.1 of [21], but the difference is that we work with the different projection π(z) and its relevant set

Acknowledgments

This research was partially supported by the United States NSF grants DMS-1923142, DMS-1914882 and DMS-2123761.

References (33)

  • DavisR.A. et al.

    Limit theory for the sample covariance and correlation functions of moving averages

    Ann. Statist.

    (1986)
  • EmbrechtsP. et al.

    Modelling Extremal Events for Insurance and Finance

    (1997)
  • de FondevilleR.

    Functional peaks-over-threshold analysis for complex extreme events

    (2018)
  • de FondevilleR. et al.

    High-dimensional peaks-over-threshold inference

    Biometrika

    (2018)
  • GentonM.G. et al.

    Multivariate max-stable spatial processes

    Biometrika

    (2015)
  • HaeuslerE. et al.

    On asymptotic normality of hill’s estimator for the exponent of regular variation

    Ann. Statist.

    (1985)
  • 1

    Both Authors, Mihyun Kim and Piotr Kokoszka, contributed equally to this work.

    View full text