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Extremal correlation coefficient for functional data

Mihyun Kim, Piotr Kokoszka

TL;DR

This work introduces an extremal-correlation coefficient for paired functional data to quantify how extreme curves co-vary in shape. Grounded in regular variation in Banach spaces and $M_0$ convergence, it defines the extremal covariance $\sigma_{XY}$ and the extremal correlation $\rho_{XY}$, and delivers a peaks-over-threshold estimator $\hat{\rho}_{n,k}$ (with $k\to\infty$ and $k/n\to0$) that is consistent under $X,Y\in RV(-\alpha,\Gamma)$ with $\alpha>2$. The authors also provide an angular-measure-based alternative $\gamma_{XY}$ for scenarios where $\alpha$ may be small, and develop practical computation steps for discrete observations. Through simulations and real-data applications to intraday sector-ETF returns and daily temperature curves, the method reveals substantial extremal dependence in both finance and climate contexts and offers a shape-focused perspective on extreme risk beyond traditional tail measures.

Abstract

We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: (i) it is designed to measure dependence between curves, (ii) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.

Extremal correlation coefficient for functional data

TL;DR

This work introduces an extremal-correlation coefficient for paired functional data to quantify how extreme curves co-vary in shape. Grounded in regular variation in Banach spaces and convergence, it defines the extremal covariance and the extremal correlation , and delivers a peaks-over-threshold estimator (with and ) that is consistent under with . The authors also provide an angular-measure-based alternative for scenarios where may be small, and develop practical computation steps for discrete observations. Through simulations and real-data applications to intraday sector-ETF returns and daily temperature curves, the method reveals substantial extremal dependence in both finance and climate contexts and offers a shape-focused perspective on extreme risk beyond traditional tail measures.

Abstract

We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: (i) it is designed to measure dependence between curves, (ii) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.
Paper Structure (17 sections, 17 theorems, 102 equations, 10 figures, 10 tables)

This paper contains 17 sections, 17 theorems, 102 equations, 10 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Regular variation in Banach spaces
  3. Bivariate regular variation in Banach spaces
  4. Extremal correlation coefficient for functional data
  5. A simulation study
  6. Applications to financial and climate functional data
  7. Extremal dependence of intraday returns on sector ETFs
  8. Extremal correlation between daily temperature curves
  9. Discussion
  10. Preliminary results
  11. Proof of Theorem \ref{['t:cs']} in Section \ref{['s:ecc']}
  12. Proof of Lemma \ref{['l:sim1']} in Section \ref{['s:pw']} and its extension to randomly sampled weights
  13. Supplementary simulation results
  14. Simulation results on the consistency of $\hat{\rho}_{n,k}$
  15. Relation to the measures $\chi$ and $\bar{\chi}$
  16. ...and 2 more sections

Key Result

lemma 1

A random element $X$ in ${\mathbb B}$ is regularly varying with index $-\alpha$, $\alpha> 0$, if and only if there exist a sequence $b^{\prime}(n) \to \infty$ and a probability measure $\Gamma$ on ${\mathbb S}_{{\mathbb B}}$ (called the angular measure) such that for any $z>0$, for some $c>0$.

Figures (10)

  • Figure 1: The first three orthonormal basis elements in $L^2[0,1]$ defined in \ref{['eq:basis']} (left-most); simulated data when $\Gamma$ concentrates on $\phi_1$(second from the left); on $\phi_2$ (third left); on $\phi_3$ (fourth left).
  • Figure 2: The CIDR of three pairs of ETFs (1.XLF and XLK, 2.XLY and XLU, 3.XLY and XLE). For each pair, the curves representing the four most extreme days are displayed, with matching colors and line types indicating curves from the same day.
  • Figure 3: Estimates of the pairwise extremal correlation coefficients of CIDRs across the nine sectors.
  • Figure 4: Estimates of the pairwise coefficients of CIDRs, calculated from closing returns (left) and from all curves including non-extreme parts (right), are displayed.
  • Figure 5: The three locations in the United States: Fort Collins, CO; Colorado Springs, CO; Austin, TX. The pairwise extremal correlation of daily temperature curves between the three locations is evaluated.
  • ...and 5 more figures

Theorems & Definitions (30)

  • definition 1
  • lemma 1
  • definition 2
  • lemma 2
  • definition 3
  • proposition 1
  • proof
  • theorem 1
  • corollary 1
  • lemma 3
  • ...and 20 more