Copula-Based Measures of Multivariate Association

TL;DR: Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube as discussed by the authors.

Abstract: ABSTRACT Time series clustering is a widely used unsupervised learning approach that identifies groups of similar time series to uncover hidden patterns in complex datasets. In recent years, this technique has gained traction in the analysis of geo‐referenced time series, where spatial information must be incorporated into the dissimilarity measure to achieve meaningful results. This paper offers a thorough review of dissimilarity‐based clustering methods with soft spatial constraints, i.e., approaches that integrate spatial context into the clustering process without enforcing strict spatial proximity within clusters. Our focus is on copula‐based clustering techniques, which effectively capture comovements among time series without requiring explicit modeling of their marginal distributions. We first introduce a general framework for copula‐based time series clustering and then explore how spatial constraints can be embedded into the clustering process. Finally, we propose a general framework, called Triple‐C , which provides two comprehensive model architectures that address this challenge through either a dissimilarity fusion step or a copula aggregation approach.

TL;DR: This paper proposes five copula-based measures of association between random vectors, capturing positive and negative relationships, and investigates their properties, nonparametric estimators, and small sample behavior through simulation and application to bond and stock indices.

TL;DR: It is shown that MI does not correctly reflect the proportion of deterministic signals hidden in noisy data, and the copula correlation (Ccor), based on the L1-distance of copula density, is shown to be equitable under both definitions.

TL;DR: The Kernel Method for Multivariate Data: Three Important Methods and Density Estimation in Action.

TL;DR: Density estimation, as discussed in this book, is the construction of an estimate of the density function from the observed data from an unknown probability density function.

TL;DR: This book discusses the fundamental properties of copulas and some of their primary applications, which include the study of dependence and measures of association, and the construction of families of bivariate distributions.

TL;DR: Rank correlation as mentioned in this paper is a measure of similarity between two rankings of the same set of individuals, and it has been used in psychological work to compare two different rankings of individuals in order to indicate similarity of taste.