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Four Measures of Association and Their Representations in Terms of Copulas

by Michel Adès, Serge B. Provost and Yishan Zang

AppliedMath 2024, 4(1), 363-382; https://doi.org/10.3390/appliedmath4010019 - 2 Mar 2024

Cited by 3 | Viewed by 1466

Four measures of association, namely, Spearman’s

ρ

, Kendall’s

τ

, Blomqvist’s

β

and Hoeffding’s

Φ^{2}

Four measures of association, namely, Spearman’s

ρ

, Kendall’s

τ

, Blomqvist’s

β

and Hoeffding’s

Φ^{2}

, are expressed in terms of copulas. Conveniently, this article also includes explicit expressions for their empirical counterparts. Moreover, copula representations of the four coefficients are provided for the multivariate case, and several specific applications are pointed out. Additionally, a numerical study is presented with a view to illustrating the types of relationships that each of the measures of association can detect. Full article

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20 pages, 514 KiB

Open AccessArticle

Modeling Bivariate Dependency in Insurance Data via Copula: A Brief Study

by Indranil Ghosh, Dalton Watts and Subrata Chakraborty

J. Risk Financial Manag. 2022, 15(8), 329; https://doi.org/10.3390/jrfm15080329 - 25 Jul 2022

Cited by 5 | Viewed by 4509

Abstract

Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate. Full article

(This article belongs to the Special Issue Financial Data Analytics and Statistical Learning)

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15 pages, 1797 KiB

Open AccessArticle

Modelling Training Adaptation in Swimming Using Artificial Neural Network Geometric Optimisation

by Justin Carrard, Petr Kloucek and Boris Gojanovic

Sports 2020, 8(1), 8; https://doi.org/10.3390/sports8010008 - 16 Jan 2020

Cited by 9 | Viewed by 4564

Abstract

This study aims to model training adaptation using Artificial Neural Network (ANN) geometric optimisation. Over 26 weeks, 38 swimmers recorded their training and recovery data on a web platform. Based on these data, ANN geometric optimisation was used to model and graphically separate adaptation from maladaptation (to training). Geometric Activity Performance Index (GAPI), defined as the ratio of the adaptation to the maladaptation area, was introduced. The techniques of jittering and ensemble modelling were used to reduce overfitting of the model. Correlation (Spearman rank) and independence (Blomqvist β) tests were run between GAPI and performance measures to check the relevance of the collected parameters. Thirteen out of 38 swimmers met the prerequisites for the analysis and were included in the modelling. The GAPI based on external load (distance) and internal load (session-Rating of Perceived Exertion) showed the strongest correlation with performance measures. ANN geometric optimisation seems to be a promising technique to model training adaptation and GAPI could be an interesting numerical surrogate to track during a season. Full article

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