Copula Correlation: An Equitable Dependence Measure and Extension of Pearson's Correlation

In Science, Reshef et al. (2011) proposed the concept of equitability for measures of dependence between two random variables. To this end, they proposed a novel measure, the maximal information coefficient (MIC). Recently a PNAS paper (Kinney and Atwal, 2014) gave a mathematical definition for equitability. They proved that MIC in fact is not equitable, while a fundamental information theoretic measure, the mutual information (MI), is self-equitable. In this paper, we show that MI also does not correctly reflect the proportion of deterministic signals hidden in noisy data. We propose a new equitability definition based on this scenario. The copula correlation (Ccor), based on the L1-distance of copula density, is shown to be equitable under both definitions. We also prove theoretically that Ccor is much easier to estimate than MI. Numerical studies illustrate the properties of the measures

Comparative Evaluation of Statistical Dependence Measures

Measuring and testing dependence between random variables is of great importance in many scientific fields. In the case of linearly correlated variables, Pearson’s correlation coefficient is a commonly used measure of the correlation strength. In the case of nonlinear correlation, several innovative measures have been proposed, such as distance-based correlation, rank-based correlations, and information theory-based correlation. This thesis focuses on the statistical comparison of several important correlations, including Spearman’s correlation, mutual information, maximal information coefficient, biweight midcorrelation, distance correlation, and copula correlation, under various simulation settings such as correlative patterns and the level of random noise. Furthermore, we apply those correlations with the overall best performance to a real genomic data set, to study the co-expression between genes in serous ovarian cancer

Copula-based methods and their application to multidimensional poverty analysis

En esta tesis, proponemos utilizar la metodología basada en cópulas para analizar la dependencia entre las dimensiones de la pobreza. Este enfoque, que ha sido recientemente introducido en el ámbito de la Economía del Bienestar, se centra en las posiciones de los individuos en las dimensiones, en lugar de en los valores específicos que esas dimensiones toman para tales individuos, y es particularmente útil cuando se mide la dependencia en contextos multivariantes, posiblemente no gaussianos y posiblemente no lineales, como los que solemos encontrar en los análisis multidimensionales de pobreza o bienestar. En particular, consideramos varios conceptos de dependencia multivariante basados en cópulas que son especialmente adecuados para el estudio de la pobreza multidimensional, a saber, los conceptos de concordancia multivariante, orthant dependence (dependencia en el ortante) y tail dependence (dependencia en las colas) multivariante.Departamento de Economía AplicadaDoctorado en Economí

Essays on multidimensional poverty measurement and the dependence among well-being dimensions

Evaluating the welfare of nations is high on the research agenda of the economists, practitioners and policy-makers. The literature contributions of the last decades triggered a multivariate perception of the well-being, which is suggested to go beyond the GDP, and created a need for more complex approaches to evaluate the welfare as well as poverty. The first essay investigates the approaches to multivariate poverty measurement and focuses on the composite index approach and the steps involved in it. An important aspect of the multivariate perspective in well-being is the dependence among the underlying indicators. There is a growing evidence in the literature that well-being dimensions are interrelated. This dependence among attributes matters for multidimensional poverty measurement, since income is no longer the only indicator to be considered. However, the reviewed approaches to multivariate poverty measurement do not commonly capture this interdependence. The second essay suggests a copula function as a flexible tool to estimate the dependence among welfare variables. Moreover, it proposes to incorporate the evaluated dependence in the composite indicator. The trade-off among attributes, which is established via the weighting of dimensions, is identified as a possible channel to include the interdependence in the composite indicator. The third essay of this dissertation defines bivariate and multivariate copula-based measures of dependence and applies them using the recent data from the EU-SILC. The results suggest that key dimensions of well-being, i.e. income, education and health, are positively interdependent. In addition, the strength of pairwise and multivariate dependence reinforced in the post-crises period in some European countries. Finally, the last essay proposes a new class of the copula-based multidimensional poverty indices by innovating over the weighting approach. The weighting scheme proposed in this dissertation incorporates the estimated copula-based dependence and contains necessary normative controls to be chosen by the practitioner. The findings of the last essay suggest that the overall poverty is driven not only by the individual shortfalls, but also I by the degree of interdependence among well-being indicators. Considering the proposed copula-based weighting scheme and the proposal of the new class of copula-based poverty indices, this dissertation contributes to the multivariate poverty measurement by suggesting the channel to enclose the dependence structure in the composite indicators. The proposed copula-based methodology will advance the multidimensional poverty analysis and the poverty-reducing policy, which can be designed to address the problem of interdependence of individual achievements

Essays on multidimensional poverty measurement and the dependence among well-being dimensions

Evaluating the welfare of nations is high on the research agenda of the economists, practitioners and policy-makers. The literature contributions of the last decades triggered a multivariate perception of the well-being, which is suggested to go beyond the GDP, and created a need for more complex approaches to evaluate the welfare as well as poverty. The first essay investigates the approaches to multivariate poverty measurement and focuses on the composite index approach and the steps involved in it. An important aspect of the multivariate perspective in well-being is the dependence among the underlying indicators. There is a growing evidence in the literature that well-being dimensions are interrelated. This dependence among attributes matters for multidimensional poverty measurement, since income is no longer the only indicator to be considered. However, the reviewed approaches to multivariate poverty measurement do not commonly capture this interdependence. The second essay suggests a copula function as a flexible tool to estimate the dependence among welfare variables. Moreover, it proposes to incorporate the evaluated dependence in the composite indicator. The trade-off among attributes, which is established via the weighting of dimensions, is identified as a possible channel to include the interdependence in the composite indicator. The third essay of this dissertation defines bivariate and multivariate copula-based measures of dependence and applies them using the recent data from the EU-SILC. The results suggest that key dimensions of well-being, i.e. income, education and health, are positively interdependent. In addition, the strength of pairwise and multivariate dependence reinforced in the post-crises period in some European countries. Finally, the last essay proposes a new class of the copula-based multidimensional poverty indices by innovating over the weighting approach. The weighting scheme proposed in this dissertation incorporates the estimated copula-based dependence and contains necessary normative controls to be chosen by the practitioner. The findings of the last essay suggest that the overall poverty is driven not only by the individual shortfalls, but also I by the degree of interdependence among well-being indicators. Considering the proposed copula-based weighting scheme and the proposal of the new class of copula-based poverty indices, this dissertation contributes to the multivariate poverty measurement by suggesting the channel to enclose the dependence structure in the composite indicators. The proposed copula-based methodology will advance the multidimensional poverty analysis and the poverty-reducing policy, which can be designed to address the problem of interdependence of individual achievements