Dependence for functional data
- Autores: Daliia Jazmín Valencia García
- Directores de la Tesis: Juan José Romo Urroz (dir. tes.), Rosa Elvira Lillo Rodríguez (dir. tes.)
- Lectura: En la Universidad Carlos III de Madrid ( España ) en 2014
- Idioma: inglés
- Tribunal Calificador de la Tesis: Francisco Javier Prieto Fernández (presid.), Alicia Nieto Reyes (secret.), Ana María Aguilera del Pino (voc.)
- Materias:
- Enlaces
- Tesis en acceso abierto en: e-Archivo
- Resumen
Measuring dependence is a basic question when dealing with functional observations. It is of great interest to know the effect that one or more functional variables can have on other ones, and even predict values of one variable from another. Although, in the functional context, this theory has not been as extensively studied,some techniques to measure dependence in functional data have already been implemented, providing a single value which represents the degree of relation between the sets of curves. However,these measures ar eusually not robust, which makes them less stable in the presence of outliers. Therefore, it is interesting to develop robust techniques that ensure high stability of the statistics. This thesis is motivated by the above issues and aims to provide measures of dependence for sets ofcurves that are more robust than those used so far. Hence, we extend non-parametric bivariate coefficients,such as Kendall’s ? and Spearman’s coefficient,to functions,i.e.to situations where the observed data are curves generated by a stochastic process. These coefficients are based on the natural data ordering, but when we work in the context of functional data,there is no such thing as a natural order between functions, meaning that we need to provide for an ordering of curves. Thus, our first task is to consider suitable ways to sort the observations. For this, we use different functional preorders, which allow us to define the coefficients in a way similar to the bivariate case. The aforementioned coefficients provide an univariate measure of the dependence between two sets of curves, which leads us to propose in the final chapter a new functional correlation coefficient that yields a representative curve of dependence between two sets of functional data. This coefficient is based on the cross-correlation function studied in the literature of functional data,which is the classic Pearson coefficient between the values of the curves indifferent time instants. We adapt theconcept of MAD and comedian to measure dependence between two sets of functions and, through them, introduce a robust alternative to the cross-correlation function, which we will call correlation median for functions. The thesis is organized as follows.In Chapter 1 we start defining what is understood as complex data in this work and show several examples. These data will be treated as functional data. Then,a review of the different approaches to analyze functional data is provided. We also offer a brief review of some of the most common measures of dependence between random variables, focusing on those where we make our contribution. This chapter also analyzes some techniques that have been extended to the functional context for calculating the dependence between two sets of curves in order to compare our results. Finally, we study the principal ordering measures for functional data which are necessary to sort the curves, and thus define the coefficients in the functional setting. In Chapter 2 we define the Kendall ? coefficient for functional observations based on the concept of functional concordance, also new in this dissertation. We study its statistical properties and provide some applications to real data, including as set portfolios infinance and microarray time series in genetics. In Chapter 3 we present a notion of Spearman’s coefficient for functional data that extends the classic bivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. Since Spearman’s coefficient for bivariate samples is based on the natural data ordering in dimension one, we need to consider a data order in the functional context. The development uses a pre-order inspired in the depth definition,but considering a down-up ordering in stead of a center-outward ordering of the sample, allowing us to introduce the notion of grade for functions to properly define the Spearman coefficient. We show some of the main characteristics of Spearman’s coefficient for functions and propose an independence test with a bootstrap methodology. We illustrate the performance of the new coefficient with both simulated and real data. The results of Chapter 4 concern a new functional correlation coefficient that is more robust than the cross-correlation function. The pair (median, MAD)is known to be a robust alternative to the pair (mean, standard deviation). Using the idea underlying the calculation of the MAD, Falk [19] defined a robust estimator for the covariance called comedian. In this chapter we adapt these concepts, the MAD and the comedian,to functional data. These measures allow us to define a robust alternative to the cross-correlation function studied in the literature of functional data, which we will call the correlation median for functions. Since the most natural extension of median in the functional context has been performed through depth measurements,the functional MAD and comedian will also b econstructed via depth. These concepts are illustrated with simulated and real data. Finally,in Chapter 5, we present some general conclusions and summarize the main contributions of the dissertation. --------------------------------------------------------------------------
