Densidades, regresión y clasificación para datos funcionales

FORZANI, LILIANA; FRAIMAN, RICARDO; LLOP, PAMELA

Viña del Mar

Congreso; XII Congreso Latinoamericano de Probabilidad y Estadística Matemática; 2012

Sociedad Latinoamericana de Probabilidad y Estadística Matemática

The work provides the development, implementation, and analysis of statistical tools, which can be applied in areas where the number of measurements per individual is very large or even infinite such as curves and images. This kind of data appear in numerous applications such as health (electrocardiograms, magnetic resonance imaging), biomechanics (analysis of human body movements), chemistry (spectrometric curves), econometrics (indices), genetics (microarrays), geophysical (satellite imagery, spatial time series), among others. The interest in studying this kind of data is related to the technological progress which allow us to measure and record continuous data. It is for this reason that the need to implement statistical tools to analyze them is growing significantly. The first topic we address in this work is the nonparametric density estimation of a stationary stochastic processs X when n independent trajectories of a continuous time stochastic processes are observed over a fixed interval [0,T]. With respect to this problem, [Labrador, 2008] defined the k-NN estimator via local time when a single sample path is observed continuously over [0,T]. On this basis, we define a nonparametric estimator of the density function of X, we prove its consistency, find its puntual and uniform convergence rates, and prove its asymptotic normality. Then we find convergence rates of the estimator defined as the extension of the one given for stationary processes to a particular class of nonstationary ones. Finally, we apply this results to obtain a new classification rule for functional data. The second problem we address is the extention of the estimator given in the first part of the work to random fields. More precisely, we define a marginal density estimator based on a dependent set of stochastic processes and find convergence rates both in stationary and nonstationary cases. Finally, we provide a general consistency result for nonparametric regression estimates in the infinite dimensional context. This result is based on the universal consistency result given by [Stone,1977] in the finite dimensional setting. We start extending Stone's result to the case of locally compact metric spaces but, since the most interesting infinite-dimensional function spaces are not locally compact, we weaken this hypothesis by asking the regression function to be, somehow regular. From this general result, we derive consistency of the k-NN and kernel estimates whenever the Besicovitch condition holds and the regression function is bounded.