Nonparametric inference with directional and linear data

Resumo

The term directional data refers to data whose support is a circumference, a sphere or, generally, an hypersphere of arbitrary dimension. This kind of data appears naturally in several applied disciplines: proteomics, environmental sciences, biology, astronomy, image analysis or text mining. The aim of this thesis is to provide new methodological tools for nonparametric inference with directional and linear data (i.e., usual Euclidean data). Nonparametric methods are obtained for both estimation and testing, for the density and the regression curves, in situations where directional random variables are present, that is, directional, directional-linear and directional-directional random variables. The main contributions of the thesis are collected in six papers briefly described in what follows. In García-Portugués et al. (2013a) different ways of estimating circular-linear and circular-circular densities via copulas are explored for an environmental application. A new directional-linear kernel density estimator is introduced in García-Portugués et al. (2013b) together with its basic properties. Three new bandwidth selectors for the kernel density estimator with directional data are given in García-Portugués (2013) and compared with the available ones. The directional-linear estimator is used in García-Portugués et al. (2014a) for constructing an independence test for directional and linear variables that is applied to study the dependence between wildfire orientation and size. In García-Portugués et al. (2014b) a central limit theorem for the integrated squared error of the directional-linear estimator is presented. This result is used to derive the asymptotic distribution of the independence test and of a goodness-of-fit test for parametric directional-linear and directional-directional densities. Finally, a local linear estimator with directional predictor and linear response is given in García-Portugués et al. (2014c) jointly with a goodness-of-fit test for parametric regression functions. References García-Portugués, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electron. J. Stat., 7:1655-1685. García-Portugués, E., Barros, A. M. G., Crujeiras, R. M., González-Manteiga, W., and Pereira, J. (2014a). A test for directional-linear independence, with applications to wildfire orientation and size. Stoch. Environ. Res. Risk Assess., 28(5):1261-1275. García-Portugués, E., Crujeiras, R. M., and González-Manteiga, W. (2013a). Exploring wind direction and SO2 concentration by circular-linear density estimation. Stoch. Environ. Res. Risk Assess., 27(5):1055-1067. García-Portugués, E., Crujeiras, R. M., and González-Manteiga, W. (2013b). Kernel density estimation for directional-linear data. J. Multivariate Anal., 121:152-175. García-Portugués, E., Crujeiras, R. M., and González-Manteiga, W. (2014b). Central limit theorems for directional and linear data with applications. Statist. Sinica, to appear. García-Portugués, E., Van Keilegom, I., Crujeiras, R. M., and González-Manteiga, W. (2014c). Testing parametric models in linear-directional regression. arXiv:1409.0506.