Bivariate splines for spatial functional regression models

Abstract

We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, in various situations where both the explanatory variable and the noise can be unbounded and dependent. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with a penalisation term. Under the assumptions that the regressors in the sample span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an application to ozone concentration forecasting over the USA that illustrates the predictive skills of the method.

Wavelet-domain regression and predictive inference in psychiatric neuroimaging

“…Spline methods for two-dimensional predictors have been studied by Marx and Eilers (2005) and Guillas and Lai (2010), and by Reiss and Ogden (2010), whose work was motivated by neuroimaging applications.…”

Spatial autoregressive and moving average Hilbertian processes

“…Recently, in the non-parametric context, Basse et al [2] analyze the properties of kernel-based density estimators formulated in the context of spatial functional random variables. Guillas and Lai [15] propose a new class of spatial functional regression models based on bivariate splines, in terms of which the surface defining the explanatory random variables is approximated. Such an approximation allows the construction of least squares estimators of the regression function with or without a penalization term.…”

Spatial autoregressive functional plug-in prediction of ocean surface temperature

“…However, the field of Spatial Functional Statistics still requires further development. We mention a few recent papers regarding the statistical inference from spatial correlated functional random variables: Guillas and Lai (2008) propose a new class of spatial functional regression models based on bivariate splines, in terms of which the surface defining the explanatory random variables is approximated. Such an approximation allows the construction of least squares estimators of the regression function with or without a penalization term.…”