مدل RKHS برای انتخاب متغیر در رگرسیون خطی کارکردی An RKHS model for variable selection in functional linear regression

1. Introduction: statement of the problem and motivation The problem under study: variable selection in functional regression The study of regression models is clearly among the leading topics in statistics. In particular, these models play a central role in the theory of statistics with functional data, often called Functional Data Analysis (FDA); see [7, 15, 16] for an overview on FDA. Throughout this paper, we will consider “functional data” consisting of independent X1 = X1(t), . . . , Xn = Xn(t) observations (trajectories) drawn from a second-order (L 2 ) stochastic process X = X(t), t ∈ [0, 1], with continuous trajectories and continuous mean and covariance functions, denoted by m = m(t) and K(s, t), respectively. All the involved random variables are supposed to be defined in a common probability space (Ω, A, Pr). We are interested on functional regression models with scalar response, of type Yi = g(Xi) + εi , where g is a real function defined on a suitable space X where the trajectories of our process are supposed to live. The random variables εi are independent errors (and also independent from the Xi) with mean zero and common variance σ 2 . More specifically, we are concerned with variable selection issues; see, [4, Sec. 1], [11] for additional information and references. Basically, a variable selection functional method is an automatic procedure that takes a function {x(t), t ∈ [0, 1]} to a finite-dimensional vector (x(t1), . . . , x(tp)). The overall idea for variable selection is to choose the variables x(ti) (or, equivalently, the “impact points” t1, . . . , tp ∈ [0, 1]; see [22]), in an “optimal way” so that the original functional problem (regression, classification, clustering,…) is replaced with the corresponding multivariate version, based on the selected variables. In the regression setting, this would amount to replace the functional model Yi = g(Xi) + εi with a finite dimensional version of type Yi = φ{X(t1), . . . , X(tp)} + ei . Nevertheless, note that still the problem is of a functional nature, since the methods to select the ti are generally based upon the full data trajectories.