Support Vector Regression Methods for Functional Data

Abstract

Many regression tasks in practice dispose in low gear instance of digitized functions as predictor variables. This has motivated the development of regression methods for functional data. In particular, Naradaya-Watson Kernel (NWK) and Radial Basis Function (RBF) estimators have been recently extended to functional nonparametric regression models. However, these methods do not allow for dimensionality reduction. For this purpose, we introduce Support Vector Regression (SVR) methods for functional data. These are formulated in the framework of approximation in reproducing kernel Hilbert spaces. On this general basis, some of its properties are investigated, emphasizing the construction of nonnegative definite kernels on functional spaces. Furthermore, the performance of SVR for functional variables is shown on a real world benchmark spectrometric data set, as well as comparisons with NWK and RBF methods. Good predictions were obtained by these three approaches, but SVR achieved in addition about 20% reduction of dimensionality.

Support vector regression for functional data in multivariate calibration problems

“…Some results that allow for the construction of some classes of nnd kernels on functional spaces were given in Ref. [17], where the Gaussian kernel appears as a particular case of these class of kernel.…”

“…Support vector Regression methods for Functional Data have been introduced recently by Hernández et al [17]. It is known that estimation methods for very general regression models have been elaborated on the basis of regularization in RKHS [18,19,21].…”

“…These nonparametric techniques require a high amount of computer memory to encode the estimates in order to make future predictions. To avoid this drawback Hernández et al [17] introduced support vector (SV) regression methods for functional data.…”

Dissimilarity representation on functional spectral data for classification

“…If the used models are correct, they are expected to perform better than the traditional techniques, as these have to learn (linear) relations from the data. More recently, a number of estimation methods for functional nonparametric classification and regression models have also been introduced, namely k‐Nearest Neighbor classifier 14, kernel methods 15–17 such as Support Vector Machine 18, 19–21.…”

A Functional Density-Based Nonparametric Approach for Statistical Calibration

“…In the case of functional regression, where one intends to estimate a random scalar variable Y from a functional variable X taking values in a functional space X , earlier works were focused on linear methods such as the functional linear model with scalar response [2][3][4][5][6][7][8] or the functional Partial Least Squares [9]. More recently, the problem has also been addressed nonparametrically with smoothing kernel estimates [10], multilayer perceptrons [11], and support vector regression [12,13]. Another point of view between these two approaches is to use a semi-parametric approach, such as the SIR (Sliced Inverse Regression, [14]) that has been extended to functional data in [15][16][17].…”