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aThe introduction of support vector regression (SVR) and least square support vector machines (LS-SVM) methods for regression purposes in the field of chemometrics has provided advantageous alternatives to the existing linear and nonlinear multivariate calibration (MVC) approaches. Relevance vector machines (RVMs) claim the advantages attributed to all the SVM-based methods over many other regression methods. Additionally, it also exhibits advantages over the standard SVM-based ones since: it is not necessary to estimate the error/margin trade-off parameter C and the insensitivity parameter in regression tasks, it is applicable to arbitrary basis functions, the algorithm gives probability estimates seamlessly and offer, additionally, excellent sparseness capabilities, which can result in a simple and robust model for the estimation of different properties. This paper presents the use of RVMs as a nonlinear MVC method capable of dealing with ill-posed problems. To study its behavior, three different chemometric benchmark datasets are considered, including both linear and non-linear solutions. RVM was compared with other calibration approaches reported in the literature. Although RVM performance is comparable with the best results obtained by LS-SVM, the final model achieved is sparser, so the prediction process is faster. Taking into account the other advantages attributed to RVMs, it can be concluded that this technique can be seen as a very promising option to solve nonlinear problems in MVC.

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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.