In general, linearity is assumed to hold in multivariate calibration (MVC), but this may not be true. We approach the MVC problem using multidimensional penalized signal regression, which can be extended with an explicit link function between linear prediction and response and in the spirit of single-index models. As the two-dimensional surface of calibration coefficients is smoothly and generally estimated with tensor product P-splines, the unknown link function is estimated using univariate Psplines. The methods presented are grounded in penalized regression, where difference penalties are placed on the rows and columns of the tensor product coefficients, as well as on the link function coefficients, each having its own tuning parameter. An application to ternary mixture data shows that a non-linearity is present. Performance comparisons are made to standard penalized signal regression, not only demonstrating the nonlinear effect, but also improvements in external prediction.

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Chemometrics and Intelligent Laboratory Systems
Erasmus MC: University Medical Center Rotterdam

Marx, B., Eilers, P., & Li, B. (2011). Multidimensional single-index signal regression. Chemometrics and Intelligent Laboratory Systems, 109(2), 120–130. doi:10.1016/j.chemolab.2011.08.006