Functional linear regression : a case study from the food industry

A problem of analyzing functions in statistics is a very recent phenomenon that has been extensively investigated. The functionality in observed data, that is, when data are generated through a process naturally described as functional, occurs in many areas of sciences. For example, time series in financial engineering, imagining records in medicine, or spectrometric wavelengths in chemometrics can be considered as a discrete approximation of a continuous set of mathematical objects. In the initial step, observed real-valued data are transposed into functional data by smoothing or interpolation technique. These functional objects can then be directly used in the framework of statistical regressions. The aim of this paper is to show an estimation of the functional linear regression, where the observed covariate is functional, and its corresponding response is scalar. In the empirical study, we utilize spectrometric data with an objective to predict the fat content in a meat sample given the spectrum of absorbances, which are recorded through the infrared analyzer, by using the estimated functional linear regression. The functional regression coefficient is estimated through the basis spline expansion, for which we evaluate a different number of basis splines and propose the best predictor estimator.