Penalized PCA approaches for B-spline expansions of smooth functional data

Functional principal component analysis (FPCA) is a dimension reduction technique that explains the dependence structure of a functional data set in terms of uncorrelated variables. In many applications the data are a set of smooth functions observed with error. In these cases the principal components are difficult to interpret because the estimated weight functions have a lot of variability and lack of smoothness. The most common way to solve this problem is based on penalizing the roughness of a function by its integrated squared d-order derivative. Two alternative forms of penalized FPCA based on B-spline basis expansions of sample curves and a simpler discrete penalty that measures the roughness of a function by summing squared d-order differences between adjacent B-spline coefficients (P-spline penalty) are proposed in this paper. The main difference between both smoothed FPCA approaches is that the first uses the P-spline penalty in the least squares approximation of the sample curves in terms of a B-spline basis meanwhile the second introduces the P-spline penalty in the orthonormality constraint of the algorithm that computes the principal components. Leave-one-out cross-validation is adapted to select the smoothing parameter for these two smoothed FPCA approaches. A simulation study and an application with chemometric functional data are developed to test the performance of the proposed smoothed approaches and to compare the results with non penalized FPCA and regularized FPCA. Many application areas have functional data that come from the observation of a random function at a discrete set of sampling points. In the majority of cases the sample curves are functions of time but in many others the argument is a different magnitude. In spectroscopy, for example, the NIR spectrum is a functional variable whose observations are measured as functions of wavelengths. The potential of functional data analysis methodologies for the chemometric analysis of spectro-scopic data was shown in [1]. A wide variety of applications with functional data in different fields were collected and analyzed by [2]. When analyzing a functional data set it is usual to have a large number of regularly spaced observations for each sample curve. Because of this a reduction dimension technique is necessary for explaining the main features of a set of sample curves in terms of a small set of uncorrelated variables. This problem was solved by generalizing principal component analysis to the case of a continuous-time stochastic process [3]. Asymptotic properties of the estimators of FPCA …