Preprocessing of centred logratio transformed density functions using smoothing splines

Abstract

With large-scale database systems, statistical analysis of data, formed by probability distributions, become an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment still represents a challenging task in functional data analysis. Namely, the usual L 2 metric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are followed by Bayes spaces of measures. The easiest possibility of expressing density functions in L 2 space is to use centred logratio transformation, nevertheless, it results in functional data with a constant integral constraint that needs to be taken into account for further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics.

Simplicial principal component analysis for density functions in Bayes spaces

“…In our setting, an explicit expression can be derived for the optimal coefficients. Indeed, they are obtained as (Machalová et al, 2015) c * = DK  ( k+1 (t)DK) ⊤ W k+1 (t)DK + λ (DK) ⊤ N kl DK  + K ⊤ D ⊤ ⊤ k+1 (t)Wy, where W = diag(w), A + denotes Moore-Penrose pseudoinverse of a matrix A, ⟨φ k+1−l −k+l , φ k+1−l −k+l ⟩ 2 · · · ⟨φ k+1−l g , φ k+1−l −k+l ⟩ 2 . .…”

“…In the case study considered in Section 6, we set the weights as well as the value of the smoothing parameter λ to one, following the default setting of Machalová et al (2015). For possible sensible determination of λ using, e.g., cross-validation, we refer to Kneip et al (2012) and Liebl (2013).…”

“…We refer to Machalová et al (2015) for further details. Let ∆τ := τ 0 = a < τ 1 < · · · < τ g < b = τ g+1 be a given sequence of knots in I.…”

“…To compute the B-splines coefficients the usual parametrization of smoothing splines applies, and the additional constraint is incorporated in the estimation algorithm as described in Machalová et al (2015). We refer to the Appendix for further details.…”

“…The raw data have been smoothed by using the procedure proposed in Machalová et al (2015), and recalled in the Appendix: the discrete clr-transforms of raw densities have been projected on a B-spline basis with support I = [0, 100] and five equally spaced knots in the years (0, 25, 50, 75, 100), with constraints to fulfil the zero-integral condition. We remark that the choice of the number and the position of the knots can follow the same strategies that are used in FDA.…”