Simplicial principal component analysis for density functions in Bayes spaces

Abstract

Probability density functions are frequently used to characterize the distributional properties\ud of large-scale database systems. As functional compositions, densities primarily carry\ud relative information. As such, standard methods of functional data analysis (FDA) are not\ud appropriate for their statistical processing. The specific features of density functions are\ud accounted for in Bayes spaces, which result from the generalization to the infinite dimensional\ud setting of the Aitchison geometry for compositional data. The aim is to build up a\ud concise methodology for functional principal component analysis of densities. A simplicial\ud functional principal component analysis (SFPCA) is proposed, based on the geometry\ud of the Bayes space B2 of functional compositions. SFPCA is performed by exploiting the\ud centred log-ratio transform, an isometric isomorphism between B2 and L2 which enables\ud one to resort to standard FDA tools. The advantages of the proposed approach with respect\ud to existing techniques are demonstrated using simulated data and a real-world example of\ud population pyramids in Upper Austria

A general framework for multivariate functional principal component analysis of amplitude and phase variation

“…We define a general class of transformations that includes existing approaches from the literature as well as alternative transformations based on density functions on and allows to compare their strengths and limitations. We offer theoretical arguments in favour of the centred log‐ratio (clr) transformation, which relates to the Bayes space of densities (see, e.g., Egozcue, Díaz‐Barrero, & Pawlowsky‐Glahn, ; Hron, Menafoglio, Templ, Hrůzová, & Filzmoser, ) and illustrate this by means of a simulated toy example. Second, we embed the methods of Hadjipantelis et al () and Lee and Jung () for the joint analysis of amplitude and phase variation into a multivariate functional principal component framework, which is based on univariate functional PCA and takes correlations of warping functions and registered functions into account.…”

Profile Monitoring of Probability Density Functions via Simplicial Functional PCA With Application to Image Data

“…Hence, the objective functional in (8) and (9) is the sample variance of the projections along the generic direction ζ, that has to be maximized to find the principal directions. Hron et al (2016) proved that minimization of (8)- (9) can be performed by solving an equivalent FPCA problem in L 2 , on a transformed dataset. More precisely, one can map the dataset of density curveŝ f 1 , .…”

“…This allows to generalize most methods in FDA -that are typically developed for data in L 2 -to the Bayes space setting. Amongst these, we here focus on Simplicial Functional Principal Component Analysis (SFPCA, Hron et al, 2016), that aims to reduce the dimensionality of a dataset of density functions. Given f 1 , ..., f M , we aim to find the directions in B 2 , denoted by ζ 1 , .…”

“…This may have a detrimental effect both on process monitoring performances and on dimensionality reduction capabilities. A number of authors (e.g., , Van den Boogaart et al, 2010, Delicado (2011, Van den Boogaart et al (2014), Menafoglio et al (2014Menafoglio et al ( , 2016aMenafoglio et al ( , 2016b, Hron et al (2016)) pointed out that PDFs can be interpreted as functional compositional data, i.e., functional observations carrying only relative information, which are usually collected in the form of constrained data integrating to a constant. Traditional FDA techniques operate in the space of square-integrable real measurable functions L 2 , whereas compositional data entails the use of a different space, known as Bayes space, B 2 , Van den Boogaart et al, 2010, Egozcue et al (2013, Van den Boogaart et al (2014)), that generalizes to the functional setting the well-known Aitchison geometry for compositional data (Aitchison, 1986;Pawlowsky-Glahn and Egozcue, 2001;Egozcue, 2009;Pawlowsky-Glahn and Buccianti, 2011).…”

“…As a matter of fact, when the functional data of interest is a PDF, f (x), two constraints have to be satisfied, namely (i) f (x) > 0 and (ii) f (x) = 1. By performing PCA on PDFs, even if the loadings should form an empirical basis for the original data, densities approximated on the basis of the retained functional principal components (FPCs) may violate the above constraints (Delicado (2011), Hron et al (2016)). This may have a detrimental effect both on process monitoring performances and on dimensionality reduction capabilities.…”