Simplicial principal component analysis for density functions in Bayes spaces

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

Object Oriented Geostatistical Simulation of Functional Compositions via Dimensionality Reduction in Bayes spaces

We address the problem of geostatistical simulation of spatial complex data, with emphasis on functional compositions (FCs). We pursue an object oriented geostatistical approach and interpret FCs as random points in a Bayes Hilbert space. This enables us to deal with data dimensionality and constraints by relying on a solid geometric basis, and to develop a simulation strategy consisting of: (i) optimal dimensionality reduction of the problem through a simplicial principal component analysis, and (ii) geostatistical simulation of random realizations of FCs via an approximate multivariate problem.We illustrate our methodology on a dataset of natural soil particle-size densities collected in an alluvial aquifer

Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric

We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, exploiting a constrained B-spline approximation. As a byproduct of our approach, we are also able to derive faster routines for previous work on PCA for distributions. By means of simulation studies, we compare our approaches to previously proposed methods, showing that our projected PCA has similar performance for a fraction of the computational cost and that the projected regression is extremely flexible even under misspecification. Several theoretical properties of the models are investigated and asymptotic consistency is proven. Two real world applications to Covid-19 mortality in the US and wind speed forecasting are discussed

Profile Monitoring of Probability Density Functions via Simplicial Functional PCA with application to Image Data

The advance of sensor and information technologies is leading to data-rich industrial environments, where large amounts of data are potentially available. This study focuses on industrial applications where image data are used more and more for quality inspection and statistical process monitoring. In many cases of interest, acquired images consist of several and similar features that are randomly distributed within a given region. Examples are pores in parts obtained via casting or additive manufacturing, voids in metal foams and light-weight components, grains in metallographic analysis, etc. The proposed approach summarizes the random occurrences of the observed features via their (empirical) probability density functions (PDFs). In particular, a novel approach for PDF monitoring is proposed. It is based on simplicial functional principal component analysis (SFPCA), which is performed within the space of density functions, that is, the Bayes space B2. A simulation study shows the enhanced monitoring performances provided by SFPCA-based profile monitoring against other competitors proposed in the literature. Finally, a real case study dealing with the quality control of foamed material production is discussed, to highlight a practical use of the proposed methodology. Supplementary materials for the article are available online

Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.Peer Reviewe

Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.Peer ReviewedPostprint (author's final draft