Gallo, Michele

MARTíN FERNANDEZ JA,

Todorov, V

Trendafilov, N

English

Università degli Studi di Napoli L'Orientale: CINECA IRIS

Tri-PLS for compositional dataGallo M.a∗, Trendafilov N.b , Todorov V.c and Martı´n-Ferna´ndez J.A.daDepartment of Human and Social Sciences, University of Naples-L’Orientale,P.zza S.Giovanni 30, Naples (It-80134), ItalybDepartment of Mathematics and Statistics, The Open University,Walton Hall, Milton Keynes MK7 6AA, UKcUnited Nations Industrial Development Organization,Wagramerstr 5 P.O. Box 300 A-1400, Vienna, Austriad Departament IMAE, Universitat de Girona,Campus Montilivi, Edif. P-IV, Girona (E-17071), Kingdom of SpainKeywords: Compositions, Aitchison geometry, Log ratio, NPLS, trilinear modelsIntroductionCompositional data (CoDa, [1] and [2]) consist of vectors of positive values summing to a unit, or in generalto some fixed constant. They can often be found in many disciplines and appear as proportions, percentages,concentrations, absolute and relative frequencies. Unfortunatly, the constant-sum constraint that characterizescompositions is frequently disregarded or improperly incorporated into statistical modeling and a misleadinginterpretation of the results is given. Due to these specifications, several difficulties arise when dealing with CoDa.The first word of warning came already in 1897 from Karl Pearson who showed the dangers of underestimatingspurious correlations.There are several approaches to incorporate CoDa into statistical modeling when it is not realistic to assume amultinomial distribution of the data. Based on the log-ratio transformations, Aitchison [1] proposed preprocessingthe compositional data by means of log-ratio transformations, and successively analyzing them in a straight-forward way by ’traditional’ methods. Following Aitchison’s approach, the high dimensionality of CoDa in manyscientific fields has encouraged the use of bilinear and trilinear decomposition models. Thus, in attempts to findadequate low-dimensional descriptions of compositional variability, CoDa are collected into two or three-wayarrays ([3], [4], [5], [6], [7]). On the other side, Hinkle and Rayens [8] examined the problems that potentiallyoccur when one performs a partial least squares (PLS) on compositional data.The principal goal of this talk is to extend the PLS regression to three-way compositional data, following theapproach proposed by Bro [9] and Bro and al. [10]. Both Candecomp/Parafac (CP - [11] [12]) and Tucker3 [13]models can be viewed as latent variables models extending principal component analysis to three-way data set.However, the most fundamental properties of PCA cannot be extended to these two models. PCA is an optimalrepresentation of a two-way array with respect to the criteria of best low-rank approximation in least squaressense and the best approximation of the data within a joint low-dimensional subspace, while Tucker3 is only thebest approximation of a three-way array within a joint low-dimensional subspace and CP is the best low-rankapproximation in a least squares sense.The proposed extension of PLS to three-way compositional data is illustrated on real data sets and a softwareimplementation will be available in the R package rrcovHD.∗Corresponding author. E-mail: mgallo@unior.it, nickolay.trendafilov@open.ac.uk, v.todorov@unido.org,josepantoni.martin@udg.edu.References[1] J. Aitchison, The statistical analysis of compositional data, Monographs on Statistics and AppliedProbability, 1986.[2] J. Aitchison, C. Barcelo´-Vidal, J. Martı´n-Ferna´ndez, and V. Pawlowsky-Glahn, “Logratio analysis andcompositional distance,” Mathematical Geology 32(3), pp. 271–275, 2000.[3] J. Aitchison, “The statistical analysis of compositional data,” Journal of the Royal Statistical Society. SeriesB (Methodological) , pp. 139–177, 1982.[4] J. Aitchison and M. Greenacre, “Biplots of compositional data,” Journal of the Royal Statistical Society:Series C (Applied Statistics) 51(4), pp. 375–392, 2002.[5] M. Gallo, “Tucker3 model for compositional data,” Commun. Statist. Theor. Meth. in press, 2013.[6] M. Engle, M. Gallo, K. Schroeder, N. Geboy, and Z. J.W., “Three-way compositional analysis of waterquality monitoring data,” Environmental and Ecological Statistics , pp. 1–17, 2013.[7] M. Gallo and A. Buccianti, “Weighted principal component analysis for compositional data: applicationexample for the water chemistry of the arno river (tuscany, central italy),” Environmetrics , 2013.[8] J. Hinkle and W. Rayens, “Partial least squares and compositional data: problems and alternatives,”Chemometrics and intelligent laboratory systems 30(1), pp. 159–172, 1995.[9] R. Bro, “Multiway calibration. multilinear pls,” J. Chemometrics 10(1), pp. 47–61, 1996.[10] R. Bro, A. Smilde, and S. de Jong, “On the difference between low-rank and subspace approximation:improved model for multi-linear pls regression,” Chemometrics and Intelligent Laboratory Systems 58(1),pp. 3–13, 2001.[11] J. Carroll and J. Chang, “Analysis of individual differences in multidimensional scaling via an n-waygeneralization of ”eckart-young? decomposition,” Psychometrika 45, pp. 283–319, 1970.[12] R. Harshman, “Foundations of the parafac procedure: Models and conditions for an ”explanatory” multi-modal factor analysis,” UCLA Working Papers Phonet , 1970.[13] L. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika 31(3), pp. 279–311,1966.

Tri-PLS for compositional data

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Tri-PLS for compositional data

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Università degli Studi di Napoli L'Orientale: CINECA IRIS