A robust Parafac model for compositional data

References

• J. Aitchison, The Statistical Analysis of Compositional Data, Monographs on Statistics and Applied Probability, Chapman & Hall Ltd., London, 1986 (Reprinted in 2003 with additional material by The Blackburn Press) London (UK).
   Google Scholar
• J. Aitchison, C. Barceló-Vidal, J. Martín-Fernández, and V. Pawlowsky-Glahn, Logratio analysis and compositional distance, Math. Geol. 32 (2000), pp. 271–275. doi: 10.1023/A:1007529726302
   Google Scholar
• J. Aitchison and M. Greenacre, Biplots of compositional data, J. R. Stat. Soc. Ser. C Appl. Stat. 51 (2002), pp. 375–392. doi: 10.1111/1467-9876.00275
   Web of Science ®Google Scholar
• C.M. Andersen and R. Bro, Practical aspects of PARAFAC modelling of fluorescence excitation-emission data, J. Chemom. 17 (2003), pp. 200–215. doi: 10.1002/cem.790
   Web of Science ®Google Scholar
• D. Billheimer, P. Guttorp, and W. Fagan, Statistical interpretation of species composition, J. Amer. Statist. Assoc. 96 (2001), pp. 1205–1214. doi: 10.1198/016214501753381850
   Web of Science ®Google Scholar
• R. Bro, Multi-way analysis in the food industry: Models, algorithms, and applications, PhD thesis, University of Amsterdam, Amsterdam, The Netherlands, 1998.
   Google Scholar
• R. Bro and H.A.L Kiers, A new efficient method for determining the number of components in PARAFAC models, J. Chemom. 17(2003), pp. 274–286. doi: 10.1002/cem.801
   Web of Science ®Google Scholar
• J.D. Carroll and P. Arabie, Multidimensional scaling, Annu. Rev. Psychol. 31 (1980), pp. 438–457. doi: 10.1146/annurev.ps.31.020180.003135
   Web of Science ®Google Scholar
• J.D. Carroll and J. Chang, Analysis of individual differences in multidimensional scaling via an n-way generalization of ‘eckart-young’ decomposition, Psychometrika 35 (1970), pp. 283–319. doi: 10.1007/BF02310791
   Web of Science ®Google Scholar
• E. Ceulemans and H. Kiers, Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method, Br. J. Math. Stat. Psychol. 59 (2006), pp. 133–150. doi: 10.1348/000711005X64817
   PubMed Web of Science ®Google Scholar
• I. Domanov and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors – Part II: Uniqueness of the overall decomposition, SIAM J. Matrix Anal. Appl. 34 (2013), pp. 876–903. doi: 10.1137/120877258
   Web of Science ®Google Scholar
• I. Domanov and L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM J. Matrix Anal. Appl. 36 (2015), pp. 1567–1589. doi: 10.1137/140970276
   Web of Science ®Google Scholar
• I. Domanov and L. De Lathauwer, Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm, Linear Algebra Appl. 513 (2017), pp. 342–375. doi: 10.1016/j.laa.2016.10.019
   Web of Science ®Google Scholar
• J.J. Egozcue, C. Barceló-Vidal, J.A. Martín-Fernández, E. Jarauta-Bragulat, J.L. Díaz-Barrero, and G. Mateu-Figueras, Elements of simplicial linear algebra and geometry, in Compositional Data Analysis: Theory and Applications, V. Pawlowsky-Glahn and A. Buccianti, eds., John Wiley & Sons, Chichester, 2011, pp. 141–157.
   Google Scholar
• J.J. Egozcue and V. Pawlowsky-Glahn, Groups of parts and their balances in compositional data analysis, Math. Geol. 57 (2005), pp. 795–828. doi: 10.1007/s11004-005-7381-9
   Google Scholar
• J.J. Egozcue and V. Pawlowsky-Glahn, Exploring compositional data with the coda-dendrogram, in IAMG 2006: The XIth annual conference of the International Association for Mathematical Geology, Liege–Belgium, 3–8 September 2006.
   Google Scholar
• J.J. Egozcue, V. Pawlowsky-Glahn, G. Mateu-Figueras, and C. Barceló-Vidal, Isometric logratio transformations for compositional data analysis, Math. Geol. 35 (2003), pp. 279–300. doi: 10.1023/A:1023818214614
   Google Scholar
• M.A. Engle, M. Gallo, K.T. Schroeder, N.J. Geboy, and J.W. Zupancic, Three-way compositional analysis of water quality monitoring data, Environ. Ecol. Stat. 21 (2014), pp. 565–581. doi: 10.1007/s10651-013-0268-x
   Web of Science ®Google Scholar
• S. Engelen, S. Frosch MØller, and M. Hubert, Automatically identifying scatter in fluorescence data using robust techniques, Chemometr. Intell. Lab. Syst. 86 (2007), pp. 35–51. doi: 10.1016/j.chemolab.2006.08.001
   Web of Science ®Google Scholar
• S. Engelen and M. Hubert, Detecting outlying samples in a parallel factor analysis model, Anal. Chim. Acta 705 (2011), pp. 155–165. doi: 10.1016/j.aca.2011.04.043
   PubMed Web of Science ®Google Scholar
• European Union-EU: Stepping up the fight against undeclared work, Report of the European Commission, Bruxelles, 2007, pp. 628.
   Google Scholar
• P. Filzmoser and K. Hron, Outlier detection for compositional data using robust methods, Math. Geosci. 40 (2008), pp. 233–248. doi: 10.1007/s11004-007-9141-5
   Web of Science ®Google Scholar
• P. Filzmoser and K. Hron, Robustness for compositional data, in Robustness and Complex Data Structures, C. Becker, R. Fried, and S. Kuhnt, eds., Springer, Heidelberg, 2013, pp. 117–131.
   Google Scholar
• P. Filzmoser and K. Hron, Exploring compositional data with the robust compositional biplot, in Advances in Latent Variables, M. Carpita, E. Brentari, and E.M. Qannari, eds., Springer, Heidelberg, 2014, pp. 219–226.
   Google Scholar
• P. Filzmoser, K. Hron, and C. Reimann, Univariate statistical analysis of environmental (compositional) data: Problems and possibilities, Sci. Total Environ. 407 (2009a), pp. 6100–6108. doi: 10.1016/j.scitotenv.2009.08.008
   PubMed Web of Science ®Google Scholar
• P. Filzmoser, K. Hron, and C. Reimann, Principal component analysis for compositional data with outliers, Environmetrics 20 (2009b), pp. 621–632. doi: 10.1002/env.966
   Web of Science ®Google Scholar
• P. Filzmoser, K. Hron, and C. Reimann, Interpretation of multivariate outliers for compositional data, Comput. Geosci. 39 (2012), pp. 77–85. doi: 10.1016/j.cageo.2011.06.014
   Web of Science ®Google Scholar
• M. Gallo, Log-ratio and parallel factor analysis: An approach to analyze three-way compositional data, in Advanced Dynamic Modeling of Economic and Social Systems, A.N. Proto, M. Squillante, and J. Kacprzyk, eds., Springer, Berlin, Heidelberg, 448, 2013, pp. 209–221.
   Google Scholar
• M. Gallo, Tucker3 model for compositional data, Comm. Statist. Theory Methods 44 (2015), pp. 4441–4453. doi: 10.1080/03610926.2013.798664
   Web of Science ®Google Scholar
• M. Gallo and A. Buccianti, Weighted principal component analysis for compositional data: Application example for the water chemistry of the Arno river (Tuscany, central Italy), Environmetrics 24 (2013), pp. 269–277. doi: 10.1002/env.2214
   Web of Science ®Google Scholar
• R.A. Harshman, Foundations of the parafac procedure: Models and conditions for an ‘explanatory’ multi-modal factor analysis, Tech. Rep. 16: 84. No. 10,085, UCLA Working Papers in Phonetics, University of California, Los Angeles, 1970.
   Google Scholar
• R.A. Harshman and M.E. Lundy, The PARAFAC Model for three-way factor analysis and multidimensional scaling, in Research Methods for Multimode Data Analysis, H.G. Law, ed., Praeger Publishers, Praeger, New York 1984, pp. 122–215.
   Google Scholar
• K. Hron, M. Templ, and P. Filzmoser, Imputation of missing values for compositional data using classical and robust methods, Comput. Statist. Data. Anal. 54 (2010), pp. 3095–3107. doi: 10.1016/j.csda.2009.11.023
   Web of Science ®Google Scholar
• M. Hubert, J. Van Kerckhoven, and T. Verdonck, Robust PARAFAC for incomplete data, J. Chemometr. 26 (2012), pp. 290–298. doi: 10.1002/cem.2452
   Web of Science ®Google Scholar
• M. Hubert, P.J. Rousseeuw, and K. Vanden Branden, ROBPCA: A new approach to robust principal components analysis, Technometrics 47 (2005), pp. 64–79. doi: 10.1198/004017004000000563
   Web of Science ®Google Scholar
• International Labour Organization-ILO, Labour inspection and undeclared work in the EU, report, (2013), ISSN: 2227–7560.
   Google Scholar
• Istituto nazionale di Statistica-ISTAT, Dossier l'economia sommersa: stime nazionali e regionali, 2010. Available at: www.3istat.it/istat/audizioni/220710/Allegato1.pdf.
   Google Scholar
• Istituto nazionale di Statistica-ISTAT, Note metodologiche, la misura della occupazione non regolare nelle stime di contabilitá nazionale [online], Roma. Available at: www.istat.it.
   Google Scholar
• H.A.L. Kiers, Some procedures for displaying results from three-way methods, J. Chemometr. 14 (2000), pp. 151–170. doi: 10.1002/1099-128X(200005/06)14:3<151::AID-CEM585>3.0.CO;2-G
   Web of Science ®Google Scholar
• P.M. Kroonenberg, Applied Multiway Data Analysis, Wiley-Interscience, Hoboken, NJ, 2008.
   Google Scholar
• J.B. Kruskal, Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Appl. 18 (1977), pp. 95–138. doi: 10.1016/0024-3795(77)90069-6
   Web of Science ®Google Scholar
• J.B. Kruskal, Rank, decomposition, and uniqueness for 3-way and N-way arrays, Multiway Data Anal. 33 (1989), pp. 7–18.
   Google Scholar
• A.M. Leroy and P.J. Rousseeuw, Robust Regression and Outlier Detection, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1987.
   Google Scholar
• R. Maronna, R.D. Martin, and V.J. Yohai, Robust Statistics: Theory and Methods, John Wiley, New York, 2006.
   Google Scholar
• J.A. Martín-Fernández, C. Barceló-Vidal, and V. Pawlowsky-Glahn, Dealing with zeros and missing values in compositional data sets using nonparametric imputation, Math. Geol. 35 (2003), pp. 253–278. doi: 10.1023/A:1023866030544
   Google Scholar
• J.A. Martín-Fernández, K. Hron, M. Templ, P. Filzmoser, and J. Palarea-Albaladejo, Model-based replacement of rounded zeros in compositional data: Classical and robust approaches, Comput. Statist. Data Anal. 56 (2012), pp. 2688–2704. doi: 10.1016/j.csda.2012.02.012
   Web of Science ®Google Scholar
• J.A. Martín-Fernández, J. Palarea-Albaladejo, and R. Olea, Dealing with zeros, in Compositional Data Analysis: Theory and Applications, V. Pawlowsky-Glahn and A. Buccianti, eds., John Wiley and Sons, Chichester, 2011, pp. 43–58.
   Google Scholar
• G. Mateu-Figueras, V. Pawlowsky-Glahn, and J.J. Egozcue, The principle of working on coordinates, in Compositional Data Analysis: Theory and applications, V. Pawlowsky-Glahn and A. Buccianti, eds., John Wiley & Sons, Chichester, 2011, pp. 31–42.
   Google Scholar
• S. Mikael and L. De Lathauwer, New uniqueness conditions for the canonical polyadic decomposition of third-order tensors, SIAM J. Matrix Anal. Appl. 36 (2015), pp. 1381–1403. doi: 10.1137/140964813
   Web of Science ®Google Scholar
• J. Palarea-Albaladejo and J.A. Martín-Fernández, A modified EM alr-algorithm for replacing rounded zeros in compositional data sets, Comput. Geosci. 34 (2008), pp. 902–917. doi: 10.1016/j.cageo.2007.09.015
   Web of Science ®Google Scholar
• J. Palarea-Albaladejo, J.A. Martín-Fernández, and J.A. Soto, Dealing with distances and transformations for fuzzy c-means clustering of compositional data, J. Classif. 29 (2012), pp. 144–169. doi: 10.1007/s00357-012-9105-4
   Web of Science ®Google Scholar
• V. Pawlowsky-Glahn and J.J. Egozcue, Geometric approach to statistical analysis on the simplex, Stoch. Environ. Res. Risk. Assess. 15 (2001), pp. 384–398. doi: 10.1007/s004770100077
   Web of Science ®Google Scholar
• V. Pawlowsky-Glahn, J.J. Egozcue, and R. Tolosana-Delgado, Modeling and Analysis of Compositional Data, Wiley, Chichester, 2015.
   Google Scholar
• J. Riu and R. Bro, Jack-knife technique for outlier detection and estimation of standard errors in parafac models, Chemometr. Intell. Lab. Syst. 65 (2003), pp. 35–49. doi: 10.1016/S0169-7439(02)00090-4
   Web of Science ®Google Scholar
• P.J. Rousseeuw and K. Van Driessen, A fast algorithm for the minimum covariance determinant estimator, Technometrics 41 (1999), pp. 212–223. doi: 10.1080/00401706.1999.10485670
   Web of Science ®Google Scholar
• P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection, Vol. 589, John Wiley & Sons, New York, 2005.
   Google Scholar
• A. Smilde, R. Bro, and P. Geladi, Multi-Way Analysis with Applications in the Chemical Sciences, John Wiley & Sons, Chichester, 2004.
   Google Scholar
• A. Stegeman, Degeneracy in Candecomp/Parafac explained for p× p× 2 arrays of rank p+1 or higher, Psychometrika 71 (2006), pp. 483–501. doi: 10.1007/s11336-004-1266-6
   Web of Science ®Google Scholar
• A. Stegeman, On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode, Linear Algebra Appl. 431 (2009), pp. 211–227. doi: 10.1016/j.laa.2009.02.025
   Web of Science ®Google Scholar
• M.E. Timmerman and H.A.L Kiers, Three-mode principal components analysis: Choosing the numbers of components and sensitivity to local optima, Br. J. Math. Statist. Psychol. 53 (2000), pp. 1–16. doi: 10.1348/000711000159132
   PubMed Web of Science ®Google Scholar
• G. Tomasi and R. Bro, A comparison of algorithms for fitting the PARAFAC model, Comput. Statist. Data. Anal. 50 (2006), pp. 1700–1734. doi: 10.1016/j.csda.2004.11.013
   Web of Science ®Google Scholar