Spatio-temporal multiway decompositions using principal tensor analysis on k-modes: The R package PTAk

Journal of Statistical Software

Volumn 34, Issue 10, 2010, Pages 1-34

  Leibovici, Didier G ^a  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Multi entries data; Multiway analysis; Multiway interaction; PTAk; R programming; Spatio temporal data; Tensor decomposition; Variance decomposition

Indexed keywords

EID: 77955134219     PISSN: 15487660     EISSN: 15487660     Source Type: Journal    
DOI: 10.18637/jss.v034.i10     Document Type: Article

References (48)

1. Baddeley AJ, Turner R (2005). "Spatstat: An R Package for Analyzing Spatial Point Patterns." Journal of Statistical Software, 12(6), 1-42. URL http://www.jstatsoft.org/v12/i06/.
2. Beckmann CF, Smith SM (2005). "Tensorial Extensions of Independent Component Analysis for Multi-Subject fMRI Analysis." NeuroImage, 25(1), 294-311.
3. Besse PC, Cardot H, Faivre R, Goulard M (2005). "Statistical Modelling of Functional Data: Research Articles." Applied Stochastic Models in Business and Industry, 21(2), 165-173.
4. Borcard D, Legendre P (2002). "All-Scale Spatial Analysis of Ecological Data by Means of Principal Coordinates of Neighbour Matrices." Ecological Modelling, 153, 51-68.
5. Borg I, Groenen P (2005). Modern Multidimensional Scaling: Theory and Applications. 2nd edition. Springer-Verlag, New York, NY.
6. Brunsdon C, Fotheringham S, Charlton M (2007). "Geographically Weighted Discriminant Analysis." Geographical Analysis, 39(4), 376-396.
7. Cailliez F, Pag-es JP (1976). Introduction _a l'Analyse Des Donn_ees. Soci_et_e de Math_ematiques Appliqu_ees et de Sciences Humaines (SMASH), Paris, France.
8. Carlier A, Kroonenberg PM (1996). "Decompositions and Biplots in Three-Way Correspondence Analysis." Psychometrika, 61(2), 355-373.
9. Carroll JD, Chang JJ (1970). "Analysis of Individual Di_erences in Multidimensional Scaling via an N-Way Generalization of 'Eckart-Young' Decomposition." Psychometrika, 35, 283-319.
10. Caussinus H, Ruiz A (1990). "Interesting Projections of Multidimensional Data by Means of Generalized Principal Components Analysis." In COMPSTAT90, pp. 121-126. Physica- Verlag, Heidelberg, Germany.
11. Chessel D, Dufour AB, Dray S, Lobry JR, Ollier S, Pavoine S, Thioulouse J (2007). Ade4: Exploratory and Euclidean Methods in Environmental Sciences. R package version 1.4-5, URL http://CRAN.R-project.org/package=ade4.
12. Comon P, Golub G, Lim LH, Mourrain B (2008). "Symmetric Tensors and Symmetric Tensor Rank." SIAM Journal on Matrix Analysis and Applications, 30(3), 1254-1279.
13. Dauxois J, Romain Y, Viguier-Pla S (1994). "Tensor Products and Statistics." Linear Algebra and Its Applications, 210, 59-88.
14. De Leeuw J, Mair P (2009). "Multidimensional Scaling Using Majorization: SMACOF in R." Journal of Statistical Software, 31(3), 1-30. URL http://www.jstatsoft.org/v31/i03/.
15. de Silva V, Lim LH (2008). "Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem." SIAM Journal on Matrix Analysis and Applications, 30(3), 1084-1127.
16. Diggle PJ (2003). Statistical Analysis of Spatial Point Patterns. Hodder Arnold, London, UK.
17. Dray S, Dufour AB (2007). "The ade4 Package: Implementing the Duality Diagram for Ecologists." Journal of Statistical Software, 22(4), 1-20. URL http://www.jstatsoft. org/v22/i04/.
18. Escou-er Y (1987). "The Duality Diagram: A Means of Better Practical Applications." In P Legendre, L Legendre (eds.), Developments in Numerical Ecology, Serie G, pp. 139-156. NATO Advanced Institute, Springer-Verlag, Berlin, Germany.
19. Faber NM, Bro R, Hopke PK (2003). "Recent Developments in CANDECOMP/PARAFAC Algorithms: A Critical Review." Chemometrics, 65(1), 119-137.
20. Faraj A (1994). "Interpretation Tools for Generalized Discriminant Analysis." In New Approaches in Classi_cation and Data Analysis, pp. 286-291. Springer-Verlag, Heidelberg, Germany.
21. Franc A (1992). Etude Alg_ebrique des Multitableaux: Apports de l'Alg_ebre Tensorielle. Ph.D. thesis, Universit_e de Montpellier II, Montpellier France
22. Gollob HF (1968). "A Statistical Model which Combines Features of Factor Analytic and Analysis of Variance Techniques. Psychometrika, 33, 73-116.
23. Harshman RA (1970). "Foundations of the PARAFAC Procedure: Models and Conditions for 'an Explanatory' Multi-Modal Factor Analysis." UCLA Working Papers in Phonetics 16, UCLA. UMI Serials in Microform, No. 10,085.
24. Heiser WJ, Bennani M (1997). "Triadic Distance Models: Axiomatization and Least Squares Representation." Journal of Mathematical Psychology, 41(2), 189-206.
25. Hill MO (1973). "Reciprocal Averaging: An Eigenvector Method of Ordination." Journal of Ecology, 61, 237-249.
26. Kaptein A, Neudecker H, Wansbeek T (1986). "An Approach to n-Mode Component Analysis." Psychometrika, 51(2), 269-275.
27. Kolda TG (2003). "A Counterexample to the Possibility of an Extension of the Eckart- Young Low-Rank Approximation Theorem for the Orthogonal Rank Tensor Decomposition." SIAM Journal on Matrix Analysis and Applications, 24(3), 762-767.
28. Kroonenberg PM (1983). Three-Mode Principal Component Analysis: Theory and Applications. DWO Press, Leiden, Netherlands.
29. Kroonenberg PM, De Leeuw J (1980). "Principal Component Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms." Psychometrika, 45, 69-97.
30. Lang S (1984). Algebra. 2nd edition. Addison-Wesley, Reading, Massachusetts
31. Le Roux B, Rouanet H (2004). Geometric Data Analysis: From Correspondence Analysis to Structured Data Analysis. Kluwer Academic Publishers, Dordrecht, The Netherlands.
32. Lebart L (1969). "Analyse Statistique de la Contiguit_ee." Publication de l'Institut de Statistiques Universitaire de Paris, XVIII, 81-112.
33. Leibovici DG (1993). Facteurs _a Mesures R_ep_et_ees et Analyses Factorielles: Applications _a un Suivi _ Epid_emiologique. Phd thesis, Universit_e de Montpellier II, Montpellier, France.
34. Leibovici DG (2000). "Multiway Multidimensional Analysis for Pharmaco-EEG Studies." Report initiated at SANOFI-RECHERCHE TR00DL2, FMRIB Centre, University of Oxford, UK. URL http://www.fmrib.ox.ac.uk/analysis/techrep/tr00dl2/tr00dl2.pdf.
35. Leibovici DG (2001). PTAk: Principal Tensor Analysis on k Modes. R package version 1.1-4, URL http://CRAN.R-project.org/package=PTAk.
36. Leibovici DG (2004). "c3s2i: Conseils Services Statistique SIG Ing_enieurie Informatique." URL http://c3s2i.free.fr/.
37. Leibovici DG (2008). "A Simple Penalised Algorithm for SVD and Multiway Functional Methods." Technical report, Centre for Geospatial Science, University of Nottingham, UK. URL http://cgs.nottingham.ac.uk/~lgzdl/.
38. Leibovici DG (2009). PTAk: Principal Tensor Analysis on k Modes. R package version 1.2-0, URL http://CRAN.R-project.org/package=PTAk.
39. Leibovici DG, Bastin L, Jackson M (2008). "Discovering Spatially Multiway Collocations. In GISRUK08 Conference, pp. 66-71. GIS Research UK, Manchester, UK.
40. Leibovici DG, Beckmann CF (2001)."An Introduction toMultiway Methods forMulti-Subject fMRI Experiments."Technical Report TR01DL1, FMRIB Centre, University of Oxford, UK. URL http://www.fmrib.ox.ac.uk/analysis/techrep/tr01dl1/tr01dl1.pdf.
41. Leibovici DG, El Ma^ache H (1997). ";Une D_ecomposition en Valeurs Singuli_eres d'un El_ement d'un Produit Tensoriel de k Espaces de Hilbert S_eparables." Compte Rendus de l'Acad_emie des Sciences I, 325(7), 779-782.
42. Leibovici DG, Quillevere G, Desconnets JC (2007). ";A Method to Classify Ecoclimatic Arid and Semi-Arid Zones in Circum-Saharan Africa Using Monthly Dynamics of Multiple Indicators." IEEE Transactions on Geoscience and Remote Sensing, 45(12), 4000-4007.
43. Leibovici DG, Sabatier R (1998). "A Singular Value Decomposition of k-Way Array for a Principal Component Analysis of Multiway Data, PTA-k." Linear Algebra and Its Applications, 269, 307-329.
44. Ni G, Wang Y (2007). ";On the Best Rank-1 Approximation to Higher-Order Symmetric Tensors." Mathematical and Computer Modelling, 46, 1345-1352.
45. R Development Core Team (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/.
46. Ripley BD (1981). Spatial Statistics. John Wiley & Sons, New York, NY
47. Rougier JC (2002). Tensor: Tensor Product of Arrays. R package version 1.4, URL http://CRAN.R-project.org/package=tensor.
48. van den Boogaart KG (2007). tensorA: Advanced Tensors Arithmetic with Named Indices. R package version 0.31, URL http://CRAN.R-project.org/package=tensorA.