Yu.N. Minaev, O.Yu. Filimonova, J.I. Minaeva
Èlektron. model. 2017, 39(1):51-74
https://doi.org/10.15407/emodel.39.01.051
ABSTRACT
The questions of representation of multidimensional (multicomponent) time series (TS) as a 3D tensor model; its following tensor decomposition with the use of procedures of PARAFAC-decomposition and higher-order singular decomposition (HOSVD) allows us to represent the entire TS (or its components — window, a fragment, a segment) in the form of a certain granule — a subset of ordered triples with properties similar to those of the second type FS called the second type psevdoFS.
The analogy is shown between the properties of the existing traces, singular values and F-norm of standard singular decomposition (2D matrix) and HOSVD used for 2D matrix. Examples showing the representation of multidimensional TS by granules — subsets of ordered triples.
KEYWORDS
singular decomposition, fuzzy multiple granule, higher-order singular decomposition, trace of a matrix, F-norm, time series.
REFERENCES
1. Esbensen, K. (2005), Analiz mnogomernyh dannyh. Izbrannye glavy [Analysis of multidimensional data. Selected chapters], Transl. from Engl. S.V. Kucheryavskiy; Ed O.E. Rodionova, Izdatelstvo IPHV RAN, Chernogolovka, Russia.
2. Dobos, L. and Abonyi, J. (2012), “On-line detection of homogeneous operation ranges by dynamic principal component analysis based time-series segmentation”, Chemical engineering science, Vol. 75, pp. 96-105.
https://doi.org/10.1016/j.ces.2012.02.022
3. Ringberg, H., Soule, Au., Rexford, J. and Diot, Ch. (2007), “Sensitivity of PCA for traffic anomaly detection”, SIGMETRICS’07, June 12-16, 2007, San Diego, California, USA. Copyright 2007 ACM 978-1-59593-639-4/07/0006
4. Skillicorn, D. (2007), Data mining and knowledge discovery series. Understanding complex datasets. Data mining with matrix decompositions, Chapman & Hall/CRC.
5. Minaev, Ju.N., Filimonova, O.Ju. and Minaeva, Ju.I. (2015), “Structured granules of fuzzy set in the problems of granular computing”, Elektronnoe modelirovanie, Vol. 37, no. 1, pp.77-95.
6. Minaev, Ju.N., Filimonova, O.Ju. and Minaeva, Ju.I. (2014), “Kronecker (tenzor) models of fuzzi-set granules”, Kibernetika i sistemnyj analiz, Vol. 50, no. 4, pp. 42-52.
7. Minaev, Ju.N., Filimonova, O.Ju. and Minaeva, Ju.I. (2013), “Tenzor models of NM-granules and their use for solving problems of fuzzy arithmetic”, Iskusstvennyj intellect, no. 2, pp. 22-31.
8. Minaev, Ju.N., Filimonova, O.Ju. andMinaeva, Ju.I. (2016), “Nonfuzzy multiple characteristics of unidimensional time series (TS)”, Elektronnoe modelirovanie, Vol. 38, no. 6, pp. 45-66.
https://doi.org/10.15407/emodel.38.06.045
9. Van Loan, Ch. (2000), “The ubiquitous Kronecker product”, Journal of computational and applied mathematics, Vol. 123, pp. 85-100.
https://doi.org/10.1016/S0377-0427(00)00393-9
10. Shen, H. and Huang, J.Z. (2008), “Sparse principal component analysis via regularized low rank matrix approximation”, Journal of multivariate analysis, Vol. 99, pp. 1015-1034.
https://doi.org/10.1016/j.jmva.2007.06.007
11. Bader, B.W. and Kolda, T.G. (2006), Tensor decompositions, the MATLAB tensor toolbox, and applications to data analysis. Tensor decompositions multilinear operators for higher-order decompositions. Technical Report SAND2006-2081, Sandia National Laboratories, April 2006. Albuquerque, New Mexico 87185 and Livermore, California 94550, available at: - http://csmr.ca.sandia. gov/~tgkolda/
12. De Silva, V. and Lim, L. Tensor rank and the ill-posedness of the best low-rank approximation problem, Institute for Computational and Mathematical Engineering, Stanford University, Stanford. CA 94305-9025, available at: arxiv.org/pdf/math/0607647.
13. Voevodin, V.V. and Voevodin, Vl.V. (2006), Entsiklopedia lineynoi algebry. Elektronnaya sistema LINEAL [Encyclopedia of linear algebra. Electron system LINEAL], BHV-Peterburg, St-Petersburg, Russia.
14. Kibangou, Al.Y. (2009), Tensor decompositions and applications. An overview and some contributions, GIPSA-N_CS. March 17, 2009, available at: www.sandia.gov/~tgkolda/.../TensorReview.pdf
15. Kofman, A. (1982), Vvedenie v teoriyu nechetkikh mnozhestv [Introduction in the theory of fuzzy sets], Transl. from French, Radio i svyaz, Moscow, Russia.
16. Van Loan, C.F. and Pitsianis, C.F. (1993), Approximation with Kronecker products, Eds M.S. Moonen et al., Linear algebra for large scale and real-time applications, Kluver Publishers, Dordrecht, the Netherlands.
17. De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000), “On the best rank-1 and rank-(R1,R2, ..., RN) approximation of higher-order tensors”, SIAM J. MATRIX ANAL. APPL. Society for Industrial and Applied Mathematics, Vol. 21, no. 4, pp. 1324-1342.
https://doi.org/10.1137/S0895479898346995
18. Costantini, R., Sbaiz, L. and S u sstrunk, S. (2008), “Higher order SVD analysis for dynamic texture synthesis”, IEEE Transactions on Image Processing, Vol. 17, no. 1, pp. 42-52.
https://doi.org/10.1109/TIP.2007.910956
19. Papalexakis, E., Faloutsos, Ch., Sidiropoulos, N. and Harpale, A. (2013), “Large scale tensor decompositions: Algorithmic developments and applications”, Bulletin of the IEEE Computer Society Technical Committee on Data Engineering, pp. 59-67.
20. Kamalja, K.K. and Khangar, N.V. (2013), “Singular value decomposition for multidimensional matrices. Researh article”, Int. Journal of engineering research and applications, Vol. 3, Iss. 6, pp. 123-129.
21. Sidiropoulos, D., Giannakis, G.B. and Bro R. (2000), “Blind PARAFAC. Receivers for DS-CDMA Systems”, IEEE Transactions on Signal Processing, Vol. 48, no. 3, pp. 810-823.
https://doi.org/10.1109/78.824675
22. Cheng, D., Qi, H. and Xue, A.A. (2007), “Survey on semi-tensor product of matrices”, J. syst. sci. and complexity, Vol. 20, pp. 304-322.
https://doi.org/10.1007/s11424-007-9027-0