FUZZY SET FEATURES OF ONE-DIMENSIONAL TIME SERIES

Yu.N. Minaev, O.Yu. Filimonova, J.I.MinaevaYu.N. Minaev, O.Yu. Filimonova, J.I.Minaeva

Èlektron. model. 2018, 38(6):45-66
https://doi.org/10.15407/emodel.38.06.045

ABSTRACT

A problem of structuring the time series (TS) (in a form of a window, fragment, segment or others structure parts) has been investigated, as well as presentation of a separate window in the form of 2D tensor  with X matrix of dimensionality m x m (m · m is the number of window elements TS) with following determination of m -vectors u, v (with certain restrictions), which for the given matrix of data X minimize a criterion ||X-Kr uvT||2F +Pλ(u,v), where trace{(X - uvT)(X - uvT)T}; Pλ(u,v)— a penalty function, -Kr — a symbol of Kronecker difference. Vectors u, v are considered as a subset of ordered pairs, where vector v plays a role of
membership function, i.e. (v →[0, 1]). The expediency of using the procedure of a singular decomposition
for this purpose is shown.

A subset of ordered pairs {u, v}, considered as psevdo FS, represents 2D tensor with the matrix of dimensionality 2 x m, allows us to shorten a body of stored information (m · m > 2 · m), to obtain hidden knowledge in the form of the spectrum of singular values and to obtain new possibilities in deciding the problems of forecasting and anomaly identifications of TS anomalies as the result of using the tensor invariants.

KEYWORDS

fuzzy set, time row, tensor decomposition, singular values, Kronecker product.

REFERENCES

1. Esbensen, K. (2005), Analiz mnogomernykh dannykh. Izbrannyye glavy [Analysis of multidimensional data, Selected chapters], Translated from English by S.V. Kucheryavsky, Ed by O.Ye. Rodionovoy, 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.
3. Ringberg, H., Soule, A., Rexford, J. and Diot Cr. (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, London, UK.
5. Minayev, Yu.N., Zhukov, I.A. and Filimonova, O.Yu. (2006), “Prediction of time deried in tensor basis”, Elektronnoe modelirovanie, Vol. 28, no. 2, pp. 18-34.
6. Laub, A.J. (2005), Matrix analysis for scientists and engineers, available at: www. c-securehost.com/SIAM/ot91.html.
7. Bader, B.W. and Kolda, T.G. (2006), SANDIA REPORT. SAND2006-7592. Efficient MATLAB computations with sparse and factored tensors, Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550, USA.
8. Shen, H. and Huang, J.Z. (2008), Sparse principal component analysis via regularized low rank matrix app-roximation, Journal of Multivariate Analysis, Vol. 99, pp.1015-1034.
9. Kibangou, A.Y. (2009), Tensor decompositions and Applications. An overview and some contribu-tions. GIPSA-N_CS. March 17, 2009, 88 ð., Internet resource.
10. Bader, W.B. 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, USA, 39 pp., available at: - http://csmr.ca.sandia.gov/~tgkolda/ .
11. Minayev, Yu.N., Filimonova, O.Yu. and Minayeva J.I. (2015), “Structured granules of fuzzy set in the problems of granule computing”, Elektronnoe modelirovanie, Vol. 37, no. 1, pp. 77-95.
12. Minayev, Yu.N., Filimonova, O.Yu. and Minayeva, J.I. (2014), “Kroneker (tenzor) models of fuzzy-set granules, Kibernetika i sistemnyi analiz, Vol. 50, no. 4, pp. 42-52.
13. Minayev, Yu.N., Filimonova, O.Yu. and Minayeva, J.I. (2013), Tensor models of NM-granules and their use for solution of problems of fuzzy arithmetic, Iskusstvennyy intellekt, no. 2, pp. 22-31.
14. Silva, V.D. and Lim, L.-H. 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.
15. Voyevodin, V.V. and Voyevodin, Vl.V. (2006), Entsiklopediya lineynoy algebry. Electronnaya sistema LINEAL [Encyclopaedia of linear algebra. Electron system LINEAL], BKHV, St-Petersburg, Russia.
16. Kofman, A. (1982), Vvedeniye v teoriyu nechetkikh mnozhestv [Introduction into the theory of fuzzy sets], Translated from French, Radio i svyaz, Moscow, Russia.
17. Van Loan, C.F. and Pitsianis, N. (1993), Approximation with Kronecker products, Eds M.S. Moonen et al., Linear Algebra for Large Scale and Real-Time Applications, Kluver Publishers, Dodrecht, the Netherlands.
18. Dompierre, P. (2010), Properties of singular value decomposition matrix computations — CPSC 5006, available at: www.cs.laurentian.ca/jdompierre/html/CPSC5006E_ F2010/ cours/ch05_ SVD _Properties.pdf
19. Witten, D.M., Tibshirani, R. and Trevor, H. (2009), A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics Vol. 10, no. 3, pp. 515-534.15. Voyevodin, V.V. and Voyevodin, Vl.V. (2006), Entsiklopediya lineynoy algebry. Electronnaya sistema LINEAL [Encyclopaedia of linear algebra. Electron system LINEAL], BKHV, St-Petersburg, Russia.

Full text: PDF (in Russian)