ANALYSIS OF SELF-SIMILARITY OF MULTIVARIATE TIME SERIES (TS) ON THE BASIS OF THE METHODS OF INTELLECTUAL ANALYSIS OF THE DATA

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

Èlektron. model. 2017, 39(4):43-68
https://doi.org/10.15407/emodel.39.04.043

ABSTRACT

Calculation methods have been proposed for the Hurst factor for univariate and multivariate TS on the basis of the main diagonals of TS tensor models. It is shown that the problem complexity determines the joint use of several mathematical theories, in particular, the tensor and multivariate matrix analysis. The examples of using the proposed methods are presented.

KEYWORDS

tensor, multivariate time series, intellectual analysis of the data, 3D matrices, diagonal 3D matrices, matrix development, self-similarity, Hurst parameter.

REFERENCES

1. Time series: Advanced methods IIa. Multivariate time series, available at: www.ucl.ac.uk/jdi/events/int-CIA-conf/ICIAC11_ Sli-des/ ICIAC11_1E_ LTompson.
2. Cichocki, A., Mandic, D., Phan, A.-H., Caiafa, C., Tensor decompositions for signal processing applications. From two-way to multiway component analysis, available at: http://www. commsp. ee.ic.ac.uk/~mandic/SPM-Cichocki-Mandic-DeLathauwer. pdf 
3. Sokolov, N.P. (1960), Prostranstvennyie matritsy i ikh prilozheniya [Space matrice and their application], Gosudarstvennoe izdatelstvo fiz.-mat. literatury, Moscow, USSR.
4. Claude, Z.B., Introduction to the general multidimensional matrix in mathematics, available at: www.ijera.com/pages/v3no6.html.
5. Solo, A., Multidimensional matrix mathematics: Notation, representation, and simplification. Parts: 1-6, Proceedings of the World Congress on Engineering (3), available at: www.ijera.com/ papers/Vol.3_issue6/ U36123129.pdf.
6. De Lathauwer, L. and Moor, B. (1998), From matrix to tensor: Multilinear algebra and signal proce-ssing, Proceedings of the 4-h IMA Int. Conf. on Mathematics in Signal Processing, Oxford, United Kingdom, Selected papers presented at pp. 1-15, J. McWhirter (Ed.), Mathematics in Signal Processing IV, Oxford University Press.
7. Skillicorn, D., Understanding complex datasets : data mining with matrix decomposi- tions. Chapman & Hall/ CRC Taylor & Francis Group 6000 Broken Sound Parkway NW. Suite 300 Boca Raton, FL 33487, 2742.
8. Cichocki, A. (2013), Tensor decompositions: A new concept in brain data analysis?, available at: arXiv:1305.0395v1 [cs.NA] May 2, 2013.
9. Lim, L.-H. (2005), Singular values and eigenvalues of tensors: A variational approach, Proceedings of the 1-st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), December 13-15, 2005, pp. 129-132.
10. Liqun, Qi. (2007), Eigenvalues and invariants of tensors, J. Math. Anal. Appl., Vol. 325, pp. 1363-1377, available at: www. elsevier. com/ locate/jmaa.
11. Kolda, T.G. and Bader, B.W. (2009), Tensor decompositions and applications, SIAM Review, Vol. 51, no. 3, pp. 455-500.
12. Kamalja, K.K. and Khangar, N.V. (2013), Singular value decomposition for multidimensional matrices, Int. Journal of Engineering Research and Applications, Vol. 3, Iss. 6, pp. 123-129.
13. Bader, B.W. and Kolda, T.G. Tensor decompositions, the MATLAB tensor toolbox, and applications to data analysis, available at: www.sandia.gov/~tgkolda/ TensorToolbox.
14. Bader, B.W. and Kolda, T.G. (2006), Multilinear operators for higher-order decompositions: Technical report SAND 2006-2081, Sandia National Laboratories, available at: pubs/pubfiles/SAND2007-6702.pdf.
15. Stegeman, A., The Parafac model for multiway data analysis, available at: http://www.ppsw.rug.nl/~stegeman.
16. Tensor toolbox is software for working with multidimensional arrays, available at: http://csmr. ca. sandia.gov/~tgkolda.
17. Kindlmann, G., Tensor invariants and their gradients, available at: USA.gk@cs.utah. edu. 
18. Bozhokin, S.V. and Parshin, D.A. (2001), Fraktaly i multifraktaly [Fractals and multi fractals], NITs Regulyarnaya i khaoticheskaya dinamika, Izhevsk, Russia.
19. Shelukhin, O.I. (2011), Multifraktaly. Infokomunikatsionnyie prilozheniya [Multifractals. Infocommunication applications], Goryachaya liniya – Telekom, Moscow, Russia.

Full text: PDF (in Russian)