TREND-CYCLE ESTIMATION USING FUZZY TRANSFORM OF HIGHER DEGREE

Document Type: Research Paper

Authors

Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Abstract

In this paper, we provide theoretical justification for the application of higher degree fuzzy transform in time series analysis. Under the assumption that a time series can be additively decomposed into a trend-cycle, a seasonal component and a random noise, we demonstrate that the higher degree fuzzy transform technique can be used for the estimation of the trend-cycle, which is one of the basic tasks in time series analysis. We prove that  high frequencies appearing in the seasonal component can be  arbitrarily suppressed and that random noise, as a stationary process, can be successfully decreased  using the fuzzy transform of higher degree with a reasonable adjustment of parameters of a generalized uniform fuzzy partition.

Keywords


[1] T. Alexandrov, S. Bianconcini, E. B. Dagum, P. Maass and T. McElroy, A review of some
modern approaches to the problem of trend extraction, In Research Report Series, Statistics
2008-3, U.S. Census Bureau, Washington, 2009.
[2] S. Cleveland and S. Devlin, Locally-weighted regression: an approach to regression analysis
by local fi tting, J. Am. Stat., Assoc. 83 (1988), 596{610.
[3] N. Golyandina and A. Zhigljavsky, Singular spectrum analysis for time series, Briefs in
Statistics, Springer, Berlin, 2013.
[4] M. Holcapek and L. Nguyen, Suppression of high frequencies in time series using fuzzy trans-
form of higher degree, Information Processing and Management of Uncertainty in Knowledge-
Based Systems: 16th International Conference, IPMU 2016, Eindhoven, The Netherlands,
Springer, (2016), 705{716.
[5] M. Holcapek, V. Novak and I. Perfi lieva, Noise reduction in time series using F-transform,
In: Proc. IEEE International Conference on Fuzzy Systems, Hyderabad, (2013), 1{8.
[6] M. Holcapek, I. Perfi lieva, V. Novak and V. Kreinovich, Necessary and sufficient conditions
for generalized uniform fuzzy partitions, Fuzzy Sets and Systems, 277 (2015), 97{121.
[7] M. Holcapek and T. Tichy, A smoothing fi lter based on fuzzy transform, Fuzzy Sets and
Systems, 180 (1) (2011), 69{97.
[8] A. H. Jazwinski, Stochastic Processes and Filtering Theory, Mineola, NY: Dover Publica-
tions, 2007.
[9] I. Kodorane and S. Asmuss, On approximation properties of spline based F-transform with
respect to fuzzy m-partition, in: G. Pasi, J. Montero, D. Ciucci (eds.), Proc. of the 8th
conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13), Atlantis
Press, (2013), 772{779.
[10] M. Kokainis and S. Asmuss, Approximation properties of higher degree F-transforms based
on B-splines, In: Proc. IEEE International Conference on Fuzzy Systems, Istanbul, (2015),
1{8.
[11] L. Nguyen and V. Novak, Filtering out high frequencies in time series using F-transform with
respect to raised cosine generalized uniform fuzzy, In: Proc. IEEE International Conference
on Fuzzy Systems, Istanbul, (2015), 1{8.
[12] V. Novak, I. Perfi lieva, M. Holcapek and V. Kreinovich, Filtering out high frequencies using
F{transform, Information Sciences, 274 (2014), 192{209.
[13] V. Novak, M. Stepnicka, A. Dvorak, I. Perfi lieva, V. Pavliska and L. Vavrckova, Analysis of
seasonal time series using fuzzy approach, Int. J. Gen. Syst., 39 (2010), 305{328.
[14] V. Novak, M. Stepnicka, I. Perfi lieva and V. Pavliska, Analysis of periodical time series
using soft computing methods, In: D. Ruan, J. Montero, J. Lu, L. Martinez, P. D'hondt, E.
E. Kerre (eds.), Computational Intelligence in Decision and Control, World Scientifi c, New
Jersey, (2008), 55{60.
[15] I. Perfi lieva, Fuzzy transforms, Peters, James F. (ed.) et al., Transactions on Rough Sets
II. Rough sets and fuzzy sets. Berlin: Springer. Lecture Notes in Computer Science 3135.
Journal Subline, (2004), 63{81.
[16] I. Perfi lieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157(8)
(2006), 993{1023.
[17] I. Perfi lieva and M. Dankova, Towards F-transform of a higher degree, in: In Proc. of
IFSA/EUSFLAT 2009, Lisbon, Portugal, (2009), 585{588.
[18] I. Perfi lieva, M. Dankova and B. Bede, Towards a higher degree F-transform, Fuzzy Sets and
Systems, 180 (1) (2011), 3{19.
[19] I. Perfi lieva and R. Valasek, Fuzzy transforms in removing noise, Innovations in Hybrid
Intelligent Systems, Springer Berlin/Heidelberg, (2005), 221{230.
[20] E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press,
Oxford, 1948.

[21] A. M. Yaglom, An introduction to the Theory of Stationary Random Functions, Revised
English ed. Translated and edited by Richard A. Silverman, Englewood Cli s, NJ: Prentice-
Hall, Inc. XIII, 1962.