# Filtering problem for sequences with periodically stationary multi-seasonal increments with spectral densities allowing canonical factorizations

### Abstract

We consider a stochastic sequence $\xi(m)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional $A{\xi}=\sum_{k=0}^{\infty}{a}(k) {\xi}(-k)$. Formulas that determine the least favourable spectral densities and the minimax (robust) spectralcharacteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.### References

C. Baek, R. A. Davis, and V. Pipiras, Periodic dynamic factor models: estimation approaches and applications, Electronic Journal of Statistics, vol. 12, no. 2, pp. 4377–4411, 2018.

R. T. Baillie, C. Kongcharoen, and G. Kapetanios, Prediction from ARFIMA models: Comparisons between MLE and semiparametric estimation procedures, International Journal of Forecasting, vol. 28, pp. 46–53, 2012.

I.V. Basawa, R. Lund, and Q. Shao, First-order seasonal autoregressive processes with periodically varying parameters, Statistics and Probability Letters, vol. 67, no. 4, p. 299–306, 2004.

G. E. P. Box, G. M. Jenkins, G. C. Reinsel, and G.M. Ljung, Time series analysis. Forecasting and control. 5rd ed., John Wiley & Sons, Hoboken, NJ, 2016.

I. I. Dubovets’ka, and M. P. Moklyachuk, Filtration of linear functionals of periodically correlated sequences, Theory of Probability and Mathematical Statistics, vol. 86, pp. 51–64, 2013.

G. Dudek, Forecasting time series with multiple seasonal cycles using neural networks with local learning, In: Rutkowski L., Korytkowski M., Scherer R., Tadeusiewicz R., Zadeh L.A., Zurada J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2013. Lecture Notes in Computer Science, vol. 7894. Springer, Berlin, Heidelberg, pp. 52–63, 2013.

J. Franke, Minimax-robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, vol. 68, no. 3, pp. 337–364, 1985.

I. I. Gikhman, and A. V. Skorokhod, The theory of stochastic processes. I., Springer, Berlin, 2004.

E. G. Gladyshev, Periodically correlated random sequences, Sov. Math. Dokl. vol, 2, pp. 385–388, 1961.

P. G. Gould, A. B. Koehler, J. K. Ord, R. D. Snyder, R. J. Hyndman, and F. Vahid-Araghi, Forecasting time-series with multiple seasonal patterns, European Journal of Operational Research, vol. 191, pp. 207–222, 2008.

U. Grenander, A prediction problem in game theory, Arkiv f¨or Matematik, vol. 3, pp. 371–379, 1957.

E. J. Hannan, Multiple time series. 2nd rev. ed., John Wiley & Sons, New York, 2009.

U. Hassler, and M.O. Pohle, Forecasting under long memory and nonstationarity, arXiv:1910.08202, 2019.

Y. Hosoya, Robust linear extrapolations of second-order stationary processes, Annals of Probability, vol. 6, no. 4, pp. 574–584, 1978.

H. Hurd, and V. Pipiras, Modeling periodic autoregressive time series with multiple periodic effects, In: Chaari F., Leskow J., Zimroz R., Wylomanska A., Dudek A. (eds) Cyclostationarity: Theory and Methods - IV. CSTA 2017. Applied Condition Monitoring, vol 16. Springer, Cham, pp. 1–18, 2020.

K. Karhunen, ¨Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, no. 37, 1947.

S. A. Kassam, and H. V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE, vol. 73, no. 3, pp. 1433–481, 1985.

A. N. Kolmogorov, Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics. Ed. by A. N. Shiryayev. Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc. Kluwer Academic Publishers, 1992.

Y. Liu, Yu. Xue, and M. Taniguchi, Robust linear interpolation and extrapolation of stationary time series in Lp, Journal of Time Series Analysis, vol. 41, no. 2, pp. 229–248, 2020.

R. Lund, Choosing seasonal autocovariance structures: PARMA or SARMA, In: Bell WR, Holan SH, McElroy TS (eds) Economic time series: modelling and seasonality. Chapman and Hall, London, pp. 63–80, 2011.

M. Luz and M. Moklyachuk, Filtering problem for functionals of stationary sequences, Statistics, Optimization and Information Computing, vol. 4, no. 1, pp. 68 – 83, 2016.

M. Luz, and M. Moklyachuk, Estimation of stochastic processes with stationary increments and cointegrated sequences, London: ISTE; Hoboken, NJ: John Wiley & Sons, 282 p., 2019.

M. Luz, and M. Moklyachuk, Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments, Statistics, Optimization and Information Computing, vol. 8, no. 3, pp. 684–721, 2020.

M. Luz, and M. Moklyachuk, Robust filtering of sequences with periodically stationary multiplicative seasonal increments, Statistics, Optimization and Information Computing, vol. 9, no. 4, pp. 1010-1030, 2021.

M. Luz, and M. Moklyachuk, Minimax prediction of sequences with periodically stationary increments observes with noise and cointegrated sequences, In: M. Moklyachuk (eds) Stochastic Processes: Fundamentals and Emerging Applications. Nova Science Publishers, New York, pp. 189–247, 2023.

M. P. Moklyachuk, Minimax filtration of linear transformations of stationary sequences, Ukrainian Mathematical Journal, vol. 43, pp. 75–81, 1991.

M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Statistics, Optimization and Information Computing, vol. 3, no. 4, pp. 348–419, 2015.

M.P. Moklyachuk, and A. Yu. Masyutka, Robust filtering of stochastic processes Theory of Stochastic Processes, vol. 13, no. 1-2, pp. 166–181, 2007.

M. Moklyachuk, M. Sidei, and O. Masyutka, Estimation of stochastic processes with missing observations, Mathematics Research Developments. Nova Science Publishers, New York, NY: Nova Science Publishers, 336 p., 2019

A. Napolitano, Cyclostationarity: New trends and applications, Signal Processing, vol. 120, pp. 385–408, 2016.

D. Osborn, The implications of periodically varying coefficients for seasonal time-series processes, Journal of Econometrics, vol.48, no. 3, pp. 373–384, 1991.

S. Porter-Hudak, An application of the seasonal fractionally differenced model to the monetary aggegrates, Journal of the American Statistical Association, vol.85, no. 410, pp. 338–344, 1990.

V. A. Reisen, E. Z. Monte, G. C. Franco, A. M. Sgrancio, F. A. F. Molinares, P. Bondond, F. A. Ziegelmann, and B. Abraham, Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations, Mathematics and Computers in Simulation, vol. 146, pp. 27–43, 2018.

R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 451 p., 1997.

C. C. Solci, V. A. Reisen, A. J. Q. Sarnaglia, and P. Bondon, Empirical study of robust estimation methods for PAR models with application to the air quality area, Communication in Statistics - Theory and Methods, vol. 48, no. 1, pp. 152–168, 2020.

H. Tsai, H. Rachinger, and E.M.H. Lin, Inference of seasonal long-memory time series with measurement error, Scandinavian Journal of Statistics, vol. 42, no. 1, pp. 137–154, 2015.

S. K. Vastola, and H. V. Poor, Robust Wiener-Kolmogorov theory, IEEE Trans. Inform. Theory, vol. 30, no. 2, pp. 316–327, 1984.

A. M. Yaglom, Correlation theory of stationary and related random processes with stationary nth increments. American Mathematical Society Translations: Series 2, vol. 8, pp. 87 –141, 1958.

A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Supplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.

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*12*(2), 343-363. https://doi.org/10.19139/soic-2310-5070-1793

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