Robust Filtering of Sequences with Periodically Stationary Multiplicative Seasonal Increments
Abstract
We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the filtering problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal filtering of the functionals. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear filtering of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.References
C. F. Ansley, and R. Kohn, Estimation, filtering and smoothing in state space models with incompletely specified initial conditions, The Annals of Statistics, vol. 13, pp. 1286–1316, 1985.
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.
P. Eiurridge, and K. F. Wallis, Seasonal adjustment and Kalman filtering: Extension to periodic variances, Journal of Forecasting, vol. 9, no. 2, p. 109–118, 1990.
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.
A. Dudek, H. Hurd, and W. Wojtowicz, PARMA methods based on Fourier representation of periodic coefficients, Wiley Interdisciplinary Reviews: Computational Statistics, vol. 8, no. 3, pp. 130–149, 2016.
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.
L. Giraitis, and R. Leipus, A generalized fractionally differencing approach in long-memory modeling, Lithuanian Mathematical Journal, vol. 35, pp. 53–65, 1995.
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 för Matematik, vol. 3, pp. 371–379, 1957.
H. Gray, Q. Cheng and W. Woodward, On generalized fractional processes, Journal of Time Series Analysis, vol. 10, no. 3, pp. 233–257, 1989.
E. J. Hannan, Multiple time series. 2nd rev. ed., John Wiley & Sons, New York, 2009.
U. Hassler, Time series analysis with long memory in view, Wiley, Hoboken, NJ, 2019.
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.
S. Johansen, and M. O. Nielsen, The role of initial values in conditional sum-of-squaresestimation of nonstationary fractional time series models, Econometric Theory, vol. 32, no. 5, pp. 1095–1139, 2016.
K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, no. 37, 1947.
S. A. Kassam, Robust hypothesis testing and robust time series interpolation and regression, Journal of Time Series Analysis, vol. 3, no. 3, pp. 185–194, 1982.
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.
P. S. Kozak, and M. P. Moklyachuk, Estimates of functionals constructed from random sequences with periodically stationary increments, Theory Probability and Mathematical Statistics, vol. 97, pp. 85–98, 2018.
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, Minimax-robust filtering problem for stochastic sequence with stationary increments, Theory Probability and Mathematical Statistics, vol. 89, pp. 127–142, 2014.
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, 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.
O. Yu. Masyutka, M. P. Moklyachuk, and M. I. Sidei, Filtering of multidimensional stationary sequences with missing observations, Carpathian Mathematical Publications, vol. 11, no. 2, pp. 361–378, 2019.
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, and M. Sidei, Filtering problem for stationary sequences with missing observations, Statistics, Optimization and Information Computing, vol. 4, no. 4, pp. 308–325, 2016.
M. Moklyachuk, M. Sidei, and O. Masyutka, Estimation of stochastic processes with missing observations, Mathematics Research Developments. Nova Science Publishers, New York, NY, 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.
W. Palma, and P. Bondon, On the eigenstructure of generalized fractional processes, Statistics and Probability Letters, vol. 65, pp. 93–101, 2003.
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, B. Zamprogno, W. Palma, and J. Arteche, A semiparametric approach to estimate two seasonal fractional parameters in the SARFIMA model, Mathematics and Computers in Simulation, vol. 98, pp. 1–17, 2014.
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 University Press, Princeton, NJ, 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.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
