Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments
AbstractWe introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. 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 estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates 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.
Andel, J. (1986). Long memory time series models. Kybernetika, 22(2), 105--123.
Arteche, J. and Robinson, P. (2002). Semiparametric inference in seasonal and cyclical long-memory processes. Journal of Time Series Analysis, 21(1), 1--25.
Baek, C., Davis, R. A. and Pipiras, V. (2018). Periodic dynamic factor models: estimation approaches and applications. Electronic Journal of Statistics, 12(2), 4377--4411.
Baillie, R. T., Kongcharoen, C. and Kapetanios, G. (2012). Prediction from ARFIMA models: Comparisons between MLE and semiparametric estimation procedures. International Journal of Forecasting, 28, 46--53.
Basawa, I. V., Lund, R. and Shao, Q. (2004). First-order seasonal autoregressive processes with periodically varying parameters. Statistics & Probability Letters, 67(4), 299--306.
Box, G. E. P., Jenkins, G. M., Reinsel, G. C. and Ljung, G.M. (2016). Time series analysis, Forecasting and control. 5rd ed., Wiley.
Dubovets'ka, I. I. and Moklyachuk, M. P. (2013). Extrapolation of periodically correlated processes from observations with noise, Theory of Probability and Mathematical Statistics, 88, 43--55.
Dudek, G. (2013) 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.
Dudek, A., Hurd, H. and Wojtowicz, W. (2016). PARMA methods based on Fourier representation of periodic coefficients, Wiley Interdisciplinary Reviews: Computational Statistics, 8(3), 130--149.
Franke, J. (1985). Minimax-robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 68, 337--364.
Gikhman, I. I. and Skorokhod, A. V. (2004). The theory of stochastic processes. I. Berlin: Springer.
Giraitis, L. and Leipus, R. (1995). A generalized fractionally differencing approach in long-memory modeling. Lithuanian Mathematical Journal, 35, 53--65.
Gladyshev, E. G. (1961). Periodically correlated random sequences. Sov. Math. Dokl. , 2, 385--388.
Gould, P. G., Koehler, A. B., Ord, J. K., Snyder, R. D., Hyndman, R. J. and Vahid-Araghi, F. (2008). Forecasting time-series with multiple seasonal patterns. European Journal of Operational Research, 191, 207--222.
Granger, C. W. J. and Joyeux, R. (1980). An intoduction to long memory time series and fractional differencing, Journal of Time Series Analysis, 1, 15--30.
Grenander, U. (1957). A prediction problem in game theory, Arkiv f"or Matematik, 6, 371--379.
Gray, H., Cheng, Q. and Woodward, W. (1989). On generalized fractional processes. Journal of Time Series Analysis, 10, 233--257.
Hannan, E. J. (1970). Multiple time series. Wiley, New York.
Hassler, U. (2019). Time series analysis with Long Memory in view. Wiley, Hoboken, NJ.
Hassler, U. and Pohle, M. O. (2019). Forecasting under long memory and nonstationarity, arXiv preprint arXiv:1910.08202.
Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68(1), 165--176.
Hosoya, Y. (1978). Robust linear extrapolations of second-order stationary processes. Annals of Probability, 6(4), 574--584.
Hurd, H. and Pipiras, V. (2020). 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, 16, Springer, Cham.
Johansen, S. and Nielsen, M. O. (2016). The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models. Econometric Theory, 32(5), 1095--1139.
Kassam, S. A. (1982). Robust hypothesis testing and robust time series interpolation and regression. Journal of Time Series Analysis, 3(3), 185--194.
Kassam, S. A. and Poor, H. V. (1985). Robust techniques for signal processing: A survey. Proceedings of the IEEE, 73, 433--481.
Karhunen, K. (1947). Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, 37.
Kolmogorov, A. N. (1992). Selected works of 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.
Kozak, P. S. and Moklyachuk, M. P. (2018). Estimates of functionals constructed from random sequences with periodically stationary increments. Theory Probability and Mathematical Statistics, 97, 85--98.
Lund, R. (2011). 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, 63--80.
Luz, M. and Moklyachuk, M. (2015). Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences. Statistics, Optimization and Information Compututing, 3, 160--188.
Luz, M. and Moklyachuk, M. (2019). Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences. Wiley - ISTE.
Moklyachuk, M. P. (1994). Stochastic autoregressive sequences and minimax interpolation. Theory
Probability and Mathematical Statistics, 48, 95--103.
Moklyachuk, M. P. (2015). Minimax-robust estimation problems for stationary stochastic sequences.
Statistics, Optimization and Information Computing, 3(4), 348--419.
Moklyachuk, M. P. and Masyutka, A. Yu. (2006). Extrapolation of multidimensional stationary processes. Random Operators and Stochastic Equations, 14(3), 233--244.
Moklyachuk, M. P. and Masyutka, A. Yu. (2011). Minimax prediction problem for multidimensional stationary stochastic processes. Communications in Statistics-Theory and Methods, 40(19-20), 3700-3710.
Moklyachuk, M. and Sidei, M. (2017). Extrapolation problem for stationary sequences with missing observations. Statistics, Optimization & Information Computing, 5(3), 212--233.
Moklyachuk, M., Sidei, M. and Masyutka, O. (2019). Estimation of stochastic processes with missing observations. Nova Science Publishers.
Napolitano, A. (2016). Cyclostationarity: New trends and applications. Signal Processing, 120, 385--408.
Osborn, D. (1991). The implications of periodically varying coefficients for seasonal time-series processes. Journal of Econometrics, 48(3), 373--384.
Palma, W. and Bondon, P. (2003). On the eigenstructure of generalized fractional processes, Statistics & Probability Letters, 65, 93--101.
Pinsker, M. S. and Yaglom, A. M. (1954). On linear extrapolaion of random processes with $n$th stationary incremens. Doklady Akademii Nauk SSSR, 94, 385--388.
Porter-Hudak, S. (1990). An application of the seasonal fractionally differenced model to the monetary aggegrates, Journal of the American Statistical Association, 85(410), 338--344.
Reisen, V. A., Zamprogno, B., Palma, W. and Arteche, J. (2014). A semiparametric approach to estimate two seasonal fractional parameters in the SARFIMA model, Mathematics and Computers in Simulation, 98, 1--17.
Reisen, V. A., Monte, E. Z., Franco, G. C., Sgrancio, A. M., Molinares, F. A. F., Bondond, P., Ziegelmann, F. A. and Abraham, B. (2018). Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations. Mathematics and Computers in Simulation, 146 , 27--43.
Rockafellar, R. T. (1997). Convex Analysis. Princeton University Press.
Solci, C. C., Reisen, V. A., Sarnaglia, A. J. Q. and Bondon, P. (2020). Empirical study of robust estimation methods for PAR models with application to the air quality area. Communication in Statistics - Theory and Methods, 48(1), 152--168,
Tsai, H., Rachinger, H. and Lin, E. M. H. (2015). Inference of seasonal long-memory time series with measurement error. Scandinavian Journal of Statistics, 42(1) , 137--154.
Vastola, S. K. and Poor, H. V. (1984). Robust Wiener-Kolmogorov theory. IEEE Trans. Inform. Theory, 30(2), 316--327.
Woodward, W., Cheng, Q. and Gray, H. (1998). A k-factor GARMA long memory model. Journal of Time Series Analysis, 19, 485--504.
Yaglom, A. M. (1955). Correlation theory of stationary and related random processes with stationary $n$th increments. Mat. Sbornik, 37(1), 141--196.
Yaglom, A. M. (1987). Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Suplementary notes and references. Springer Series in Statistics. New York etc.: Springer-Verlag.
- 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).