Non-Stationary Stochastic Processes Estimation: Vector Stationary Increments, Periodically Stationary Multi-Seasonal Increments

The problem of forecasting future values of economic and physical processes, the problem of restoring lost information, cleaning signals or other data observations from noise, is magnified in an information-laden word. Methods of stochastic processes estimation depend on two main factors.

The first factor is construction of a model of the process being investigated.

The second factor is the available information about the structure of the process under consideration. In this book, we propose results of the investigation of the problem of mean square optimal estimation (extrapolation, interpolation, and filtering) of linear functionals

depending on unobserved values of stochastic sequences and processes

with periodically stationary and long memory multiplicative seasonal increments.

Formulas for calculating the mean square errors and the spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where

spectral structure of the considered sequences and processes are exactly known.

In the case where spectral densities of the sequences and processes are not known exactly while some sets of admissible spectral densities are given, we apply the minimax-robust method of estimation.