Efficient Schur parametrization of near-stationary stochastic processes

International Journal of Electronics and Telecommunications

“…They are based on and following from the concepts of 'low-rank' [12], [10], resulting in a hierarchical classification of nonstationary processes in terms of their 'distance' from stationarity, block-Toeplitz or other structured matrices. In this applicationsoriented paper originating from [21], [23] we show, for a subclass of second-order near-stationary time-seriess, which we call 'p-stationary' whose etimates of the covariance matrices are block-Toeplitz, how their low displacment-rank is reflected in the structures of the corresponding Schur parametrization schemas. allowing for a considerable complexity reduction in a uniform way.…”

“…we can immediately see that < z i+1 y|z k+1 y > T = ĥi+1,k+1 = ĥi,k =< z i y|z k y > T resulting in the Toeplitz estimate ĤT of the covariance matrix. Hence, the idea of p-stationary class stochastic processes, introduced in [21], can also be employed for the underlying time-series. Firstly, let us observe that the sample-product space and the space of random variables are isometrically isomorphic.…”

“…Complexity reduction can be introduced if the matrix is 'structured', and many approaches have been proposed, to mention block-Toeplitz cases, staircase extension idea, low displacement-rank and/or α-stationarity concepts. The last treatment of the problem has been summarized in [21] as that work has been inspired up to some extend by those ideas, allowing to propose and consider a class of stochastic processes whose covariance matrices are block-Toeplitz (which we called p-stationary).…”

“…

Following the results presented in [21], we present an efficient approach to the Schur parametrization/modeling of a subclass of second-order time-series which we term p-stationary time-series, yielding a uniform hierarchy of algorithms suitable for efficient implementations and being a good starting point for nonlinear generalizations to higher-order non-Gaussian nearstationary time-series.

…”