GENERALIZED FRACTIONAL GAUSSIAN NOISE PREDICTION BASED ON THE WALSH FUNCTIONS

Abstract

Context. Some of the authors’ recent papers were devoted to the Kolmogorov-Wiener filter for telecommunication traffic prediction in some stationary models, such as the fractional Gaussian noise model, the power-law structure function model, and the GFSD (Gaussian fractional sum-difference) model. Recently, the so-called generalized fractional Gaussian noise model was proposed for stationary telecommunication traffic description in some cases. So, in this paper the theoretical fundamentals of the continuous Kolmogorov-Wiener filter used for the prediction of the generalized fractional Gaussian noise are investigated.  Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding Wiener– Hopf integral equation with the kernel equal to the generalized fractional Gaussian noise correlation function. Method. A truncated Walsh function expansion is proposed in order to obtain the corresponding solution. This expansion is a special case of the Galerkin method, in the framework of which the unknown function is sought as a truncated series in orthogonal functions. The integral brackets and the results for the mean absolute percentage errors, which are a measure of discrepancy between the left-hand side and the right-hand side of the Wiener-Hopf integral equation, are calculated numerically on the basis of the Wolfram Mathematica package. Results. The investigation is made for approximations up to sixty four Walsh functions. Different model parameters are investigated. It is shown that for different model parameters the proposed method is convergent and leads to small mean absolute percentage errors for approximations of rather large numbers of Walsh functions. Conclusions. The paper is devoted to a theoretical construction of the continuous Kolmogorov-Wiener filter weight function for the prediction of a stationary random process described by the generalized fractional Gaussian noise model. As is known, this model may give a good description of some actual telecommunication traffic data in systems with packet data transfer. The corresponding weight function is sought on the basis of the truncated Walsh function expansion method. The corresponding discrepancy errors are small and the method is convergent.