Demixing multivariate-operator self-similar processes

Operator self-similarity naturally extends the concepts of univariate self-similarity and scale invariance to multivariate data. Beyond a vector of Hurst parameters, operator self-similarity models also involve a mixing matrix. The present contribution aims at estimating the collection of Hurst parameters in the case where the mixing matrix is not diagonal. To the best of our knowledge, this has never been achieved. In addition, the mixing matrix is also identified. The devised procedure relies on a source separation methodology, since the underlying components of the operator self-similar process are assumed to have a diagonal pre-mixing covariance structure. The principle behind the demixing procedure is illustrated based on synthetic 4-variate operator self-similar processes, with a priori prescribed and controlled Hurst parameters and mixing matrix. Identification and estimation performance for both Hurst parameters and mixing matrices are shown to be very satisfactory, using large size Monte Carlo simulations.