In this paper we consider a discrete scale invariant (DSI) process {X(t), t ∈ R + } with scale l > 1. We consider to have some fix number of observations in every scale, say T , and to get our samples at discrete points α k , k ∈ W where α is obtained by the equality l = α T and W = {0, 1, . . .}. So we provide a discrete time scale invariant (DT-SI) process X(·) with parameter space {α k , k ∈ W}. We find the spectral representation of the covariance function of such DT-SI process. By providing harmonic like representation of multi-dimensional self-similar processes, spectral density function of them are presented. We assume that the process {X(t), t ∈ R + } is also Markov in the wide sense and provide a discrete time scale invariant Markov (DT-SIM) process with the above scheme of sampling. We present an example of DT-SIM process, simple Brownian motion, by the above sampling scheme and verify our results. Finally we find the spectral density matrix of such DT-SIM process and show that its associated T -dimensional self-similar Markov process is fully specified by {R H j (1), R H j (0), j = 0, 1, . . . , T − 1} where R H j (τ ) is the covariance function of jth and (j + τ )th observations of the process.
We study locally self-similar processes (LSSPs) in Silverman's sense. By deriving the minimum mean-square optimal kernel within Cohen's class counterpart of time-frequency representations, we obtain an optimal estimation for the scale invariant Wigner spectrum (SIWS) of Gaussian LSSPs. The class of estimators is completely characterized in terms of kernels, so the optimal kernel minimizes the mean-square error of the estimation. We obtain the SIWS estimation for two cases: global and local, where in the local case, the kernel is allowed to vary with time and frequency. We also introduce two generalizations of LSSPs: the locally self-similar chrip process and the multicomponent locally self-similar process, and obtain their optimal kernels. Finally, the performance and accuracy of the estimation is studied via simulation.
By considering special sampling of discrete scale invariant (DSI) processes we provide a sequence which is in correspondence to multi-dimensional self-similar process. By imposing Markov property we show that the covariance functions of such discrete scale invariant Markov (DSIM) sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion with drift and scale invariant autoregressive model of order one are presented and these properties are investigated. By simulating DSIM sequences we provide visualization of their behavior and investigate these results. Finally we present a new method to estimate Hurst parameter of DSI processes and show that it has much better performance than maximum likelihood method for simulated data.
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