Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding

The problem of synthesizing multivariate stationary series Y[n]=(Y1[n],…,YP[n])T, n∈Zn∈Z, with prescribed non-Gaussian marginal distributions, and a targeted covariance structure, is addressed. The focus is on constructions based on a memoryless transformation Yp[n]=fp(Xp[n]) of a multivariate stationary Gaussian series X[n]=(X1[n],…,XP[n])T. The mapping between the targeted covariance and that of the Gaussian series is expressed via Hermite expansions. The various choices of the transforms fp for a prescribed marginal distribution are discussed in a comprehensive manner. The interplay between the targeted marginal distributions, the choice of the transforms fp, and on the resulting reachability of the targeted covariance, is discussed theoretically and illustrated on examples. Also, an original practical procedure warranting positive definiteness for the transformed covariance at the price of approximating the targeted covariance is proposed, based on a simple and natural modification of the popular circulant matrix embedding technique. The applications of the proposed methodology are also discussed in the context of network traffic modeling. Matlab codes implementing the proposed synthesis procedure are publicly available at http://www.hermir.org.