A Random Matrix Approach to VARMA Processes

We apply random matrix theory to derive spectral density of large sample covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2) processes. In particular, we consider a limit where the number of random variables N and the number of consecutive time measurements T are large but the ratio N/T is fixed. In this regime the underlying random matrices are asymptotically equivalent to Free Random Variables (FRV). We apply the FRV calculus to calculate the eigenvalue density of the sample covariance for several VARMA-type processes. We explicitly solve the VARMA(1,1) case and demonstrate a perfect agreement between the analytical result and the spectra obtained by Monte Carlo simulations. The proposed method is purely algebraic and can be easily generalized to q1>1 and q2>1.Comment: 16 pages, 6 figures, submitted to New Journal of Physic

Wiener splines

We describe an alternative way of constructing interpolating B-spline curves, surfaces or volumes in Fourier space which can be used for visualization. In our approach the interpolation problem is considered from a signal processing point of view and is reduced to finding an inverse B-spline filter sequence. The Fourier approach encompasses some advantageous features, such as successive approximation, compression, fast convolution and hardware support. In addition, optimal Wiener filtering can be applied to remove noise and distortions from the initial data points and to compute a smooth, least-squares fitting ‘Wiener spline’. Unlike traditional fitting methods, the described algorithm is simple and easy to implement. The performance of the presented method is illustrated by some examples showing the restoration of surfaces corrupted by various types of distortions