Analysis Of A Long Memory Circular Convolution Model

TL;DR

This paper introduces a stochastic model using circulant matrices and normal vectors to generate long memory time series with power law spectral density, connecting spectral analysis to directional statistics.

Contribution

It presents a novel model linking circulant matrix eigenanalysis to long memory time series and provides practical R code for exploration.

Findings

Produces long memory time series with power law spectral density

Eigenanalysis estimates spectral trend and intrinsic dimension

R code enables practical exploration of the model

Abstract

A stochastic model, the product of a circulant matrix and a random normal vector, is shown to produce an evolutive long memory time series with a power law spectral density. The distribution of the time series, a beta location scale family of distributions, provides a connection to the unit centered spherical distribution and directional statistics. The eigenanalysis of the deterministic circulant matrix is shown to provide estimates of the discrete Fourier spectral trend, the intrinsic dimension, the probability density shape parameter of the resulting time series, the condition number of the matrix and a principle component analysis. Examples of the R code, used as the constructive exploratory element of the work are given as constructive elements of the paper. The R code may be copied, pastedinto a R editor, and explored.