Stationary time series resulting from certain positive definite kernels and simulation via high-order vector autoregressive models

Abstract

Vector auto-regressive models have traditionally been used to model and forecast multivariate time series data, predicting future values based on previous observations. In this thesis, we introduce some multivariate time series with power-law decaying covariance matrix functions, and then construct a VAR model in order to generate approximate data from that time series. A fast model is developed to solve for the VAR(p) coefficients, implementing a block-Toeplitz equation solver to enable the choice of large p, avoiding the memory and speed issues with solving large systems via Gaussian elimination. An approximate error bound is established, demonstrating the quality of the simulation for large p. Finally, we explore inverting the VAR model to obtain a VMA (vector moving average) model.