Bayesian inference on the order of stationary vector autoregressions with application to multivariate modelling of electroencephalography data

Abstract

Vector autoregressions (VARs) are widely used for modelling multivariate time series. VARs have an associated order p; given observations at the preceding p time points, the variable at time t is conditionally independent of all earlier history. The model order is therefore intrinsic to the characterisation of the process. It is common to assume a VAR is stationary, which requires the means, variances and covariances of the process to be constant over time. This can be enforced by imposing the stationarity condition which restricts the parameter space of the autoregressive coefficients to the stationary region. However, implementing this constraint is difficult as the stationary region has a complex geometry. Fortunately, pioneering recent work has provided a solution for enforcing stationarity in autoregressions of fixed order p based on a reparameterisation in terms of a set of interpretable and unconstrained transformed partial autocorrelation matrices. In this research, focus is placed on the difficult problem of allowing p to be unknown, developing priors and computational inference that take full account of order uncertainty. To this end, a comparison of existing approaches for determining the order of station ary univariate autoregressions is provided. An approach employing shrinkage priors for partial autocorrelations is then generalised for the multivariate case, using the cumula tive shrinkage and multiplicative gamma process priors to increasingly shrink the partial autocorrelation matrices with increasing lag. Identifying the lag beyond which these ma trices become equal to zero then determines p. Methods for identifying whether a partial autocorrelation matrix is effectively zero are developed. The work is illustrated through application to neural activity data. In particular, a detailed discussion of methods to decompose a VAR into latent processes is provided, which is then used to investigate ultradian rhythms in the brain. Relationships between different regions of the brain are investigated through Granger causality plots.