Bayesian inference of time varying parameters in autoregressive processes

In the autoregressive process of first order AR(1), a homogeneous correlated time series ut is recursively constructed as ut = q ut−1 + σ t, using random Gaussian deviates t and fixed values for the correlation coefficient q and for the noise amplitude σ. To model temporally heterogeneous time series, the coefficients qt and σt can be regarded as time-dependent variables by themselves, leading to the time-varying autoregressive processes TVAR(1). We assume here that the time series ut is known and attempt to infer the temporal evolution of the ’superstatistical’ parameters qt and σt. We present a sequential Bayesian method of inference, which is conceptually related to the Hidden Markov model, but takes into account the direct statistical dependence of successively measured variables ut. The method requires almost no prior knowledge about the temporal dynamics of qt and σt and can handle gradual and abrupt changes of these superparameters simultaneously. We compare our method with a Maximum Likelihood estimate based on a sliding window and show that it is superior for a wide range of window sizes.