Continuous-Time Autoregressive Models: Sampling, Uniqueness and Estimation
Abstract: Continuous-time autoregressive (AR) models are widely used in astronomy, geology, quantitative finance, control theory and signal processing, to name a few. In practice, the available data is discrete, and one is often required to estimate continuous-time parameters from sampled data. The intertwining relations between the continuous-time model and its sampled version play an important role in such estimation tasks, and this is the main concern of the presented work. In particular, it will be shown that almost every continuous-time AR model is uniquely determined by its sampled version on a uniform grid. This is achieved by removing a set of measure zero from the collection of all AR models and by investigating the asymptotic behaviour of the remaining set of autocorrelation functions. This uniqueness property is further exploited for introducing an estimation algorithm that recovers continuous-time AR parameters from sampled data, while imposing no constraint on the sampling interval value. The usefulness of the algorithm will be then demonstrated for both Gaussian and non-Gaussian AR processes, as well as for image processing tasks.