Inference on fractal processes using multiresolution approximation

We consider Bayesian inference via Markov chain Monte Carlo for a variety of fractal Gaussian processes on the real line. These models have unknown parameters in the covariance matrix, requiring inversion of a new covariance matrix at each Markov chain Monte Carlo iteration. The processes have no suitable independence properties so this becomes computationally prohibitive. We surmount these difficulties by developing a computational algorithm for likelihood evaluation based on a 'multiresolution approximation' to the original process. The method is computationally very efficient and widely applicable, making likelihood-based inference feasible for large datasets. A simulation study indicates that this approach leads to accurate estimates for underlying parameters in fractal models, including fractional Brownian motion and fractional Gaussian noise, and functional parameters in the recently introduced multifractional Brownian motion. We apply the method to a variety of real datasets and illustrate its application to prediction and to model selection.