Biometrika Advance Access published online on May 23, 2007
Biometrika, doi:10.1093/biomet/asm025
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Copyright © 2007 Biometrika Trust
Article |
Inference on fractal processes using multiresolution approximation
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, U.K.
Instituto Español de Oceanografía, Cabo Estai-Canido, 36200 Vigo, Spain
kjf{at}st-and.ac.uk
carmen.fernandez{at}vi.ieo.es
Received for publication 1 February 2005.
Revision received 1 July 2006.
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Abstract |
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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.
Key Words: Fractal process • Fractional Brownian motion • Graphical model • Markov chain Monte Carlo • Multiresolution approximation • Temporal modelling