A Bayesian approach for the joint estimation of the multifractality parameter and integral scale based on the Whittle approximation

Multifractal analysis is a powerful tool used in signal processing. Multifractal models are essentially characterized by two parameters, the multifractality parameter c₂ and the integral scale A (the time scale beyond which multifractal properties vanish). Yet, most applications concentrate on estimating c₂ while the estimation of A is in general overlooked, despite the fact that A potentially conveys important information. Joint estimation of c₂ and A is challenging due to the statistical nature of multifractal processes (i.e. the strong dependence and non-Gaussian nature), and has barely been considered. The present contribution addresses these limitations and proposes a Bayesian procedure for the joint estimation of (c₂, A). Its originality resides, first, in the construction of a generic multivariate model for the statistics of wavelet leaders for multifractal multiplicative cascade processes, and second, in the use of a suitable Whittle approximation for the likelihood associated with the model. The resulting model enables Bayesian estimators for (c₂, A) to also be computed for large sample size. Performance is assessed numerically for synthetic multifractal processes and illustrated for wind-tunnel turbulence data. The proposed procedure significantly improves estimation of c₂ and yields, for the first time, reliable estimates for A.