Multi-scale invariant fields: estimation and prediction

Abstract

Extending the concept of multi-selfsimilar random field, we study multi-scale invariant (MSI) fields with component-wise discrete scale invariant properties. Assuming scale parameters as λ i > 1, i = 1, …, d and the parameter space as (1, ∞) d , the first scale rectangle is referred to the rectangle (1, λ 1) ×⋯× (1, λ d ). We show that the covariance functions of the sampled Markov MSI field are characterized by the variances and covariances of samples inside the first scale rectangle. As an example of an MSI field, a two-dimensional simple fractional Brownian sheet (sfBss) is demonstrated. In addition, real data for precipitation in the area of Brisbane, Australia, covering a period of two days (25 and 26 January 2013), are examined. We show that precipitation in this area demonstrates MSI properties, and estimate it as a simple MSI field with stationary increments inside scale intervals. This structure enables us to predict the precipitation in terms of surface and time. We apply the mean absolute percentage error as a measure for the accuracy of the predictions.