Fractal estimation using models on multiscale trees

Abstract

1291since both ITL and ilf -711 are even. Assume that g ( 7 ) # 0. Then one can easily see, from (A.l I), that g ( j ) = 0 for all j # 7 . In other words, the optimum g ( k ) has the form g ( k ) = ab(k -7 ) . Thus Fractal Estimation Using Models on Multiscale TreesPaul W. Fieguth and Alan S. Willsky Abstract-In this correspondence, we estimate the Hurst parameter H of fractional Brownian motion (or, by extension, the fractal exponent 9 of stochastic processes having 1/ f +'-like spectra) by applying a recently introduced multiresolution framework. This framework admits an efficient likelihood function evaluation, allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease. In addition to yielding results that compare well with other proposed methods, and in contrast with other approaches, our method is directly applicable with, at most, very simple modification in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2-D random fields.

A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing

“…The utility of this class of models is twofold. First, the class has been shown to provide useful models for a wide variety of random processes and fields, such as 1-D Markov processes and two-dimensional (2-D) Markov random fields (MRF's) [19] and self-similar and fractal processes that can be used to model natural phenomena arising in geophysics [8], [9]. Second, and most importantly, just as the Markov property associated with 1-D autoregressive models leads to a highly efficient estimation algorithm, (the Kalman filter), the multiscale models satisfy a Markov property in scale and space which leads to an efficient estimation algorithm.…”

Fast, Resolution-Consistent Spatial Prediction of Global Processes From Satellite Data

“…In the last decade, there has been a lot of research interest in multiresolution methods, including multiresolution representations of signals based on wavelet transforms (e.g., Daubechies 1992;Mallat 1989;Meyer 1992), and multiresolution stochastic models linking coarser-scale variables to ner-scale variables in an autoregressive manner via trees (e.g., Chou et al 1994;Luettgen andWillsky 1995a, 1995b;Fieguth and Willsky 1996). An advantage of using these methods is that many signals naturally have multiscale features.…”

“…Pioneers in this area have been A. S. Willsky and colleagues (e.g., Chou et al 1994;Luettgen and Willsky 1995a, 64 H.-C. HUANG, N. CRESSIE, AND J. GABROSEK 1995b;Fieguth and Willsky 1996); in Daoudi, Frakt, and Willsky (1999), these models are referred to as multiscale autoregressive (MAR) models. In our application to global spatial prediction of total column ozone, we focus on a speci c subclass of these models and introduce the natural requirement of "mass balance."…”

Dynamic programming and the graphical representation of error-correcting codes

“…Because of their computational advantages, tree-structured dependencies are attractive modeling tools (e.g., [32], [33]). They also come up naturally when working with production systems, which define the so-called "context-free" grammars studied in formal linguistics ( [4]).…”