Multiresolution stochastic models for the efficient solution of large-scale space-time estimation problems

Abstract

The successful application of a recently developed treebased multiscale estimation framework [l] to large-scale static estimation problems has proven the statistical flexibility and the computational efficiency of the framework. However, the estimation of processes evolving in time remains a considerable challenge. In this paper, we address this challenge by investigating multiscale models for the steady-state estimation of dynamic systems. In particular, we build such models for 1-D and 2-D heat diffusion processes, making use of a canonical correlations realization technique.

A generalized Levinson algorithm for covariance extension with application to multiscale autoregressive modeling

“…That the choice of maximizes the determinant of can be seen as follows. 11 Using (25), (26a), and the fact that we have that Applying (26b) and using Lemma 4, we have (27) Hence, the choice of maximizes the determinant of .…”

“…MAR models generalize state-space models of time series, evolving in scale rather than in time. They have been effectively applied to a wide variety of signal and image processing problems [8], [9], [13]- [17], [27], [30], [34]- [38], [42], [48], [49] and their success stems, in part, from the efficiency of the statistical inference algorithms to which they lead [4], [44]. Recent approaches to the MAR model identification problem [10], [18], [19], [31] rely on complete knowledge of the second-order statistics of the process to be modeled.…”

A methodology for merging multisensor precipitation estimates based on expectation‐maximization and scale‐recursive estimation

“…This step can be involved even in simple dynamics such as diffusion processes. Research on SRE of dynamic fields includes that of Ho et al [1996]. The idea behind their approach is that the multiscale models for the updated and predicted estimation errors are propagated through time in the same way that Kalman filter propagates the error covariances, but in a more computationally efficient manner, i.e., without computing or storing the full error covariance matrix.…”

A Canonical Correlations Approach to Multiscale Stochastic Realization

“…In contrast to traditional 2-D optimal estimation formulations based on Markov random fields, which lead to iterative algorithms having a per-pixel computational complexity that typically grows with image size, these multiscale algorithms are noniterative and have a per-pixel complexity independent of image size [18], [19]. Substantial computational savings can thus result, as evidenced by exploitation of the multiscale framework in many applications, including computer vision (e.g., calculation of optical flow [19]), remote sensing (e.g., optimal interpolation of sea level variations in the North Pacific Ocean from satellite measurements, treating the ocean as either static [11], or more recently as dynamic [12]) and geophysics (e.g., characterizing spatial variations in hydraulic conductivity from both point and nonlocal measurements [6]). In exactly the same way that Kalman filtering requires the prior specification of a state-space model before least-square estimation or likelihood calculation can be carried out, so does multiscale statistical processing require such prior modeling.…”

“…For instance, in many image-processing applications (such as de-noising problems [15], or terrain segmentation [17]), the finest-scale process is a pixel-by-pixel representation of the image, the measurements are noisy observations of some subset of the pixels, and the objective is either to estimate the value of each image pixel [15] or to calculate the likelihood of the observations [17]. Similarly, in distributed parameter estimation problems (such as those arising in remote sensing [12] or geophysics [6]), the finest-scale process is a sampled version of the parameter of interest (e.g., hydraulic conductivity [6]), the measurements are noisy observations related to the parameter, and the objective is to estimate the parameter. For all of these problems, the multiscale framework provides an efficient statistical approach for obtaining estimates and error covariances or calculating likelihoods, even though every other aspect of these problems involves only the finest scale.…”