Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models

. In this paper we develop a stochastic realization theory for multiscale autoregressive (MAR) processes that leads to computationally efficient realization algorithms. The utility of MAR processes has been limited by the fact that the previously known general purpose realization algorithm, based on canonical correlations, leads to model inconsistencies and has complexity quartic in problem size. Our realization theory and algorithms addresses these issues by focusing on the estimation-theoretic concept of predictive efficiency and by exploiting the scale-recursive structure of so-called internal MAR processes. Consequently, our realized models are consistent and our algorithm has complexity quadratic in problem size. We also introduce an approximation to obtain an algorithm that has complexity linear in problem size. Keywords: multiscale autoregressive models, stochastic realization, graphical models, predictive efficiency, state-space, Markov models 1. Introduction The power of the m..

Scalable Filtering Methods For High-Dimensional Spatio-Temporal Data

We propose a family of filtering methods for deriving the filtering distribution in the context of a high-dimensional state-space model. In the first chapter, we develop and describe in detail the basic method, which can be used in a linear case with Gaussian data. In the second chapter, we show how this method can be extended to incorporate non-Gaussian observations and non-linear temporal evolution models. We discuss how two algorithms, the multi-resolution decomposition and the incomplete Cholesky decomposition, can be used to quickly update the filtering distribution at each time step of the filtering procedures

Multi-sensor large scale land surface data assimilation using ensemble approaches

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2006.Includes bibliographical references (p. 223-234).One of the ensemble Kalman filter's (EnKF) attractive features in land surface applications is its ability to provide distributional information. The EnKF relies on normality approximations that improve its efficiency but can also compromise the accuracy of its distributional estimates. The effects of these approximations are evaluated by comparing the conditional marginal distributions and moments estimated by the EnKF to those obtained from an SIR particle filter, which gives exact solutions for large ensemble sizes. The results show that overall the EnKF appears to provide a good approximation for nonlinear, non-normal land surface problems. A difficulty in land data assimilation problems results from the high dimensionality of states created by spatial discretization over large computational grids. The high dimensionality can be reduced by exploiting the fact that soil moisture field may have significant spatial correlation structure especially after extensive rainfall while it may have local structure determined by soil and vegetation variability after prolonged drydown. This is confirmed by SVD of the replicate matrix produced in an ensemble forecasting experiment. Local EnKF's are suitable for problems during dry periods but give less accurate results after rainfall.(cont.) The most promising option is to develop a generalized method that reflects structural changes in the ensemble. A highly efficient ensemble multiscale filter (EnMSF) is then proposed to solve large scale nonlinear estimation problems with arbitrary uncertainties. At each prediction step realizations of the state variables are propagated. At update times, joint Gaussian distribution of states and measurements are assumed and the Predictive Efficiency method is used to identify a multiscale tree to approximate statistics of the propagated ensemble. Then a two-sweep update is performed to estimate the state variables using all the data. By controlling the tree parameters, the EnMSF can reduce sampling error while keep long range correlation in the ensemble. Applications of the EnMSF to Navier-Stokes equation and a nonlinear diffusion problem are demonstrated. Finally, the EnMSF is successfully applied to soil moisture and surface fluxes estimation over the Great Plains using synthetic multiresolution L-band passive and active microwave soil moisture measurements following HYDROS specifications.by Yuhua Zhou.Ph.D

Stochastic realization theory for exact and approximate multiscale models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. [245]-252).The thesis provides a detailed analysis of the independence structure possessed by multiscale models and demonstrates that such an analysis provides important insight into the multiscale stochastic realization problem. Multiscale models constitute a broad class of probabilistic models which includes the well--known subclass of multiscale autoregressive (MAR) models. MAR models have proven useful in a variety of different application areas, due to the fact that they provide a rich set of tools for various signal processing tasks. In order to use these tools, however, a MAR or multiscale model must first be constructed to provide an accurate probabilistic description of the particular application at hand. This thesis addresses this issue of multiscale model identification or realization. Previous work in the area of MAR model identification has focused on developing algorithms which decorrelate certain subsets of random vectors in an effort to design an accurate model. In this thesis, we develop a set-theoretic and graph-theoretic framework for better understanding these types of realization algorithms and for the purpose of designing new such algorithms.(cont.) The benefit of the framework developed here is that it separates the realization problem into two understandable parts - a dichotomy which helps to clarify the relationship between the exact realization problem, where a multiscale model is designed to exactly satisfy a probabilistic constraint, and the approximate realization problem, where the constraint is only approximately satisfied. The first part of our study focuses on developing a better understanding of the independence structure exhibited by multiscale models. As a result of this study, we are able to suggest a number of different sequential procedures for realizing exact multiscale models. The second part of our study focuses on approximate realization, where we define a relaxed version of the exact multiscale realization problem. We show that many of the ideas developed for the exact realization problem may be used to better understand the approximate realization problem and to develop algorithms for solving it. In particular, we propose an iterative procedure for solving the approximate realization problem, and we show that the parameterized version of this procedure is equivalent to the well-known EM algorithm. Finally, a specific algorithm is developed for realizing a multiscale model which matches the statistics of a Gaussian random process.by Dewey S. Tucker.Ph.D