Topic: Multiresolution Modeling from Data and Partial Specifications
By: Dewey Tucker (MIT), Alan S. Willsky (MIT)
In this research, we study a recently developed class of models, called multiscale models, which is well-suited to represent a wide variety of random processes. The advantage of this type of model is in providing an extremely efficient estimation algorithm which is a generalization of the traditional Kalman filter and Rauch-Tung-Striebel smoother. Multiscale models and the associated estimation algorithm have been shown to be useful in a number of applications including image processing, remote sensing, and geophysics. In particular, we focus on the problem of constructing multiscale models from data and partial covariance information. Previous research in this area has provided an algorithm which builds a multiscale model to match the covariance of a given process of interest. However, this approach requires complete knowledge of the covariance and the capability to store it, and for problems of even moderate size, this type of complete characterization is an overwhelmingly large data storage problem. To circumvent this problem, we seek methods to construct multiscale models based solely on realizations (sample paths) of the process, with no assumed knowledge of the covariance. In addition, we examine the problem of positive definite covariance extension with the specific goal of constructing multiscale models from partial covariance information.
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