Multiscale representations of Markov random fields

Includes bibliographical references (p. 54-58).Supported by the Draper Laboratory IR&D Program. DL-H-418524 Supported by the Office of Naval Research. N00014-91-J-1004 Supported by the Army Research Office of Research. DAAL03-92-G-0115 Supported by the Air Force Office of Scientific Research. F49620-91-C-0047 AFOSR-92-J-0002by Mark R. Luettgen ... [et al.]

Extrapolation of Stationary Random Fields

We introduce basic statistical methods for the extrapolation of stationary random fields. For square integrable fields, we set out basics of the kriging extrapolation techniques. For (non--Gaussian) stable fields, which are known to be heavy tailed, we describe further extrapolation methods and discuss their properties. Two of them can be seen as direct generalizations of kriging.Comment: 52 pages, 25 figures. This is a review article, though Section 4 of the article contains new results on the weak consistency of the extrapolation methods as well as new extrapolation methods for $\alpha$-stable fields with $0<\alpha\leq 1

Fractals in Geoscience and Remote Sensing

Abstract not availableNA-NOT AVAILABL

A statistical mechanical model of economics

Statistical mechanics pursues low-dimensional descriptions of systems with a very large number of degrees of freedom. I explore this theme in two contexts. The main body of this dissertation explores and extends the Yard Sale Model (YSM) of economic transactions using a combination of simulations and theory. The YSM is a simple interacting model for wealth distributions which has the potential to explain the empirical observation of Pareto distributions of wealth. I develop the link between wealth condensation and the breakdown of ergodicity due to nonlinear diffusion effects which are analogous to the geometric random walk. Using this, I develop a deterministic effective theory of wealth transfer in the YSM that is useful for explaining many quantitative results. I introduce various forms of growth to the model, paying attention to the effect of growth on wealth condensation, inequality, and ergodicity. Arithmetic growth is found to partially break condensation, and geometric growth is found to completely break condensation. Further generalizations of geometric growth with growth in- equality show that the system is divided into two phases by a tipping point in the inequality parameter. The tipping point marks the line between systems which are ergodic and systems which exhibit wealth condensation. I explore generalizations of the YSM transaction scheme to arbitrary betting functions to develop notions of universality in YSM-like models. I find that wealth condensation is universal to a large class of models which can be divided into two phases. The first exhibits slow, power-law condensation dynamics, and the second exhibits fast, finite-time condensation dynamics. I find that the YSM, which exhibits exponential dynamics, is the critical, self-similar model which marks the dividing line between the two phases. The final chapter develops a low-dimensional approach to materials microstructure quantification. Modern materials design harnesses complex microstructure effects to develop high-performance materials, but general microstructure quantification is an unsolved problem. Motivated by statistical physics, I envision microstructure as a low-dimensional manifold, and construct this manifold by leveraging multiple machine learning approaches including transfer learning, dimensionality reduction, and computer vision breakthroughs with convolutional neural networks

Supervised and unsupervised segmentation of textured images by efficient multi-level pattern classification

This thesis proposes new, efficient methodologies for supervised and unsupervised image segmentation based on texture information. For the supervised case, a technique for pixel classification based on a multi-level strategy that iteratively refines the resulting segmentation is proposed. This strategy utilizes pattern recognition methods based on prototypes (determined by clustering algorithms) and support vector machines. In order to obtain the best performance, an algorithm for automatic parameter selection and methods to reduce the computational cost associated with the segmentation process are also included. For the unsupervised case, the previous methodology is adapted by means of an initial pattern discovery stage, which allows transforming the original unsupervised problem into a supervised one. Several sets of experiments considering a wide variety of images are carried out in order to validate the developed techniques.Esta tesis propone metodologías nuevas y eficientes para segmentar imágenes a partir de información de textura en entornos supervisados y no supervisados. Para el caso supervisado, se propone una técnica basada en una estrategia de clasificación de píxeles multinivel que refina la segmentación resultante de forma iterativa. Dicha estrategia utiliza métodos de reconocimiento de patrones basados en prototipos (determinados mediante algoritmos de agrupamiento) y máquinas de vectores de soporte. Con el objetivo de obtener el mejor rendimiento, se incluyen además un algoritmo para selección automática de parámetros y métodos para reducir el coste computacional asociado al proceso de segmentación. Para el caso no supervisado, se propone una adaptación de la metodología anterior mediante una etapa inicial de descubrimiento de patrones que permite transformar el problema no supervisado en supervisado. Las técnicas desarrolladas en esta tesis se validan mediante diversos experimentos considerando una gran variedad de imágenes