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These lecture notes study the interplay between randomness and geometry of graphs. The first part of the notes reviews several basic geometric concepts, before moving on to examine the manifestation of the underlying geometry in the behavior of random processes, mostly percolation and random walk. The study of the geometry of infinite vertex transitive graphs, and of Cayley graphs in particular, is fairly well developed. One goal of these notes is to point to some random metric spaces modeled by graphs that turn out to be somewhat exotic, that is, they admit a combination of properties not encountered in the vertex transitive world. These include percolation clusters on vertex transitive graphs, critical clusters, local and scaling limits of graphs, long range percolation, CCCP graphs obtained by contracting percolation clusters on graphs, and stationary random graphs, including the uniform infinite planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).
Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematics. --- Geometry. --- Probabilities. --- Graph theory. --- Physics. --- Statistics. --- Continuum mechanics. --- Probability Theory and Stochastic Processes. --- Mathematical Methods in Physics. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Continuum Mechanics and Mechanics of Materials. --- Graph Theory. --- Stochastic processes. --- Geometric probabilities. --- Probabilities --- Random processes --- Distribution (Probability theory. --- Mathematical physics. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Physical mathematics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Euclid's Elements --- Statistics . --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Extremal problems
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“Particle Filters for Random Set Models” presents coverage of state estimation of stochastic dynamic systems from noisy measurements, specifically sequential Bayesian estimation and nonlinear or stochastic filtering. The class of solutions presented in this book is based on the Monte Carlo statistical method. The resulting algorithms, known as particle filters, in the last decade have become one of the essential tools for stochastic filtering, with applications ranging from navigation and autonomous vehicles to bio-informatics and finance. While particle filters have been around for more than a decade, the recent theoretical developments of sequential Bayesian estimation in the framework of random set theory have provided new opportunities which are not widely known and are covered in this book. These recent developments have dramatically widened the scope of applications, from single to multiple appearing/disappearing objects, from precise to imprecise measurements and measurement models. This book is ideal for graduate students, researchers, scientists and engineers interested in Bayesian estimation.
Estimation theory. --- Random sets. --- Stochastic processes. --- Random sets --- Stochastic processes --- Estimation theory --- Electrical & Computer Engineering --- Engineering & Applied Sciences --- Applied Physics --- Telecommunications --- Electrical Engineering --- Estimating techniques --- Random processes --- Engineering. --- Mathematical statistics. --- Artificial intelligence. --- Information theory. --- Computational intelligence. --- Signal, Image and Speech Processing. --- Information and Communication, Circuits. --- Probability and Statistics in Computer Science. --- Artificial Intelligence (incl. Robotics). --- Computational Intelligence. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Communication theory --- Communication --- Cybernetics --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Construction --- Industrial arts --- Technology --- Statistical methods --- Least squares --- Mathematical statistics --- Geometric probabilities --- Set theory --- Mathematics. --- Computer science. --- Artificial Intelligence. --- Informatics --- Science --- Math --- Signal processing. --- Image processing. --- Speech processing systems. --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication)
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