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Random set theory is a fascinating branch of mathematics that amalgamates techniques from topology, convex geometry, and probability theory. Social scientists routinely conduct empirical work with data and modelling assumptions that reveal a set to which the parameter of interest belongs, but not its exact value. Random set theory provides a coherent mathematical framework to conduct identification analysis and statistical inference in this setting and has become a fundamental tool in econometrics and finance. This is the first book dedicated to the use of the theory in econometrics, written to be accessible for readers without a background in pure mathematics. Molchanov and Molinari define the basics of the theory and illustrate the mathematical concepts by their application in the analysis of econometric models. The book includes sets of exercises to accompany each chapter as well as examples to help readers apply the theory effectively.

Econometrics. --- Random sets. --- Geometric probabilities --- Set theory --- Economics, Mathematical --- Statistics --- Econometrics --- Random sets

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Algebra --- Functional analysis --- Hilbert space. --- Random sets. --- Invariants. --- Calculus of tensors. --- Espace de Hilbert --- Ensembles aléatoires --- Analyse multidimensionnelle --- Calcul tensoriel --- 51 <082.1> --- Mathematics--Series --- Hilbert, Espaces de --- Invariants --- Ensembles aléatoires --- Calculus of tensors --- Hilbert space --- Random sets --- Geometric probabilities --- Set theory --- Banach spaces --- Hyperspace --- Inner product spaces --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis --- Hilbert, Espaces de. --- Ensembles aléatoires. --- Calcul tensoriel.

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This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.

Probabilities. --- Computer mathematics. --- Geometry. --- Mathematical models. --- Applied mathematics. --- Engineering mathematics. --- Probability Theory and Stochastic Processes. --- Computational Science and Engineering. --- Mathematical Modeling and Industrial Mathematics. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Models, Mathematical --- Simulation methods --- Mathematics --- Euclid's Elements --- Computer mathematics --- Electronic data processing --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Random sets. --- Geometric probabilities --- Set theory --- Models matemàtics --- Models (Matemàtica) --- Models experimentals --- Models teòrics --- Mètodes de simulació --- Anàlisi de sistemes --- Mètode de Montecarlo --- Modelització multiescala --- Models economètrics --- Models lineals (Estadística) --- Models multinivell (Estadística) --- Models no lineals (Estadística) --- Programació (Ordinadors) --- Simulació per ordinador --- Teoria de màquines --- Models biològics

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Stochastic geometry is a relatively new branch of mathematics. Although its predecessors such as geometric probability date back to the 18th century, the formal concept of a random set was developed in the beginning of the 1970s. Theory of Random Sets presents a state of the art treatment of the modern theory, but it does not neglect to recall and build on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference of the 1990s. The book is entirely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time, fixes terminology and notation that are often varying in the current literature to establish it as a natural part of modern probability theory, and to provide a platform for future development. An extensive, searchable bibliography to accompany the book is freely available via the web. The book will be an invaluable reference for probabilists, mathematicians in convex and integral geometry, set-valued analysis, capacity and potential theory, mathematical statisticians in spatial statistics and image analysis, specialists in mathematical economics, and electronic and electrical engineers interested in image analysis.

Random sets. --- Set theory. --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Logic, Symbolic and mathematical --- Mathematics --- Geometric probabilities --- Set theory --- Distribution (Probability theory. --- Mathematics. --- Statistics. --- Computer engineering. --- Economic theory. --- Probability Theory and Stochastic Processes. --- Game Theory, Economics, Social and Behav. Sciences. --- Theoretical, Mathematical and Computational Physics. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Electrical Engineering. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Economic theory --- Political economy --- Social sciences --- Economic man --- Computers --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Design and construction --- Probabilities. --- Game theory. --- Mathematical physics. --- Statistics . --- Electrical engineering. --- Electric engineering --- Engineering --- Physical mathematics --- Physics --- Games, Theory of --- Theory of games --- Mathematical models --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Simultaneous Localisation and Map (SLAM) building algorithms, which rely on random vectors to represent sensor measurements and feature maps are known to be extremely fragile in the presence of feature detection and data association uncertainty. Therefore new concepts for autonomous map representations are given in this book, based on random finite sets (RFSs). It will be shown that the RFS representation eliminates the necessity of fragile data association and map management routines. It fundamentally differs from vector based approaches since it estimates not only the spatial states of features but also the number of map features which have passed through the field(s) of view of a robot's sensor(s), an attribute which is necessary for SLAM. The book also demonstrates that in SLAM, a valid measure of map estimation error is critical. It will be shown that under an RFS-SLAM representation, a consistent metric, which gauges both feature number as well as spatial errors, can be defined. The concepts of RFS map representations are accompanied with autonomous SLAM experiments in urban and marine environments. Comparisons of RFS-SLAM with state of the art vector based methods are given, along with pseudo-code implementations of all the RFS techniques presented. John Mullane received the B.E.E. degree from University College Cork, Ireland, and Ph.D degree from Nanyang Technological University (NTU), Singapore. Ba-Ngu Vo is Winthrop Professor and Chair of Signal Processing, University of Western Australia (UWA). He received joint Bachelor degrees (Science and Elec. Eng.), UWA, and Ph.D., Curtin University. Martin Adams is Professor in autonomous robotics research, University of Chile. He holds bachelors, masters and doctoral degrees from Oxford University. Ba-Tuong Vo is Assistant Professor, UWA. He received his B.Sc, B.E and Ph.D. degrees from UWA.

Robotics --- Mobile robots --- Mappings (Mathematics) --- Mechanical Engineering --- Engineering & Applied Sciences --- Mechanical Engineering - General --- Finite groups. --- Random sets. --- SLAM (Computer program language) --- Robots --- Mathematics. --- Control systems. --- Simulation Language for Alternative Modeling (Computer program language) --- Groups, Finite --- Robot control --- Engineering. --- Artificial intelligence. --- Robotics. --- Automation. --- Robotics and Automation. --- Artificial Intelligence (incl. Robotics). --- Automatic factories --- Automatic production --- Computer control --- Engineering cybernetics --- Factories --- Industrial engineering --- Mechanization --- Assembly-line methods --- Automatic control --- Automatic machinery --- CAD/CAM systems --- Automation --- Machine theory --- 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 --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Construction --- Industrial arts --- Technology --- FORTRAN (Computer program language) --- Modeling languages (Computer science) --- Geometric probabilities --- Set theory --- Group theory --- Modules (Algebra) --- Artificial Intelligence.