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6.262 Discrete Stochastic Processes (MIT) 6.262 Discrete Stochastic Processes (MIT)

Description

Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance. Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Subjects

probability | probability | Poisson processes | Poisson processes | finite-state Markov chains | finite-state Markov chains | renewal processes | renewal processes | countable-state Markov chains | countable-state Markov chains | Markov processes | Markov processes | countable state spaces | countable state spaces | random walks | random walks | large deviations | large deviations | martingales | martingales

License

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16.410 Principles of Autonomy and Decision Making (MIT) 16.410 Principles of Autonomy and Decision Making (MIT)

Description

This course surveys a variety of reasoning, optimization, and decision-making methodologies for creating highly autonomous systems and decision support aids. The focus is on principles, algorithms, and their applications, taken from the disciplines of artificial intelligence and operations research. Reasoning paradigms include logic and deduction, heuristic and constraint-based search, model-based reasoning, planning and execution, reasoning under uncertainty, and machine learning. Optimization paradigms include linear, integer and dynamic programming. Decision-making paradigms include decision theoretic planning, and Markov decision processes. This course is offered both to undergraduate (16.410) students as a professional area undergraduate subject, in the field of aerospace information This course surveys a variety of reasoning, optimization, and decision-making methodologies for creating highly autonomous systems and decision support aids. The focus is on principles, algorithms, and their applications, taken from the disciplines of artificial intelligence and operations research. Reasoning paradigms include logic and deduction, heuristic and constraint-based search, model-based reasoning, planning and execution, reasoning under uncertainty, and machine learning. Optimization paradigms include linear, integer and dynamic programming. Decision-making paradigms include decision theoretic planning, and Markov decision processes. This course is offered both to undergraduate (16.410) students as a professional area undergraduate subject, in the field of aerospace information

Subjects

autonomy | autonomy | decision | decision | decision-making | decision-making | reasoning | reasoning | optimization | optimization | autonomous | autonomous | autonomous systems | autonomous systems | decision support | decision support | algorithms | algorithms | artificial intelligence | artificial intelligence | a.i. | a.i. | operations | operations | operations research | operations research | logic | logic | deduction | deduction | heuristic search | heuristic search | constraint-based search | constraint-based search | model-based reasoning | model-based reasoning | planning | planning | execution | execution | uncertainty | uncertainty | machine learning | machine learning | linear programming | linear programming | dynamic programming | dynamic programming | integer programming | integer programming | network optimization | network optimization | decision analysis | decision analysis | decision theoretic planning | decision theoretic planning | Markov decision process | Markov decision process | scheme | scheme | propositional logic | propositional logic | constraints | constraints | Markov processes | Markov processes | computational performance | computational performance | satisfaction | satisfaction | learning algorithms | learning algorithms | system state | system state | state | state | search treees | search treees | plan spaces | plan spaces | model theory | model theory | decision trees | decision trees | function approximators | function approximators | optimization algorithms | optimization algorithms | limitations | limitations | tradeoffs | tradeoffs | search and reasoning | search and reasoning | game tree search | game tree search | local stochastic search | local stochastic search | stochastic | stochastic | genetic algorithms | genetic algorithms | constraint satisfaction | constraint satisfaction | propositional inference | propositional inference | rule-based systems | rule-based systems | rule-based | rule-based | model-based diagnosis | model-based diagnosis | neural nets | neural nets | reinforcement learning | reinforcement learning | web-based | web-based | search trees | search trees

License

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6.231 Dynamic Programming and Stochastic Control (MIT) 6.231 Dynamic Programming and Stochastic Control (MIT)

Description

This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed. This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed.

Subjects

dynamic programming | dynamic programming | | stochastic control | | stochastic control | | mathematics | optimization | | | mathematics | optimization | | algorithms | | algorithms | | probability | | probability | | Markov chains | | Markov chains | | optimal control | optimal control | stochastic control | stochastic control | mathematics | mathematics | optimization | optimization | algorithms | algorithms | probability | probability | Markov chains | Markov chains

License

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16.410 Principles of Autonomy and Decision Making (MIT) 16.410 Principles of Autonomy and Decision Making (MIT)

Description

This course surveys a variety of reasoning, optimization, and decision-making methodologies for creating highly autonomous systems and decision support aids. The focus is on principles, algorithms, and their applications, taken from the disciplines of artificial intelligence and operations research. Reasoning paradigms include logic and deduction, heuristic and constraint-based search, model-based reasoning, planning and execution, reasoning under uncertainty, and machine learning. Optimization paradigms include linear, integer and dynamic programming. Decision-making paradigms include decision theoretic planning, and Markov decision processes. This course is offered both to undergraduate (16.410) students as a professional area undergraduate subject, in the field of aerospace information This course surveys a variety of reasoning, optimization, and decision-making methodologies for creating highly autonomous systems and decision support aids. The focus is on principles, algorithms, and their applications, taken from the disciplines of artificial intelligence and operations research. Reasoning paradigms include logic and deduction, heuristic and constraint-based search, model-based reasoning, planning and execution, reasoning under uncertainty, and machine learning. Optimization paradigms include linear, integer and dynamic programming. Decision-making paradigms include decision theoretic planning, and Markov decision processes. This course is offered both to undergraduate (16.410) students as a professional area undergraduate subject, in the field of aerospace information

Subjects

autonomy | autonomy | decision | decision | decision-making | decision-making | reasoning | reasoning | optimization | optimization | autonomous | autonomous | autonomous systems | autonomous systems | decision support | decision support | algorithms | algorithms | artificial intelligence | artificial intelligence | a.i. | a.i. | operations | operations | operations research | operations research | logic | logic | deduction | deduction | heuristic search | heuristic search | constraint-based search | constraint-based search | model-based reasoning | model-based reasoning | planning | planning | execution | execution | uncertainty | uncertainty | machine learning | machine learning | linear programming | linear programming | dynamic programming | dynamic programming | integer programming | integer programming | network optimization | network optimization | decision analysis | decision analysis | decision theoretic planning | decision theoretic planning | Markov decision process | Markov decision process | scheme | scheme | propositional logic | propositional logic | constraints | constraints | Markov processes | Markov processes | computational performance | computational performance | satisfaction | satisfaction | learning algorithms | learning algorithms | system state | system state | state | state | search treees | search treees | plan spaces | plan spaces | model theory | model theory | decision trees | decision trees | function approximators | function approximators | optimization algorithms | optimization algorithms | limitations | limitations | tradeoffs | tradeoffs | search and reasoning | search and reasoning | game tree search | game tree search | local stochastic search | local stochastic search | stochastic | stochastic | genetic algorithms | genetic algorithms | constraint satisfaction | constraint satisfaction | propositional inference | propositional inference | rule-based systems | rule-based systems | rule-based | rule-based | model-based diagnosis | model-based diagnosis | neural nets | neural nets | reinforcement learning | reinforcement learning | web-based | web-based | search trees | search trees

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.867 Machine Learning (MIT) 6.867 Machine Learning (MIT)

Description

6.867 is an introductory course on machine learning which provides an overview of many techniques and algorithms in machine learning, beginning with topics such as simple perceptrons and ending up with more recent topics such as boosting, support vector machines, hidden Markov models, and Bayesian networks. The course gives the student the basic ideas and intuition behind modern machine learning methods as well as a bit more formal understanding of how and why they work. The underlying theme in the course is statistical inference as this provides the foundation for most of the methods covered.  6.867 is an introductory course on machine learning which provides an overview of many techniques and algorithms in machine learning, beginning with topics such as simple perceptrons and ending up with more recent topics such as boosting, support vector machines, hidden Markov models, and Bayesian networks. The course gives the student the basic ideas and intuition behind modern machine learning methods as well as a bit more formal understanding of how and why they work. The underlying theme in the course is statistical inference as this provides the foundation for most of the methods covered. 

Subjects

machine learning | machine learning | perceptrons | perceptrons | boosting | boosting | support vector machines | support vector machines | Markov | Markov | hidden Markov models | hidden Markov models | HMM | HMM | Bayesian networks | Bayesian networks | statistical inference | statistical inference | regression | regression | clustering | clustering | bias | bias | variance | variance | regularization | regularization | Generalized Linear Models | Generalized Linear Models | neural networks | neural networks | Support Vector Machine | Support Vector Machine | SVM | SVM | mixture models | mixture models | kernel density estimation | kernel density estimation | gradient descent | gradient descent | quadratic programming | quadratic programming | EM algorithm | EM algorithm | orward-backward algorithm | orward-backward algorithm | junction tree algorithm | junction tree algorithm | Gibbs sampling | Gibbs sampling

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.262 Discrete Stochastic Processes (MIT)

Description

Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Subjects

probability | Poisson processes | finite-state Markov chains | renewal processes | countable-state Markov chains | Markov processes | countable state spaces | random walks | large deviations | martingales

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc. Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations

License

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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference. This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference.

Subjects

probabilistic systems | probabilistic systems | probabilistic systems analysis | probabilistic systems analysis | applied probability | applied probability | uncertainty | uncertainty | uncertainty modeling | uncertainty modeling | uncertainty quantification | uncertainty quantification | analysis of uncertainty | analysis of uncertainty | uncertainty analysis | uncertainty analysis | sample space | sample space | random variables | random variables | transform techniques | transform techniques | simple random processes | simple random processes | probability distribution | probability distribution | Markov process | Markov process | limit theorem | limit theorem | statistical inference | statistical inference

License

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14.381 Statistical Method in Economics (MIT) 14.381 Statistical Method in Economics (MIT)

Description

This course is divided into two sections, Part I and Part II.  Part I provides an introduction to statistical theory and can be found by visiting 14.381 Fall 2013. Part II, found here, prepares students for the remainder of the econometrics sequence. The emphasis of the course is to understand the basic principles of statistical theory. A brief review of probability will be given; however, this material is assumed knowledge. The course also covers basic regression analysis. Topics covered include probability, random samples, asymptotic methods, point estimation, evaluation of estimators, Cramer-Rao theorem, hypothesis tests, Neyman Pearson lemma, Likelihood Ratio test, interval estimation, best linear predictor, best linear approximation, conditional expectation function, buil This course is divided into two sections, Part I and Part II.  Part I provides an introduction to statistical theory and can be found by visiting 14.381 Fall 2013. Part II, found here, prepares students for the remainder of the econometrics sequence. The emphasis of the course is to understand the basic principles of statistical theory. A brief review of probability will be given; however, this material is assumed knowledge. The course also covers basic regression analysis. Topics covered include probability, random samples, asymptotic methods, point estimation, evaluation of estimators, Cramer-Rao theorem, hypothesis tests, Neyman Pearson lemma, Likelihood Ratio test, interval estimation, best linear predictor, best linear approximation, conditional expectation function, buil

Subjects

statistical theory | statistical theory | econometrics | econometrics | regression analysis | regression analysis | probability | probability | random samples | random samples | asymptotic methods | asymptotic methods | point estimation | point estimation | evaluation of estimators | evaluation of estimators | Cramer-Rao theorem | Cramer-Rao theorem | hypothesis tests | hypothesis tests | Neyman Pearson lemma | Neyman Pearson lemma | Likelihood Ratio test | Likelihood Ratio test | interval estimation | interval estimation | best linear predictor | best linear predictor | best linear approximation | best linear approximation | conditional expectation function | conditional expectation function | building functional forms | building functional forms | regression algebra | regression algebra | Gauss-Markov optimality | Gauss-Markov optimality | finite-sample inference | finite-sample inference | consistency | consistency | asymptotic normality | asymptotic normality | heteroscedasticity | heteroscedasticity | autocorrelation | autocorrelation

License

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6.438 Algorithms for Inference (MIT) 6.438 Algorithms for Inference (MIT)

Description

This is a graduate-level introduction to the principles of statistical inference with probabilistic models defined using graphical representations. The material in this course constitutes a common foundation for work in machine learning, signal processing, artificial intelligence, computer vision, control, and communication. Ultimately, the subject is about teaching you contemporary approaches to, and perspectives on, problems of statistical inference. This is a graduate-level introduction to the principles of statistical inference with probabilistic models defined using graphical representations. The material in this course constitutes a common foundation for work in machine learning, signal processing, artificial intelligence, computer vision, control, and communication. Ultimately, the subject is about teaching you contemporary approaches to, and perspectives on, problems of statistical inference.

Subjects

inference | inference | algorithm | algorithm | graphical model | graphical model | factor graph | factor graph | markov chain | markov chain | Gaussian model | Gaussian model | loopy belief propagation | loopy belief propagation | EM algorithm | EM algorithm | statistical inference | statistical inference | probabilistic graphical model | probabilistic graphical model | Hidden Markov model | Hidden Markov model | linear dynamical systems | linear dynamical systems | Sum-product algorithm | Sum-product algorithm | junction tree algorithm | junction tree algorithm | Forward-backward algorithm | Forward-backward algorithm | Kalman filtering | Kalman filtering | smoothing | smoothing | Variational method | Variational method | mean-field theory | mean-field theory | Min-sum algorithm | Min-sum algorithm | Viterbi algorithm | Viterbi algorithm | parameter estimation | parameter estimation | learning structure | learning structure

License

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations | linear algebra | linear algebra | determinants | determinants | eigenvalues | eigenvalues | similarity | similarity | least-squares approximations | least-squares approximations | stability of differential equations | stability of differential equations | networks | networks

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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14.382 Econometrics I (MIT) 14.382 Econometrics I (MIT)

Description

This course focuses on the specification and estimation of the linear regression model. The course departs from the standard Gauss-Markov assumptions to include heteroskedasticity, serial correlation, and errors in variables. Advanced topics include generalized least squares, instrumental variables, nonlinear regression, and limited dependent variable models. Economic applications are discussed throughout the course. This course focuses on the specification and estimation of the linear regression model. The course departs from the standard Gauss-Markov assumptions to include heteroskedasticity, serial correlation, and errors in variables. Advanced topics include generalized least squares, instrumental variables, nonlinear regression, and limited dependent variable models. Economic applications are discussed throughout the course.

Subjects

Economics | Economics | econometrics | econometrics | linear regression model | linear regression model | Gauss-Markov | Gauss-Markov | heteroskedasticity | heteroskedasticity | serial correlation | serial correlation | errors | errors | variables | variables | generalized least squares | generalized least squares | instrumental variables | instrumental variables | nonlinear regression | nonlinear regression | limited dependent variable models | limited dependent variable models

License

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6.231 Dynamic Programming and Stochastic Control (MIT)

Description

This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed.

Subjects

dynamic programming | | stochastic control | | mathematics | optimization | | algorithms | | probability | | Markov chains | | optimal control | stochastic control | mathematics | optimization | algorithms | probability | Markov chains

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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16.410 Principles of Autonomy and Decision Making (MIT)

Description

This course surveys a variety of reasoning, optimization, and decision-making methodologies for creating highly autonomous systems and decision support aids. The focus is on principles, algorithms, and their applications, taken from the disciplines of artificial intelligence and operations research. Reasoning paradigms include logic and deduction, heuristic and constraint-based search, model-based reasoning, planning and execution, reasoning under uncertainty, and machine learning. Optimization paradigms include linear, integer and dynamic programming. Decision-making paradigms include decision theoretic planning, and Markov decision processes. This course is offered both to undergraduate (16.410) students as a professional area undergraduate subject, in the field of aerospace information

Subjects

autonomy | decision | decision-making | reasoning | optimization | autonomous | autonomous systems | decision support | algorithms | artificial intelligence | a.i. | operations | operations research | logic | deduction | heuristic search | constraint-based search | model-based reasoning | planning | execution | uncertainty | machine learning | linear programming | dynamic programming | integer programming | network optimization | decision analysis | decision theoretic planning | Markov decision process | scheme | propositional logic | constraints | Markov processes | computational performance | satisfaction | learning algorithms | system state | state | search treees | plan spaces | model theory | decision trees | function approximators | optimization algorithms | limitations | tradeoffs | search and reasoning | game tree search | local stochastic search | stochastic | genetic algorithms | constraint satisfaction | propositional inference | rule-based systems | rule-based | model-based diagnosis | neural nets | reinforcement learning | web-based | search trees

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.445 Introduction to Stochastic Processes (MIT) 18.445 Introduction to Stochastic Processes (MIT)

Description

This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix. This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.

Subjects

probability | probability | Stochastic Processes | Stochastic Processes | Markov chains | Markov chains | random walks | random walks | martingales | martingales | Galton-Watsom tree | Galton-Watsom tree | linear algebra | linear algebra

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.867 Machine Learning (MIT)

Description

6.867 is an introductory course on machine learning which provides an overview of many techniques and algorithms in machine learning, beginning with topics such as simple perceptrons and ending up with more recent topics such as boosting, support vector machines, hidden Markov models, and Bayesian networks. The course gives the student the basic ideas and intuition behind modern machine learning methods as well as a bit more formal understanding of how and why they work. The underlying theme in the course is statistical inference as this provides the foundation for most of the methods covered. 

Subjects

machine learning | perceptrons | boosting | support vector machines | Markov | hidden Markov models | HMM | Bayesian networks | statistical inference | regression | clustering | bias | variance | regularization | Generalized Linear Models | neural networks | Support Vector Machine | SVM | mixture models | kernel density estimation | gradient descent | quadratic programming | EM algorithm | orward-backward algorithm | junction tree algorithm | Gibbs sampling

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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6.436J Fundamentals of Probability (MIT) 6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations. This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations.

Subjects

sample space | sample space | random variables | random variables | expectations | expectations | transforms | transforms | Bernoulli process | Bernoulli process | Poisson process | Poisson process | Markov chains | Markov chains | limit theorems | limit theorems | measure theory | measure theory

License

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6.895 Computational Biology: Genomes, Networks, Evolution (MIT) 6.895 Computational Biology: Genomes, Networks, Evolution (MIT)

Description

This course focuses on the algorithmic and machine learning foundations of computational biology, combining theory with practice. We study the principles of algorithm design for biological datasets, and analyze influential problems and techniques. We use these to analyze real datasets from large-scale studies in genomics and proteomics. The topics covered include:Genomes: Biological Sequence Analysis, Hidden Markov Models, Gene Finding, RNA Folding, Sequence Alignment, Genome Assembly.Networks: Gene Expression Analysis, Regulatory Motifs, Graph Algorithms, Scale-free Networks, Network Motifs, Network Evolution.Evolution: Comparative Genomics, Phylogenetics, Genome Duplication, Genome Rearrangements, Evolutionary Theory, Rapid Evolution. This course focuses on the algorithmic and machine learning foundations of computational biology, combining theory with practice. We study the principles of algorithm design for biological datasets, and analyze influential problems and techniques. We use these to analyze real datasets from large-scale studies in genomics and proteomics. The topics covered include:Genomes: Biological Sequence Analysis, Hidden Markov Models, Gene Finding, RNA Folding, Sequence Alignment, Genome Assembly.Networks: Gene Expression Analysis, Regulatory Motifs, Graph Algorithms, Scale-free Networks, Network Motifs, Network Evolution.Evolution: Comparative Genomics, Phylogenetics, Genome Duplication, Genome Rearrangements, Evolutionary Theory, Rapid Evolution.

Subjects

Genomes: Biological sequence analysis | Genomes: Biological sequence analysis | hidden Markov models | hidden Markov models | gene finding | gene finding | RNA folding | RNA folding | sequence alignment | sequence alignment | genome assembly | genome assembly | Networks: Gene expression analysis | Networks: Gene expression analysis | regulatory motifs | regulatory motifs | graph algorithms | graph algorithms | scale-free networks | scale-free networks | network motifs | network motifs | network evolution | network evolution | Evolution: Comparative genomics | Evolution: Comparative genomics | phylogenetics | phylogenetics | genome duplication | genome duplication | genome rearrangements | genome rearrangements | evolutionary theory | evolutionary theory | rapid evolution | rapid evolution

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Técnicas de Investigación Operativa en Ingeniería Técnicas de Investigación Operativa en Ingeniería

Description

El curso trata de mostrar una visión amplia de las técnicas que se recogen bajo la denominación de Investigación Operativa (IO), agrupándolas en tres grandes bloques (métodos deterministas, métodos probabilísticos y otros métodos como simulación, toma de decisiones, participación pública). Dentro de cada bloque se expone brevemente en qué consiste cada método y se acompaña con ejemplos y ejercicios de aplicación. El curso trata de mostrar una visión amplia de las técnicas que se recogen bajo la denominación de Investigación Operativa (IO), agrupándolas en tres grandes bloques (métodos deterministas, métodos probabilísticos y otros métodos como simulación, toma de decisiones, participación pública). Dentro de cada bloque se expone brevemente en qué consiste cada método y se acompaña con ejemplos y ejercicios de aplicación.

Subjects

Precio sombra | Precio sombra | Programación entera | Programación entera | Algoritmos genéticos | Algoritmos genéticos | Grafos | Grafos | Procesos aleatorios | Procesos aleatorios | Subgrafo | Subgrafo | Métodos heurísticos | Métodos heurísticos | Matriz estocástica | Matriz estocástica | Función objetivo | Función objetivo | Recurrencia | Recurrencia | Rango de optimalidad | Rango de optimalidad | Problema de rutas | Problema de rutas | PERT | PERT | Información | Información | Restricciones | Restricciones | Simulación | Simulación | Comunidades de opinión | Comunidades de opinión | Método Promethee | Método Promethee | Función de energía | Función de energía | Combinatoria | Combinatoria | Métodos de optimización | Métodos de optimización | Coste fijo | Coste fijo | Teorema de Arrow | Teorema de Arrow | Programación dinámica | Programación dinámica | Monte ideal | Monte ideal | Muestreador de Gibbs | Muestreador de Gibbs | Minimax | Minimax | Riesgo | Riesgo | Métodos de inteligencia artificial | Métodos de inteligencia artificial | Problema de recubrimiento | Problema de recubrimiento | Estacionariedad | Estacionariedad | Método Electre | Método Electre | Redes | Redes | Grafo árbol | Grafo árbol | Modelos deterministas | Modelos deterministas | Forestal | Forestal | Gestión | Gestión | Camino más corto | Camino más corto | Métodos determinísticos | Métodos determinísticos | Solución óptima | Solución óptima | Simplex | Simplex | Problema de la mochila | Problema de la mochila | Métodos estadísticos | Métodos estadísticos | Conocimiento | Conocimiento | Función de valor | Función de valor | Asignación | Asignación | Protoagentes | Protoagentes | Métodos meta-heurísticos | Métodos meta-heurísticos | Indicadores | Indicadores | Multigrafo | Multigrafo | Investigación de operaciones | Investigación de operaciones | Matriz de adyacencia | Matriz de adyacencia | Flujo | Flujo | Algoritmo de Floyd-Warshall | Algoritmo de Floyd-Warshall | Optimización | Optimización | Región factible | Región factible | Mercado de trabajo | Mercado de trabajo | Tabu-search | Tabu-search | Sistemas complejos | Sistemas complejos | Utilidad | Utilidad | Multiatributo | Multiatributo | Simulated annealing | Simulated annealing | Bucle | Bucle | Algoritmos enumerativos | Algoritmos enumerativos | Probabilidad de transición | Probabilidad de transición | Tasa de transición | Tasa de transición | Modelos de decisión | Modelos de decisión | Cadena homogénea | Cadena homogénea | Espacio de estados | Espacio de estados | Tiempo de vida | Tiempo de vida | Colas | Colas | Sostenibilidad | Sostenibilidad | Organización estratégica | Organización estratégica | Árbol de decisión | Árbol de decisión | Decisiones colectivas | Decisiones colectivas | Preferencias | Preferencias | Maximin | Maximin | Diagrama de estados | Diagrama de estados | Problema de la diligencia | Problema de la diligencia | Modelos probabilísticos | Modelos probabilísticos | Métodos de superclasificación | Métodos de superclasificación | Problema de asignación | Problema de asignación | Nodos | Nodos | Función de supervivencia | Función de supervivencia | Ecuaciones de Chapman-Kolmogorov | Ecuaciones de Chapman-Kolmogorov | Formulación de funciones | Formulación de funciones | Grafo completo | Grafo completo | Valoración de alternativas | Valoración de alternativas | Riesgo de la decisión | Riesgo de la decisión | Curva de la bañera | Curva de la bañera | Distribución exponencial | Distribución exponencial | Programación no lineal | Programación no lineal | Técnicas de optimización | Técnicas de optimización | Método GRASP | Método GRASP | Algoritmo de Dijskstra | Algoritmo de Dijskstra | Métodos monetarios | Métodos monetarios | Proceso de Poisson | Proceso de Poisson | Suma de variables aleatorias | Suma de variables aleatorias | Control de proyectos | Control de proyectos | Problema de emparejamiento | Problema de emparejamiento | Procesos estocásticos | Procesos estocásticos | Problema de partición | Problema de partición | Estructuración jerárquica | Estructuración jerárquica | Problema de empaquetado | Problema de empaquetado | Toma de decisiones | Toma de decisiones | Investigación operativa | Investigación operativa | Grafo etiquetado | Grafo etiquetado | Fiabilidad | Fiabilidad | Tasa de fallos | Tasa de fallos | Markov | Markov | Problema del viajante | Problema del viajante | Análisis de sensibilidad | Análisis de sensibilidad | Programación lineal | Programación lineal | Multicriterio | Multicriterio | Problema del transporte | Problema del transporte

License

Copyright 2009, by the Contributing Authors http://creativecommons.org/licenses/by-nc-sa/3.0/

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6.867 Machine Learning (MIT) 6.867 Machine Learning (MIT)

Description

6.867 is an introductory course on machine learning which gives an overview of many concepts, techniques, and algorithms in machine learning, beginning with topics such as classification and linear regression and ending up with more recent topics such as boosting, support vector machines, hidden Markov models, and Bayesian networks. The course will give the student the basic ideas and intuition behind modern machine learning methods as well as a bit more formal understanding of how, why, and when they work. The underlying theme in the course is statistical inference as it provides the foundation for most of the methods covered. 6.867 is an introductory course on machine learning which gives an overview of many concepts, techniques, and algorithms in machine learning, beginning with topics such as classification and linear regression and ending up with more recent topics such as boosting, support vector machines, hidden Markov models, and Bayesian networks. The course will give the student the basic ideas and intuition behind modern machine learning methods as well as a bit more formal understanding of how, why, and when they work. The underlying theme in the course is statistical inference as it provides the foundation for most of the methods covered.

Subjects

machine learning algorithms | machine learning algorithms | statistical inference | statistical inference | representation | representation | generalization | generalization | model selection | model selection | linear/additive models | linear/additive models | active learning | active learning | boosting | boosting | support vector machines | support vector machines | hidden Markov models | hidden Markov models | Bayesian networks | Bayesian networks | classification | classification | linear regression | linear regression | modern machine learning methods | modern machine learning methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference. This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference.

Subjects

probabilistic systems | probabilistic systems | probabilistic systems analysis | probabilistic systems analysis | applied probability | applied probability | uncertainty | uncertainty | uncertainty modeling | uncertainty modeling | uncertainty quantification | uncertainty quantification | analysis of uncertainty | analysis of uncertainty | uncertainty analysis | uncertainty analysis | sample space | sample space | random variables | random variables | transform techniques | transform techniques | simple random processes | simple random processes | probability distribution | probability distribution | Markov process | Markov process | limit theorem | limit theorem | statistical inference | statistical inference

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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1.010 Uncertainty in Engineering (MIT) 1.010 Uncertainty in Engineering (MIT)

Description

This course gives an introduction to probability and statistics, with emphasis on engineering applications. Course topics include events and their probability, the total probability and Bayes' theorems, discrete and continuous random variables and vectors, uncertainty propagation and conditional analysis. Second-moment representation of uncertainty, random sampling, estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation), and simple and multiple linear regression. Concepts illustrated with examples from various areas of engineering and everyday life. This course gives an introduction to probability and statistics, with emphasis on engineering applications. Course topics include events and their probability, the total probability and Bayes' theorems, discrete and continuous random variables and vectors, uncertainty propagation and conditional analysis. Second-moment representation of uncertainty, random sampling, estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation), and simple and multiple linear regression. Concepts illustrated with examples from various areas of engineering and everyday life.

Subjects

fundamentals of probability | fundamentals of probability | random processes | random processes | statistics | statistics | decision analysis | decision analysis | random variables and vectors | random variables and vectors | uncertainty propagation | uncertainty propagation | conditional distributions | conditional distributions | second-moment analysis | second-moment analysis | system reliability | system reliability | Bayes theorem | Bayes theorem | total probability theorem | total probability theorem | Bayesian analysis and risk-based decision | Bayesian analysis and risk-based decision | estimation of distribution parameters | estimation of distribution parameters | hypothesis testing | hypothesis testing | simple and multiple linear regressions | simple and multiple linear regressions | Poisson and Markov processes | Poisson and Markov processes

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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1.203J Logistical and Transportation Planning Methods (MIT) 1.203J Logistical and Transportation Planning Methods (MIT)

Description

The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics. The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics.

Subjects

1.203 | 1.203 | 6.281 | 6.281 | 15.073 | 15.073 | 16.76 | 16.76 | ESD.216 | ESD.216 | logistics | logistics | transportation | transportation | hypercube models | hypercube models | barrier example | barrier example | operations research | operations research | spatial queues | spatial queues | queueing models | queueing models | network models | network models | TSP | TSP | heuristics | heuristics | geometrical probabilities | geometrical probabilities | Markov | Markov | quantitative techniques | quantitative techniques | transportation systems analysis | transportation systems analysis | urban service systems | urban service systems | emergency services | emergency services | random variables | random variables | multi-server queueing theory | multi-server queueing theory | spatial location theory | spatial location theory | network analysis | network analysis | graph theory | graph theory | simulation | simulation | urban OR | urban OR

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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1.151 Probability and Statistics in Engineering (MIT) 1.151 Probability and Statistics in Engineering (MIT)

Description

This class covers quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, random processes, statistics, and decision analysis are covered, along with random variables and vectors, uncertainty propagation, conditional distributions, and second-moment analysis. System reliability is introduced. Other topics covered include Bayesian analysis and risk-based decision, estimation of distribution parameters, hypothesis testing, simple and multiple linear regressions, and Poisson and Markov processes. There is an emphasis placed on real-world applications to engineering problems. This class covers quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, random processes, statistics, and decision analysis are covered, along with random variables and vectors, uncertainty propagation, conditional distributions, and second-moment analysis. System reliability is introduced. Other topics covered include Bayesian analysis and risk-based decision, estimation of distribution parameters, hypothesis testing, simple and multiple linear regressions, and Poisson and Markov processes. There is an emphasis placed on real-world applications to engineering problems.

Subjects

fundamentals of probability | fundamentals of probability | random processes | random processes | statistics | statistics | decision analysis | decision analysis | random variables and vectors | random variables and vectors | uncertainty propagation | uncertainty propagation | conditional distributions | conditional distributions | second-moment analysis | second-moment analysis | system reliability | system reliability | Bayesian analysis and risk-based decision | Bayesian analysis and risk-based decision | estimation of distribution parameters | estimation of distribution parameters | hypothesis testing | hypothesis testing | simple and multiple linear regressions | simple and multiple linear regressions | Poisson and Markov processes | Poisson and Markov processes

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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1.203J Logistical and Transportation Planning Methods (MIT) 1.203J Logistical and Transportation Planning Methods (MIT)

Description

The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics. The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics.

Subjects

logistics | logistics | transportation | transportation | hypercube models | hypercube models | barrier example | barrier example | operations research | operations research | spatial queues | spatial queues | queueing models | queueing models | network models | network models | TSP | TSP | heuristics | heuristics | geometrical probablities | geometrical probablities | Markov | Markov | 1.203 | 1.203 | 6.281 | 6.281 | 15.073 | 15.073 | 16.76 | 16.76 | ESD.216 | ESD.216

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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