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DeGroot Learning
DeGroot learning refers to a rule-of-thumb type of social learning process. The idea was stated in its general form by the American statistician Morris H. DeGroot; antecedents were articulated by John R. P. French and Frank Harary. The model has been used in physics, computer science and most widely in the theory of social networks. Setup and the learning process Take a society of n agents where everybody has an opinion on a subject, represented by a vector of probabilities p(0) = (p_1(0), \dots, p_n(0) ) . Agents obtain no new information based on which they can update their opinions but they communicate with other agents. Links between agents (who knows whom) and the weight they put on each other's opinions is represented by a trust matrix T where T_ is the weight that agent i puts on agent j 's opinion. The trust matrix is thus in a one-to-one relationship with a weighted, directed graph where there is an edge between i and j if and only if T_ > 0 . The trust ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morris DeGroot
Morris Herman DeGroot (June 8, 1931 – November 2, 1989) was an American statistician. Biography Born in Scranton, Pennsylvania, DeGroot graduated from Roosevelt University and earned master's and doctor's degrees from the University of Chicago. DeGroot joined Carnegie Mellon in 1957 and became a University Professor, the school's highest faculty position. He was the founding editor of the review journal '' Statistical Science''. Academic works He wrote six books, edited four volumes and authored over one hundred papers. Most of his research was on the theory of rational decision-making under uncertainty. His ''Optimal Statistical Decisions'', published in 1970, is still recognized as one of the great books in the field. His courses on statistical decision theory taught at Carnegie-Mellon influenced Edward C. Prescott and Robert Lucas, Jr., influential figures in the development of new classical macroeconomics and real business-cycle theory. DeGroot's undergraduate text, ''P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues And Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic roo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Random Variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially random or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence (probability Theory)
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DeGroot Learning Non-Convergent
DeGroot is an agglutinated form of the Dutch surname De Groot. It may refer to: People * Bruce DeGroot (born 1963), American politician * Chad DeGroot (born 1974), American freestyle BMX rider *Diede de Groot (born 1986), Dutch Tennis Grand Slam Champion *Dudley DeGroot (1899–1970), American athlete and coach * Gerard DeGroot, author of the 2008 book '' The Sixties Unplugged'' * Jeff DeGroot (born 1985), American soccer player * Morris H. DeGroot (1931–1989), American statistician Characters *Gerald and Karen DeGroot, characters on the American television show ''Lost'' *LuAnn DeGroot, main character of the comic strip '' Luann'' *Tavish Finnegan DeGroot, the real name of the Demoman class from ''Team Fortress 2 ''Team Fortress 2'' is a 2007 multiplayer first-person shooter, first-person shooter game developed and published by Valve Corporation. It is the sequel to the 1996 ''Team Fortress'' Mod (video gaming), mod for ''Quake (video game), Quake'' and ...'' See also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DeGroot Learning Convergent
DeGroot is an agglutinated form of the Dutch surname De Groot. It may refer to: People * Bruce DeGroot (born 1963), American politician * Chad DeGroot (born 1974), American freestyle BMX rider *Diede de Groot (born 1986), Dutch Tennis Grand Slam Champion *Dudley DeGroot (1899–1970), American athlete and coach * Gerard DeGroot, author of the 2008 book '' The Sixties Unplugged'' * Jeff DeGroot (born 1985), American soccer player * Morris H. DeGroot (1931–1989), American statistician Characters *Gerald and Karen DeGroot, characters on the American television show ''Lost'' *LuAnn DeGroot, main character of the comic strip '' Luann'' *Tavish Finnegan DeGroot, the real name of the Demoman class from ''Team Fortress 2 ''Team Fortress 2'' is a 2007 multiplayer first-person shooter, first-person shooter game developed and published by Valve Corporation. It is the sequel to the 1996 ''Team Fortress'' Mod (video gaming), mod for ''Quake (video game), Quake'' and ...'' See also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron–Frobenius Theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory ( ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics ( Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines ( PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Graph
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer ''k'' > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph ''G'' is called the ''period'' of ''G''. Graphs that cannot be aperiodic In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every ''k'' divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by ''n'', the length of that cycle. Testing for aperiodicity Suppose that ''G'' is strongly connected and that ''k'' divides the lengths of all cycles in ''G''. Consider the result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
