Stochastic nature

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  • The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L2 (T2 ). Unlike previous 0 works, this class is independent of the viscosity and the strength of the noise.

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  • We have attempted to explain the concepts which have been used and developed to model the stochastic dynamics of natural and biological systems. While the theory of stochastic differential equations and stochastic processes provide an attractive framework with an intuitive appeal to many problems with naturally induced variations, the solutions to such models are an active area of research, which is in its infancy. Therefore, this book should provide a large number of areas to research further.

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  • Stochastic gradient descent (SGD) uses approximate gradients estimated from subsets of the training data and updates the parameters in an online fashion. This learning framework is attractive because it often requires much less training time in practice than batch training algorithms. However, L1-regularization, which is becoming popular in natural language processing because of its ability to produce compact models, cannot be efficiently applied in SGD training, due to the large dimensions of feature vectors and the fluctuations of approximate gradients. ...

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  • We introduce a stochastic grammatical channel model for machine translation, that synthesizes several desirable characteristics of both statistical and grammatical machine translation. As with the pure statistical translation model described by Wu (1996) (in which a bracketing transduction grammar models the channel), alternative hypotheses compete probabilistically, exhaustive search of the translation hypothesis space can be performed in polynomial time, and robustness heuristics arise naturally from a language-independent inversiontransduction model. ...

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  • A description will be given of a procedure to asslgn the most likely probabilitles to each of the rules of a given context-free grammar. The grammar developed by S. Kuno at Harvard University was picked as the basis and was successfully augmented with rule probabilities. A brief exposition of the method with some preliminary results, w h e n u s e d as a device for disamblguatingparsing English texts picked from natural corpus, will be given.

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  • To study PP attachment disambiguation as a benchmark for empirical methods in natural language processing it has often been reduced to a binary decision problem (between verb or noun attachment) in a particular syntactic configuration. A parser, however, must solve the more general task of deciding between more than two alternatives in many different contexts. We combine the attachment predictions made by a simple model of lexical attraction with a full-fledged parser of German to determine the actual benefit of the subtask to parsing.

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  • In some computer applications of linguistics (such as maximum-likelihood decoding of speech or handwriting), the purpose of the language-handling component (Language Model) is to estimate the linguistic (a priori) probability of arbitrary natural-language sentences.

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  • In Data Oriented Parsing (DOP), an annotated corpus is used as a stochastic grammar. An input string is parsed by combining subtrees from the corpus. As a consequence, one parse tree can usually be generated by several derivations that involve different subtrces. This leads to a statistics where the probability of a parse is equal to the sum of the probabilities of all its derivations. In (Scha, 1990) an informal introduction to DOP is given, while (Bed, 1992a) provides a formalization of the theory. ...

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  • Basic principles underlying the transactions of financial markets are tied to probability and statistics. Accordingly it is natural that books devoted to mathematical finance are dominated by stochastic methods. Only in recent years, spurred by the enormous economical success of financial derivatives, a need for sophisticated computational technology has developed. For example, to price an American put, quantitative analysts have asked for the numerical solution of a free-boundary partial differential equation.

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  • Mastering the concepts of probability can cast new light on situations in which randomness and chance appear to rule. With an emphasis on why probability works and how it can be applied, Henk Tijms introduces the reader to the world of probability in an informal way. Lotteries and casino games provide a natural source of motivation, and he carefully discusses these with many worked examples to illustrate the key concepts from probability theory. Henk Tijms has been a professor in Operations Research at the Vrije University in Amsterdam since 1976.

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  • Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief, in constructing and exploiting natural differentiable structures on abstract probability spaces; in other words, Stochastic Calculus of Variations proceeds from a merging of differential calculus and probability theory. As optimization under a random environment is at the heart of mathematical finance, and as differential calculus is of paramount importance for the search of extrema, it is not surprising that Stochastic Calculus of Variations appears in mathematical finance.

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  • In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. ...

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  • The second result underlines the interest of a two-factor model rather than a one factor model to represent the stochastic dynamics of interest rates. For example, Balduzzi, Das and Foresi (1998) show the presence of a stochastic central tendency in the dynamics of the short and long term interest rates which explains that the level of the short rate is not the only relevant variable to describe its conditional mean.

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  • We should now develop some intuition before we take up the rigorous development in the next section. The elements ω ∈ Ω are the possible states of nature and one among them, say δ, is the true state of nature. The true state of nature is unknown and controls the outcome of all random experiments. An event A ∈ F occurs or does not occur according as δ ∈ A or δ ∈ A, that is, according as the random variable 1A assumes the value one or zero at δ. To gain information about the true state of nature we determine by means of experiments whether...

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  • We prove that almost every nonregular real quadratic map is ColletEckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step in achieving the same results for more general families of unimodal maps.

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  • Dedicated to the memory of Gert Kjærg˚ Pedersen ard Abstract In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the follow(n) (n) ing extension of Voiculescu’s random matrix result: Let (X1 , . . . , Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu’s random matrix paper...

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  • Measurement of efficiency started with Farrell (1957) who, following Debreu (1951) and Koopmas (1951), proposed a division of efficiency into two components: technical efficiency, which represents a firm’s ability to produce a maximum level of output from a given level of inputs, and allocative efficiency, which is the ability of a firm to use inputs in optimal proportions, given their respective prices and available technology. The combination of these two measures yields the level of economic efficiency.

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  • This volume is devoted to the study of feedback control of so-calledlinear plant/nonlinear instrumentation(LPNI) systems. Such systems appear naturally in situations where the plant can be viewed as linear but the instrumentation, that is, actuators and sensors, can not.

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  • This paper describes the conversion of a Hidden Markov Model into a sequential transducer that closely approximates the behavior of the stochastic model. This transformation is especially advantageous for part-of-speech tagging because the resulting transducer can be composed with other transducers that encode correction rules for the most frequent tagging errors. The speed of tagging is also improved. The described methods have been implemented and successfully tested on six languages.

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  • We suggested the new infrared mechanism of dimensional transmutation that is omitted in the conventional approach and leads effectively to the stochastization of the Faddeev-Popov functional. We have proved the possibilities of such a stochastization in the Abelian version of the collective excitation and showed that the quantization of infrared − → fields k 2 = 0 leads to one of the version of the ”confinement propagator” .

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