Distribution functions for the one‐dimensional radii of gyration were calculated for random flight chains of 2, 4, and 10 statistical segments, and were compared with approximate distributions calculated with the “l...
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Distribution functions for the one‐dimensional radii of gyration were calculated for random flight chains of 2, 4, and 10 statistical segments, and were compared with approximate distributions calculated with the “long‐chain” eigenvalues. The correct distributions are broader and flatter than their long‐chain counterparts and have larger moments. The true distributions are not well represented by the long‐chain approximation when the number of statical segmentsis less than 10. But, considering the rate at which the two distributions approach each other with increasing , we feel confident that the long‐chain eigenvalues would be adequate for . Indeed, for many practical considerations, one might take the long‐chain distributions as reasonable approximations to the correct ones, even for .
Let p be the probabilitydensity of a probability distribution P on the real line R with respect to the Lebesgue measure. The characteristic function (p) over cap of p is defined as [GRAPHICS] We consider probability ...
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Let p be the probabilitydensity of a probability distribution P on the real line R with respect to the Lebesgue measure. The characteristic function (p) over cap of p is defined as [GRAPHICS] We consider probability densities p which are their own characteristic functions, that means [GRAPHICS] By linear combination of Hermitian functions we find a family of probability densities which are solutions of this integral equation. These solutions are entire functions of order 2 and type 1/2. This is contradictory to Corollary 3 in [J. L. Teugels, Bull. Soc. Math Belg., 23 (1971), pp. 236-262.]. Furthermore, we characterize the general solution of the integral equation (1) within the convex cone of probability density functions.
The purpose of this article is to study the first‐order statistics (i.e., probability density functions and moments) of finite sums of cosines and of sines, each having nonuniformly distributed random phases governed...
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The purpose of this article is to study the first‐order statistics (i.e., probability density functions and moments) of finite sums of cosines and of sines, each having nonuniformly distributed random phases governed by a von Mises probabilitydensity function. The number of sinusoids N is, first, taken to be deterministic, but arbitrary, and, second, taken to be a discrete random variable governed by a negative binomial distribution (which includes the Poisson distribution as a special case). Since the phase probability density functions are nonuniform, the probability density functions of the cosine sum and the sine sum are different. The probability density functions of these cosine and sine sums are evaluated using Fourier series expansions whose Fourier coefficients are sampled values of the corresponding characteristic functions. Representative numerical calculations have been carried out to illustrate the general features. The corresponding moments have also been evaluated. Finally, the behavior of the probability density functions and moments when N approaches infinity in the deterministic situation and 〈N〉 approaches infinity in the random situation is investigated, the former involving the theory of stable probability density functions, the latter involving the theory of infinitely divisible probability density functions.
Recent developments in the probabilistic and statistical analysis of probability density functions are reviewed. densityfunctions are treated as data objects for which suitable notions of the center of distribution a...
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Recent developments in the probabilistic and statistical analysis of probability density functions are reviewed. densityfunctions are treated as data objects for which suitable notions of the center of distribution and variability are discussed. Special attention is given to nonlinear methods that respect the constraints densityfunctions must obey. Regression, time series and spatial models are discussed. The exposition is illustrated with data examples. A supplementary vignette contains expanded versions of data analyses with accompanying codes. (C) 2021 EcoSta Econometrics and Statistics. Published by Elsevier B.V. All rights reserved.
A FORTRAN IV program is described, which may be run interactively with tutorial assistance or in batch and which allows a user to selectively fit any of seven probability density functions (p.d.f.'s) or a combinat...
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This paper presents a set of probability density functions for EURIBOR outturns in three months' time, estimated from the prices of options on EURIBOR futures. It is the first official and freely available dataset...
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This paper presents a set of probability density functions for EURIBOR outturns in three months' time, estimated from the prices of options on EURIBOR futures. It is the first official and freely available dataset to span the complete history of EURIBOR futures options, thus comprising over ten years of daily data, from 13 January 1999 onwards. Time series of the statistical moments of these option-implied probability density functions are documented until April 2010. Particular attention is given to how these probability density functions, and their associated summary statistics, reacted to the unfolding financial crisis between 2007 and 2009. The latter shows how option-implied probability density functions can be used as an uncertainty measure for monetary policy and financial stability analysis purposes.
We present an algorithm for grouping families of probability density functions(pdfs). We exploit the fact that under the squar-root re-parametrization, the space of pdfs forms a Riemannian manifold, namely the unit Hi...
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ISBN:
(纸本)9783540874782
We present an algorithm for grouping families of probability density functions(pdfs). We exploit the fact that under the squar-root re-parametrization, the space of pdfs forms a Riemannian manifold, namely the unit Hilbert sphere. An immediate consequence of the re-parametrization is that different families of pdfs form different submanifolds of the unit Hilbert sphere. Therefore, the problem of clustering pdfs reduces to the problem of clustering multiple submanifolds on the unit Hilbert sphere. We solve this problem by first learning a low-dimensional representation of the pdfs using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces. Then, by assuming that the pdfs from different groups are separated, we show that the null space of a matrix built from the local representation gives that segmentation of the pdfs. We also apply of our approach to the texture segmentation problem in computer vision.
There has been great interest in creating probabilistic programming languages to simplify the coding of statistical tasks;however, there still does not exist a formal language that simultaneously provides (1) continuo...
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ISBN:
(纸本)9781450310833
There has been great interest in creating probabilistic programming languages to simplify the coding of statistical tasks;however, there still does not exist a formal language that simultaneously provides (1) continuous probability distributions, (2) the ability to naturally express custom probabilistic models, and (3) probability density functions (PDFs). This collection of features is necessary for mechanizing fundamental statistical techniques. We formalize the first probabilistic language that exhibits these features, and it serves as a foundational framework for extending the ideas to more general languages. Particularly novel are our type system for absolutely continuous (AC) distributions (those which permit PDFs) and our PDF calculation procedure, which calculates PDFs for a large class of AC distributions. Our formalization paves the way toward the rigorous encoding of powerful statistical reformulations.
The behavior of probability density functions (pdfs) of the velocity and temperature fluctuations in the unstable surface layer is systematically studied within the framework of similarity theory. The pdfs of horizont...
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The behavior of probability density functions (pdfs) of the velocity and temperature fluctuations in the unstable surface layer is systematically studied within the framework of similarity theory. The pdfs of horizontal velocity fluctuations are nearly Gaussian for a wide range of atmospheric stability conditions. Thus, the standard deviations of horizontal velocity fluctuations are sufficient to predict the pdfs. Using mixed-layer similarity, these standard deviations can be estimated from mean meteorological conditions. The pdfs of vertical velocity fluctuations are non-Gaussian. We introduce the truncated stable distributions and find that they can be better for fitting the tails of measured pdfs than other non-Gaussian pdfs commonly used in the airborne dispersion models. This implies that the truncated stable distributions can be better for describing the small probability but high-impact events. Similar to the pdfs of the horizontal velocity fluctuations, the standard deviations of vertical velocity fluctuations are also sufficient to predict the pdfs. Using Monin-Obukhov similarity, their standard deviations can also be estimated from mean meteorological conditions. The pdfs of temperature fluctuations are much more complicated than those of velocity fluctuations. We show that the left tails of the temperature pdfs are likely to be power law and the right tails of temperature pdfs are Gaussian. Although complicated for pdfs, the Monin-Obukhov similarity is also valid to the standard deviations of temperature fluctuations.
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