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Probability density function

About: Probability density function is a research topic. Over the lifetime, 22321 publications have been published within this topic receiving 422885 citations. The topic is also known as: probability function & PDF.


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Journal ArticleDOI
TL;DR: The Liouville Equation (Liouville equation) as discussed by the authors provides a framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts, in which the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics.
Abstract: The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems. The purpose of this ...

116 citations

Journal ArticleDOI
TL;DR: In this article, a three-parameter normal tail approximation to a non-normal distribution function is proposed, where the distribution function, the probability density function and its derivative are matched at the approximation point with the approximating function.

115 citations

Journal ArticleDOI
TL;DR: The log-normal law serves as an excellent mathematical model for particle size distribution analysis to the extent that the various mathematical terms are properly interpreted.

115 citations

Journal ArticleDOI
TL;DR: In this paper, the authors address the empirical bandwidth choice problem in cases where the range of dependence may be virtually arbitrarily long and provide surprising evidence that, even for some strongly dependent data sequences, the asymptotically optimal bandwidth for independent data is a good choice.
Abstract: We address the empirical bandwidth choice problem in cases where the range of dependence may be virtually arbitrarily long. Assuming that the observed data derive from an unknown function of a Gaussian process, it is argued that, unlike more traditional contexts of statistical inference, in density estimation there is no clear role for the classical distinction between short- and long-range dependence. Indeed, the "boundaries" that separate different modes of behaviour for optimal bandwidths and mean squared errors are determined more by kernel order than by traditional notions of strength of dependence, for example, by whether or not the sum of the covariances converges. We provide surprising evidence that, even for some strongly dependent data sequences, the asymptotically optimal bandwidth for independent data is a good choice. A plug-in empirical bandwidth selector based on this observation is suggested. We determine the properties of this choice for a wide range of different strengths of dependence. Properties of cross-validation are also addressed.

115 citations

Journal ArticleDOI
03 Nov 2003
TL;DR: In this paper, the authors analyzed a set of random frequency modulated (FM) signals for wideband radar imaging and assessed their resolution capability and sidelobe distribution on the range-Doppler plane.
Abstract: The authors analysed a set of random frequency modulated (FM) signals for wideband radar imaging and assessed their resolution capability and sidelobe distribution on the range–Doppler plane. To this effect deterministic, bounded, nonlinear iterated maps were first considered. The initial condition of each chaotic map was assigned to a random variable to obtain statistically independent samples with invariant probability density function. The resulting sequences, which have white time–frequency representations, are used to construct wideband stochastic FM signals. These FM signals are ergodic and stationary. The autocorrelation, spectrum and the ambiguity surface associated with each of the FM signals were characterised. It was also demonstrated that the ambiguity surface of an FM signal generated via a chaotic map with uniform sample distribution and tail-shifted chaotic attractor is comparable to the ambiguity function of a Gaussian FM signal.

115 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023382
2022906
2021906
20201,047
20191,117
20181,083