Topic
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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TL;DR: In this article, a method for estimating definite and indefinite integrals over a density function, such as a local density of states, defined by a three-term recurrence relation, is presented.
Abstract: A method is presented for estimating definite and indefinite integrals over a density function, such as a local density of states, defined by a three-term recurrence relation. This may be generated, for example, by the 'recursion method' applied to some Hamiltonian, and properties of the approximation are given, and the results derived, in that context.
114 citations
TL;DR: In this article, the validity of posterior probability statements follows from probability calculus when the likelihood is the density of the observations, and a more intuitive definition of validity is introduced, based on coverage of posterior sets.
Abstract: SUMMARY The validity of posterior probability statements follows from probability calculus when the likelihood is the density of the observations. To investigate other cases, a second, more intuitive definition of validity is introduced, based on coverage of posterior sets. This notion of validity suggests that the likelihood must be the density of a statistic, not necessarily sufficient, for posterior probability statements to be valid. A convenient numerical method is proposed to invalidate the use of certain likelihoods for Bayesian analysis. Integrated, marginal, and conditional likelihoods, derived to avoid nuisance parameters, are also discussed.
114 citations
TL;DR: A generalized Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (PolSAR) data is presented and a Bayesian classification scheme is proposed, which can be used in both supervised and unsupervised modes.
Abstract: In this paper, we present a generalized Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (PolSAR) data. Our starting point is to demonstrate that the scale mixture of Gaussian (SMoG) distribution model is suitable for modeling PolSAR data. We show that the distribution of the sample covariance matrix for the SMoG model is given as a generalization of the Wishart distribution and present this expression in integral form. We then derive the closed-form solution for one particular SMoG distribution, which is known as the multivariate K-distribution. Based on this new distribution for the sample covariance matrix, termed as the K -Wishart distribution, we propose a Bayesian classification scheme, which can be used in both supervised and unsupervised modes. To demonstrate the effect of including non-Gaussianity, we present a detailed comparison with the standard Wishart classifier using airborne EMISAR data.
114 citations
TL;DR: In this paper, a probabilistic and distribution-free class-modeling technique is developed from potential function discriminant analysis, where the class boundary is built either by the sample percentile of the probability density estimated by means of potential functions, or by the estimate of the equivalent determinant of the variance covariance matrix.
Abstract: A probabilistic and distribution-free class-modelling technique is developed from potential function discriminant analysis. In the multidimensional space of variables the class boundary is built either by the sample percentile of the probability density estimated by means of potential functions, or by the estimate of the ‘equivalent’ determinant of the variance–covariance matrix. The equivalent determinant is that of a hypothetical multivariate normal distribution whose mean probability density was obtained by potential functions. The bases of this modelling rule are evaluated by means of Monte Carlo experiments. The results on four datasets are used to measure the performances of this method, which equal and sometimes exceed the performances of parametric class-modelling methods based on linear and quadratic discriminant analysis which were used for comparison.
113 citations
TL;DR: In this article, the transport equation for the joint probability density function of velocity and scalars is shown to provide a good basis for modeling turbulent reactive flows, and closed approximations are presented for the terms involving the fluctuating pressure and viscous and diffusive mixing.
Abstract: The transport equation for the joint probability density function of velocity and scalars is shown to provide a good basis for modeling turbulent reactive flows. As in the equation for the probability density function of the scalars alone, nonlinear reaction schemes can be treated without approximation. The advantage of considering the joint probability density function equation is that convection (by both the mean and fluctuating velocities) appears in closed form. Consequently, the gradient‐diffusion assumption for turbulent transport is avoided. Closure approximations are presented for the terms involving the fluctuating pressure and viscous and diffusive mixing. These models can be expected to be reliable since they are compatible with accurate and proven Reynolds‐stress models. The resulting modeled transport equation for the joint probability density function can be solved by the Monte‐Carlo method for inhomogeneous flows with complex reactions.
113 citations