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.

The 3D skeleton: tracing the filamentary structure of the Universe

Abstract: The skeleton formalism, which aims at extracting and quantifying the filamentary structure of our Universe, is generalized to 3D density fields. A numerical method for computing a local approximation of the skeleton is presented and validated here on Gaussian random fields. It involves solving equation (H∇p x Vp) = 0, where Vp and H are the gradient and Hessian matrix of the field. This method traces well the filamentary structure in 3D fields such as those produced by numerical simulations of the dark matter distribution on large scales, and is insensitive to monotonic biasing. Two of its characteristics, namely its length and differential length, are analysed for Gaussian random fields. Its differential length per unit normalized density contrast scales like the probability distribution function of the underlying density contrast times the total length times a quadratic Edgeworth correction involving the square of the spectral parameter. The total length-scales like the inverse square smoothing length, with a scaling factor given by 0.21 (5.28 + n) where n is the power index of the underlying field. This dependency implies that the total length can be used to constrain the shape of the underlying power spectrum, hence the cosmology. Possible applications of the skeleton to galaxy formation and cosmology are discussed. As an illustration, the orientation of the spin of dark haloes and the orientation of the flow near the skeleton is computed for cosmological dark matter simulations. The flow is laminar along the filaments, while spins of dark haloes within 500 kpc of the skeleton are preferentially orthogonal to the direction of the flow at a level of 25 per cent.

A Comparison of Probabilistic Forecasts from Bred, Singular-Vector, and Perturbed Observation Ensembles

Abstract: The statistical properties of analysis and forecast errors from commonly used ensemble perturbation methodologies are explored. A quasigeostrophic channel model is used, coupled with a 3D-variational data assimilation scheme. A perfect model is assumed. Three perturbation methodologies are considered. The breeding and singular-vector (SV) methods approximate the strategies currently used at operational centers in the United States and Europe, respectively. The perturbed observation (PO) methodology approximates a random sample from the analysis probability density function (pdf) and is similar to the method performed at the Canadian Meteorological Centre. Initial conditions for the PO ensemble are analyses from independent, parallel data assimilation cycles. Each assimilation cycle utilizes observations perturbed by random noise whose statistics are consistent with observational error covariances. Each member’s assimilation/forecast cycle is also started from a distinct initial condition. Relativ...

Distribution- and correlation-functions for a laser amplitude

Abstract: In this paper a contribution to the nonlinear theory of laser noise is given. The lasing field is treated as a classical random variable, the noise is introduced by the concept of fluctuating dipoles. In order to obtain correlation functions the method of distribution functions is employed. The distribution functions are calculated by the Fokker-Planck equation.

On the Product of Independent Complex Gaussians

Abstract: In this paper, we derive the joint (amplitude, phase) distribution of the product of two independent non-zero-mean Complex Gaussian random variables. We call this new distribution the complex Double Gaussian distribution. This probability distribution function (PDF) is a doubly infinite summation over modified Bessel functions of the first and second kind. We analyze the behavior of this sum and show that the number of terms needed for accuracy is dependent upon the Rician k-factors of the two input variables. We derive an upper bound on the truncation error and use this to present an adaptive computational approach that selects the minimum number of terms required for accuracy. We also present the PDF for the special case where either one or both of the input complex Gaussian random variables is zero-mean. We demonstrate the relevance of our results by deriving the optimal Neyman-Pearson detector for a time reversal detection scheme and computing the receiver operating characteristics through Monte Carlo simulations, and by computing the symbol error probability (SEP) for a single-channel M-ary phase-shift-keying (M-PSK) communication system.

A Note on the Reward Function for PHD Filters with Sensor Control

Abstract: The context is sensor control for multi-object Bayes filtering in the framework of partially observed Markov decision processes (POMDPs). The current information state is represented by the multi-object probability density function (pdf), while the reward function associated with each sensor control (action) is the information gain measured by the alpha or Renyi divergence. Assuming that both the predicted and updated state can be represented by independent identically distributed (IID) cluster random finite sets (RFSs) or, as a special case, the Poisson RFSs, this work derives the analytic expressions of the corresponding Renyi divergence based information gains. The implementation of Renyi divergence via the sequential Monte Carlo method is presented. The performance of the proposed reward function is demonstrated by a numerical example, where a moving range-only sensor is controlled to estimate the number and the states of several moving objects using the PHD filter.