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 paper, it was shown that for the 2t-variate joint and the transition probability density function, strongly consistent estimates for the initial and the one-step transition distribution functions of the process were obtained.
113 citations
TL;DR: In this article, a hierarchy of inhomogeneous, nonstationary stochastic models of material transport is formulated, and its properties are described, and the transport models from the hierarchy sequence provide progressively more skillful simulations of the subgrid-scale transport by mesoscale eddies, which are typically not resolved in coarse-grid representations of the ocean circulation.
Abstract: A hierarchy of inhomogeneous, nonstationary stochastic models of material transport is formulated, and its properties are described. The transport models from the hierarchy sequence provide progressively more skillful simulations of the subgrid-scale transport by mesoscale eddies, which are typically not resolved in coarse-grid representations of the ocean circulation. The stochastic transport models yield random motion of individual passive particles, and the probability density function of the particle population can be interpreted as the concentration of a passive tracer. Performance of the models is evaluated by (a) estimating their parameters from Eulerian and Lagrangian statistics of a fluid-dynamic reference solution, (b) solving for the transport, and (c) comparing the stochastic and fluid-dynamic transports. The reference solution represents midlatitude oceanic gyres, and it is found by solving steadily forced, quasigeostrophic equations of motion at large Reynolds number. The gyres are characterized by abundant coherent structures, such as swift, meandering currents, strong vortices, eddies, and planetary waves. The common, nondiffusive spreading of material (i.e., single-particle dispersion that is a nonlinear function of time) is induced by all these structures on intermediate times and by inhomogeneity and lateral boundaries on longer times. The higher-order members of the hierarchy are developed specially for simulating nondiffusive transports by turbulence in the presence of organized fluid patterns. The simplest, but least skillful, member of the hierarchy is the commonly used diffusion model. In terms of the random particle motion, the diffusion is equivalent to the random walk (Markov-0) process for particle positions. The higher-order members of the hierarchy are the Markov-1 (a.k.a. Langevin or random acceleration), Markov-2, and Markov-3 models, which are jointly Markovian for particle position and its time derivatives. Each model in the hierarchy incorporates all features of the models below it. The Markov-1 model simulates short-time ballistic behavior associated with exponentially decaying Lagrangian velocity correlations, but on large times it is overly dispersive because it does not account for trapping of material by the coherent structures. The Markov-2 model brings in the capability to simulate intermediate-time, subdiffusive (slow) spreading associated with such trappings and with both decaying and oscillating Lagrangian velocity correlations. The Markov-3 model is also capable of simulating intermediate-time, superdiffusive (fast) spreading associated with sustained particle drifts combined with the trapping phenomenon and with the related asymmetry of the decaying and oscillating Lagrangian velocity correlations.
113 citations
TL;DR: In this article, a Galerkin projection procedure is used to derive a set of ordinary differential equations which can be solved numerically to determine the coefficients in the series, which are then used to solve a non-Markovian oscillator response.
113 citations
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TL;DR: This work proposes a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility, and demonstrates that neural spline flows improve density estimation, variational inference, and generative modeling of images.
Abstract: A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.
113 citations
TL;DR: An approximation to the probability density function of the received power of a spatially correlated multiple-beam system in terms of the parameters of the single-channel gamma-gamma function is proposed.
Abstract: By means of numerical simulations we analyze the statistical properties of the power fluctuations induced by the incoherent superposition of multiple transmitted laser beams in a terrestrial free-space optical communication link. The measured signals arising from different transmitted optical beams are found to be statistically correlated. This channel correlation increases with receiver aperture and propagation distance. We find a simple scaling rule for the spatial correlation coefficient in terms of the propagation distance and we are able to predict the scintillation reduction in previously reported experiments with good accuracy. We propose an approximation to the probability density function of the received power of a spatially correlated multiple-beam system in terms of the parameters of the single-channel gamma-gamma function. A bit-error-rate evaluation is also presented to demonstrate the improvement of a multibeam system over its single-beam counterpart.
113 citations