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.

Self‐Consistent‐Field Approach to Turbulence Theory

Abstract: A self‐consistent‐field type perturbation theory is developed to treat the dynamics of stationary and homogeneous turbulence. The method consists of expanding the full probability distribution function about the product of the exact univariate distributions of all the Fourier modes. The theory is used in second order to find expressions for the turbulent energy spectrum and associated response frequencies. The results for the energy spectrum are identical to a simplified form of the direct‐interaction approximation of Kraichnan and closely resemble the results of the generalized randomphase approximation of Edwards. The relation of the present method to both the above approaches is discussed.

Scale: Landscape attributes and geographical information systems

Abstract: The roles and limitations of geographical information systems (GISs) in scaling hydrological models over heterogeneous land surfaces are outlined. Scaling is defined here as the extension of small-scale process models, which may be directly parameterized and validated, to larger spatial extents. A process computation can be successfully scaled if this extension can be carried out with minimal bias. Much of our understanding of land surface hydrological processes as currently applied within distributed models has been derived in conjunction with 'point' or 'plot' experiments, in which spatial variations and patterns of the controlling soil, canopy and meteorological factors are not defined. In these cases, prescription of model input parameters can be accomplished by direct observation. As the spatial extent is expanded beyond these point experiments to catchment or larger watershed regions, the direct extension of the point models requires an estimation of the distribution of the model parameters and process computations over the heterogeneous land surface. If the distribution of the set of spatial variables required for a given hydrological model (e.g. surface slope, soil hydraulic conductivity) can be described by a joint density function, f(x), where x = x 1 ,x 2 ,x 3 ,... are the model variables, then a GIS may be evaluated as a tool for estimating this function. In terms of the scaling procedure, the GIS is used to replace direct measurement or sampling of f(x) as the area of simulation is increased beyond the extent over which direct sampling of the distribution is feasible. The question to be asked is whether current GISs and current available spatial data sets are sufficient to adequately estimate these density functions.

Smoothing Spline Density Estimation: Theory

Abstract: In this article, a class of penalized likelihood probability density estimators is proposed and studied. The true log density is assumed to be a member of a reproducing kernel Hilbert space on a finite domain, not necessarily univariate, and the estimator is defined as the unique unconstrained minimizer of a penalized log likelihood functional in such a space. Under mild conditions, the existence of the estimator and the rate of convergence of the estimator in terms of the symmetrized Kullback-Leibler distance are established. To make the procedure applicable, a semiparametric approximation of the estimator is presented, which sits in an adaptive finite dimensional function space and hence can be computed in principle. The theory is developed in a generic setup and the proofs are largely elementary. Algorithms are yet to follow.

Overcoming stiffness in stochastic simulation stemming from partial equilibrium: A multiscale Monte Carlo algorithm

Abstract: In this paper the problem of stiffness in stochastic simulation of singularly perturbed systems is discussed. Such stiffness arises often from partial equilibrium or quasi-steady-state type of conditions. A multiscale Monte Carlo method is discussed that first assesses whether partial equilibrium is established using a simple criterion. The exact stochastic simulation algorithm (SSA) is next employed to sample among fast reactions over short time intervals (microscopic time steps) in order to compute numerically the proper probability distribution function for sampling the slow reactions. Subsequently, the SSA is used to sample among slow reactions and advance the time by large (macroscopic) time steps. Numerical examples indicate that not only long times can be simulated but also fluctuations are properly captured and substantial computational savings result.