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.

Optimal Transformations for Multiple Regression: Application to Permeability Estimation From Well Logs

Abstract: Conventional Imtrltiple regression for permeability estimation from well logs requires a functional relationship to be presumed Due to the inexact nature of the relationship between petrophysical variables, it is not always possible to identify the underlying functional form between dependent and independent variables in advance When large variations in metrological properties arc exhibited, parametric regression often fails or leads to unstable and erroneous results, especially for multi variate cases In this paper we describe a nonparametric approach for estimating optimal transformations of petrophysical data to obtain the maximum correlation between observed variables The approach does not require a priori assumptions of a functional form and the optimal transformations are derived solely based on the data set An iterative procedure involving the ul[ernaling conditional expec[a[ion (ACE) forms the basis of our approach The power of ACE is illustrated using synthetic as well as field examples The results clearly demonstrate improved permeability estimation by ACE compared to conventional parametric regression methods Introduction A critical aspect of reservoir description involves estimating References and illustrations at end of paper permeability in uncored wells based on well logs and other known petrophysical attributes A common approach is to develop a permeability-porosity relationship by regressing on data from cored wells and then, to predict permeability in uncored wells from well logs 1’2 Multiple regression is used when large variations in metrological properties exist (e g a wide range in grain sizes, high degree of cementation, diagenetic alteration etc) and a simple permeability-porosity relationship no longer holds good However, there are several Iim itations to such an approach Many of these arise from the inexact nature of the relationship between petrophysical variables and u priori assumptions regarding functional forms used to model the data -all leading to biased estimates When prediction of permeability extremes is a major concern, the high and low values are enhanced through a weighting scheme in the regression Besides being subjective in nature, such weighting can cause the prediction to become unstable which leads to erroneous results Most importantly, conventional regression assumes independent variables to be free of error, which is highly optimistic for geologic and petrophysical data, Jensen and Lake? introduced power transformations for optimization of regression-based permeability y-porosity predictions The underlying theory is that if the joint probability distribution function (jpdf) of two variables is binorrnal, (he relationship will be linear,3 Several methods exist to estimate the exponents for power transformation One method, described by Emerson and Stoto4 and adopted by Jensen and Lake,2 is based on symmetrizing the probability distribution function (pdf) Another method is a trial-anderror approach based on a normal probability plot of the data By power transforming permeability and porosity separately the authors are able to improve permeability-porosity correlations However, using a trial-and-error method for selecting exponents for power transformation is time consuming, and symmetrizing the p,d f does not necessarily guarantee a binormal distribution of transformed variables In addition, there are no indications as to whether power transformations will work for multivariate cases

Characterizations and simulations of a class of stochastic processes to model anomalous diffusion

Abstract: In this paper, we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions. This class, called generalized grey Brownian motion (ggBm), is made up of self-similar with stationary increments processes (H-sssi) and depends on two real parameters α ∈ (0, 2) and β ∈ (0, 1]. It includes fractional Brownian motion when α ∈ (0, 2) and β = 1, and time-fractional diffusion stochastic processes when α = β ∈ (0, 1). The latter have a marginal probability density function governed by timefractional diffusion equations of order β. The ggBm is defined through the explicit construction of the underlying probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite-dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of theM-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that the ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the H-sssi nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differential equation of a fractional type.

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

Abstract: In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.

Multidimensional density shaping by sigmoids

Abstract: An estimate of the probability density function of a random vector is obtained by maximizing the output entropy of a feedforward network of sigmoidal units with respect to the input weights. Classification problems can be solved by selecting the class associated with the maximal estimated density. Newton's optimization method, applied to the estimated density, yields a recursive estimator for a random variable or a random sequence. A constrained connectivity structure yields a linear estimator, which is particularly suitable for "real time" prediction. A Gaussian nonlinearity yields a closed-form solution for the network's parameters, which may also be used for initializing the optimization algorithm when other nonlinearities are employed. A triangular connectivity between the neurons and the input, which is naturally suggested by the statistical setting, reduces the number of parameters. Applications to classification and forecasting problems are demonstrated.

Bayesian sequential Gaussian simulation of lithology with non-linear data

Abstract: A multivariate stochastic simulation application that involves the mapping of a primary variable from a combination for sparse primary data and more densely sampled secondary data The method is applicable when the relationship between the simulated primary variable and one or more secondary variables is non-linear The method employs a Bayesian updating rule to build a local posterior distribution for the primary variable at each simulated location The posterior distribution is the product of a Gaussian kernel function obtained by simple kriging of the primary variable and a secondary probability function obtained directly from a scatter diagram between primary and secondary variables