THE MARGINAL PROBABILITY DENSITY FUNCTIONS OF WISHART PROBABILITY DENSITY FUNCTION

2018 ◽  
Vol 54 (3) ◽  
pp. 239-326
Author(s):  
Yongshun Xiao
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Density Functions ◽  
Density Functions ◽  
Marginal Probability

scholarly journals Probability Density Functions for Prediction Using Normal and Exponential Distribution

2021 ◽  
pp. 40-52
Author(s):  
Kunio Takezawa
Keyword(s):  
Probability Density Function ◽  
Normal Distribution ◽  
Probability Density ◽  
Density Function ◽  
Exponential Distribution ◽  
Predictive Ability ◽  
Probability Density Functions ◽  
Density Functions ◽  
Specific Distribution ◽  
Using Data

When data are found to be realizations of a specific distribution, constructing the probability density function based on this distribution may not lead to the best prediction result. In this study, numerical simulations are conducted using data that follow a normal distribution, and we examine whether probability density functions that have shapes different from that of the normal distribution can yield larger log-likelihoods than the normal distribution in the light of future data. The results indicate that fitting realizations of the normal distribution to a different probability density function produces better results from the perspective of predictive ability. Similarly, a set of simulations using the exponential distribution shows that better predictions are obtained when the corresponding realizations are fitted to a probability density function that is slightly different from the exponential distribution. These observations demonstrate that when the form of the probability density function that generates the data is known, the use of another form of the probability density function may achieve more desirable results from the standpoint of prediction.

scholarly journals An inequality for probability density functions arising from a distinguishability problem

1998 ◽  
Vol 39 (3) ◽  
pp. 350-354
Author(s):  
Boris Guljaš ◽  
C. E. M. Pearce ◽  
Josip Pečarić
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Real Line ◽  
Integral Inequality ◽  
Probability Density Functions ◽  
Density Functions ◽  
The Real ◽  
Image Position

AbstractAn integral inequality is established involving a probability density function on the real line and its first two derivatives. This generalizes an earlier result of Sato and Watari. If f denotes the probability density function concerned, the inequality we prove is thatunder the conditions β > α 1 and 1/(β+1) < γ ≤ 1.

scholarly journals A Characterization Related to the Equilibrium Distribution Associated with a Polynomial Structure

2010 ◽  
Vol 47 (01) ◽  
pp. 293-299 ◽  
Author(s):  
Shaul K. Bar-Lev ◽  
Onno Boxma ◽  
Gérard Letac
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Equilibrium Distribution ◽  
Renewal Theory ◽  
Probability Density Functions ◽  
Density Functions ◽  
Surprising Result ◽  
Polynomial Structure ◽  
Equilibrium Probability

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class C f of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in C f then the equilibrium probability density x ↦ P(X &gt; x) / E(X) also belongs to C f : this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

Study on Nonlinear Vibration Characteristic Evaluation of Nonlinear Vibration System With Gaps by Transition Probability Density Function

2004 ◽  
Author(s):  
Masanori Shintani ◽  
Hiroyuki Ikuta ◽  
Hajime Takada
Keyword(s):  
Probability Density Function ◽  
Nonlinear Vibration ◽  
Probability Density ◽  
Density Function ◽  
Transition Probability ◽  
Probability Density Functions ◽  
Force Characteristic ◽  
Density Functions ◽  
Restoring Force ◽  
Transition Probability Density

In this paper, the transition probability density functions between response velocity and response displacement in nonlinear vibration systems which have the restoring force characteristic of a cubic equation are governed by the Fokker-Planck Equation. The experimental probability density functions are compared with analytical results. The analytical model of the cubic equation as Duffing Equation is proposed by the restoring force characteristic of the nonlinear vibration system with gaps in the experiments. However, a slight difference for the frequency range of the transfer function was shown by simulation results. Then, it is considered using transition probability density functions in the response characteristic. For stationary random input waves, the probability density function between the response displacement and the response velocity are easily estimated by the Fokker-Planck Equation and the Duffing Equation. The slight difference of the transfer function of the response acceleration is evaluated by the scattering of the restoring force characteristic estimated by the probability density function and self-natural frequency curve. The R.M.S. value and the transfer function of the experimental results are compared with the analytical results. It is thought that the estimation of the probability density function of the response has validity. It is thought that the evaluation of the nonlinear vibration characteristics by the probability density function is valid.

Selective interaction of two independent recurrent processes

1965 ◽  
Vol 2 (02) ◽  
pp. 286-292 ◽  
Author(s):  
M. Ten Hoopen ◽  
H. A. Reuver
Keyword(s):  
Time Series ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Density Functions ◽  
Density Functions ◽  
Selective Interaction ◽  
Mutually Independent

SummaryConsidered are two mutually independent recurrent processes each consisting of a time series of unitary stimuli. The durations of the intervals between the stimuli in each series are independent of each other and identically distributed with probability density functionsφ(t) andψ(t). Every stimulus of theψ(t) process annihilates the next stimulus of theφ(t) process. The probability density function of the intervals of the transformedφ(t) process is derived for the case where either theφ(t) or theψ(t) process is Poisson.

Normalized structure factor analysis with theCentroMKprogram

2013 ◽  
Vol 46 (1) ◽  
pp. 88-92 ◽  
Author(s):  
Marcin Kowiel
Keyword(s):  
Factor Analysis ◽  
Probability Density Function ◽  
Probability Density ◽  
Crystal Structures ◽  
Density Function ◽  
Structure Factor ◽  
Probability Density Functions ◽  
Density Functions ◽  
Exact Probability ◽  
Space Group

Statistical analysis of the normalized structure factorEis important during space-group determination. Several approaches to solve this problem have been described in the literature. In this paper, the most popular approach, the ideal asymptotic probability density function developed by Wilson, is compared with the more accurate exact probability density functions described by Shmueli and co-workers. Furthermore, a new computer program,CentroMK, for normalized structure factor analysis, is presented. The program is capable of plotting histograms of the normalized structure factors and exact probability density functions. Moreover, the program calculates five estimators helpful during the space-group determination: 〈|E|〉, 〈|E2 − 1|〉, %E> 2, %E< 0.25 and the discrepancyRfunction. The two approaches and the error rates of the five listed estimators are compared for nearly 30 600 crystal structures obtained fromActa Crystallographica Section E.It is shown that within a space group the means 〈|E|〉 and 〈|E2 − 1|〉 of real crystal structures show high variability. The comparison shows that decisions based on the exact probability density function are more accurate, the computing time is reasonable, and estimators 〈|E|〉, %E< 0.25 andRare the most accurate and should be preferred during space-group determination.

Selective interaction of two independent recurrent processes

1965 ◽  
Vol 2 (2) ◽  
pp. 286-292 ◽  
Author(s):  
M. Ten Hoopen ◽  
H. A. Reuver
Keyword(s):  
Time Series ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Density Functions ◽  
Density Functions ◽  
Selective Interaction ◽  
Mutually Independent

SummaryConsidered are two mutually independent recurrent processes each consisting of a time series of unitary stimuli. The durations of the intervals between the stimuli in each series are independent of each other and identically distributed with probability density functions φ (t) and ψ (t). Every stimulus of the ψ (t) process annihilates the next stimulus of the φ (t) process. The probability density function of the intervals of the transformed φ (t) process is derived for the case where either the φ (t) or the ψ (t) process is Poisson.

A novel distribution regression approach for data loss compensation in structural health monitoring

2017 ◽  
Vol 17 (6) ◽  
pp. 1473-1490 ◽  
Author(s):  
Zhicheng Chen ◽  
Yuequan Bao ◽  
Hui Li ◽  
Billie F Spencer
Keyword(s):  
Probability Density Function ◽  
Structural Health Monitoring ◽  
Probability Density ◽  
Density Function ◽  
Health Monitoring ◽  
Probability Density Functions ◽  
Density Functions ◽  
Warping Function ◽  
Structural Health ◽  
Regression Approach

Structural health monitoring has arisen as an important tool for managing and maintaining civil infrastructure. A critical problem for all structural health monitoring systems is data loss or data corruption due to sensor failure or other malfunctions, which bring into question in subsequent structural health monitoring data analysis and decision-making. Probability density functions play a very important role in many applications for structural health monitoring. This article focuses on data loss compensation for probability density function estimation in structural health monitoring using imputation methods. Different from common data, continuous probability density functions belong to functional data; the conventional distribution-to-distribution regression technique has significant potential in missing probability density function imputation; however, extrapolation and directly borrowing shape information from the covariate probability density function are the main challenges. Inspired by the warping transformation of distributions in the field of functional data analysis, a new distribution regression approach for imputing missing correlated probability density functions is proposed in this article. The warping transformation for distributions is a mapping operation used to transform one probability density function to another by deforming the original probability density function with a warping function. The shape mapping between probability density functions can be characterized well by warping functions. Given a covariate probability density function, the warping function is first estimated by a kernel regression model; then, the estimated warping function is used to transform the covariate probability density function and obtain an imputation for the missing probability density function. To address issues with poor performance when the covariate probability density function is contaminated, a hybrid approach is proposed that fuses the imputations obtained by the warping transformation approach with the conventional distribution-to-distribution regression approach. Experiments based on field monitoring data are conducted to evaluate the performance of the proposed approach. The corresponding results indicate that the proposed approach can outperform the conventional method, especially in extrapolation. The proposed approach shows good potential to provide more reliable estimation of distributions of missing structural health monitoring data.

scholarly journals A Characterization Related to the Equilibrium Distribution Associated with a Polynomial Structure

2010 ◽  
Vol 47 (1) ◽  
pp. 293-299
Author(s):  
Shaul K. Bar-Lev ◽  
Onno Boxma ◽  
Gérard Letac
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Equilibrium Distribution ◽  
Renewal Theory ◽  
Probability Density Functions ◽  
Density Functions ◽  
Surprising Result ◽  
Polynomial Structure ◽  
Equilibrium Probability

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class Cf of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to Cf: this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

On the entropy to texture index relationship in quantitative texture analysis

2007 ◽  
Vol 40 (2) ◽  
pp. 371-375 ◽  
Author(s):  
R. Hielscher ◽  
H. Schaeben ◽  
D. Chateigner
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Texture Analysis ◽  
Sharp Inequality ◽  
Probability Density Functions ◽  
Density Functions ◽  
Orientation Space ◽  
Reconstruction Software ◽  
Orientation Density

This communication demonstrates a sharp inequality between the L^{2}-norm and the entropy of probability density functions. This inequality is applied to texture analysis, and the relationship between the entropy and the texture index of an orientation density function is characterized. More precisely, the orientation space is shown to allow for texture index and entropy variations of orientation probability density functions between an upper and a lower bound for the entropy. In this way, it is proved that there is no functional relationship between entropy and texture index of an orientation probability density function as conjectured previously on the basis of practical numerical texture analyses using the widely used pole-to-orientation probability density function reconstruction softwareWIMV, known by the initials of its authors and their ancestors (Williams–Imhof–Matthies–Vinel). Synthetic orientation probability density functions were then synthesized, covering a large domain of variation for texture index and entropy, and used to check the numerical results of the same software package.

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