Approximation of joint PDFs by discrete distributions generated with Monte Carlo methods

A flamelet-based Eulerian transported PDF method for the modeling and simulation of supersonic combustion

The non-line-of-sight (NLOS) propagation caused by intermittently blocking of the direct path between the base station and the mobile station is the main difficulty for mobile user location in cellular wireless communication system. So it is indispensable to investigate the techniques about the NLOS error identification and correction. In this paper, we suggest a robust and accurate approach using probability density function (PDF) estimator to identify the NLOS situation, and then an improved biased Kalman filter which depends on PDF estimator as the nonlinearity weighting coefficient in recursive operation is used to mitigate the NLOS error for time of arrival (TOA) measurements. As the PDF estimator is calculated from actual TOA samples, it is adaptive for the complex practical NLOS environments. The simulation results also indicate that with the less prior information about communication environments, the approach we present has a good location performance even in severe NLOS situations.

In this paper, we propose a fast and versatile blind source separation including closed-form estimation of sources' probability density functions (PDFs), where the ICA's activation function is automatically adapted to various noise conditions. In the proposed method, closed-form second-order ICA and closed-form PDF estimation are introduced as a computational-cost-efficient preprocessing to extract sources' PDFs. Compared with various type of conventional ICAs, e.g., fixed activation-function type and ML-based type, our proposed algorithm can give a faster and higher convergence. Experimental assessment reveals that the proposed method is versatile for handling non-speech sound sources.

The error estimate of Borgonovo's moment-independent index δi is considered, and it shows that the possible computational complexity of δi is mainly due to the probability density function (PDF) estimate because the PDF estimate is an ill-posed problem and its convergence rate is quite slow. So it reminds us to compute Borgonovo's index using other methods. To avoid the PDF estimate, δi, which is based on the PDF, is first approximatively represented by the cumulative distribution function (CDF). The CDF estimate is well posed and its convergence rate is always faster than that of the PDF estimate. From the representation, a stable approach is proposed to compute δi with an adaptive procedure. Since the small probability multidimensional integral needs to be computed in this procedure, a computational strategy named asymptotic space integration is introduced to reduce a high-dimensional integral to a one-dimensional integral. Then we can compute the small probability multidimensional integral by adaptive numerical integration in one dimension with an improved convergence rate. From the comparison of numerical error analysis of some examples, it can be shown that the proposed method is an effective approach to uncertainty importance measure computation.

Probability density function (PDF) estimation is a fundamentally important problem for statistical pattern recognition. Independent component analysis (ICA) can be applied to the feature vectors so that the PDF estimation of a high dimensional vector can be converted to the PDF estimation of several 1-dimensional variables. However in practice we find that this PDF is in poor generalization ability for pattern classification because of the implied noise. Hence, this paper proposes an improvement of ICA based PDF estimation method. A latent variable model is built to separate the noise from the feature vector so that the pattern information and the noise can be dealt with respectively. Based on the latent variable model, a modified ICA based PDF is deduced. The validity of our proposed method is demonstrated by the experiments of off-line handwritten numeral recognition.

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We establish and prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite location-scale mixtures of an essentially bounded PDF can approximate any PDF in in the norm.

This paper aims to study a class of differential equations with parametric Gaussian colored noise. We present the general framework to get the solvability conditions of the approximate stationary probability density function, which is determined by the Fokker-Planck-Kolmogorov (FPK) equations. These equations are derived using the stochastic averaging method and the operator theory with the perturbation technique. An illustrative example is proposed to demonstrate the procedure of our proposed method. The analytical expression of approximate stationary probability density function is obtained. Numerical simulation is carried out to verify the analytical results and excellent agreement can be easily found. The FPK equation for the probability density function of order e0 is used to examine the almost-sure stability for the amplitude process. Finally, the stability in probability of the amplitude process is investigated by Lin and Cai’s method.

The main part of the noise in digital images arises when taking pictures or transmission. There is noise in the images captured by the image sensors of the real world. Noise, based on its causes can have different probability density functions. For example, such a model is called the Poisson distribution function of the random nature of photon arrival process that is consistent with the distribution of pixel values measured. The parameters of the noise probability density function (PDF) can be achieved to some extent the properties of the sensor. But, we need to estimate the parameters for imaging settings. If we assume that the PDF of noise is approximately Gaussian, then we need only to estimate the mean and variance because the Gaussian PDF with only two parameters is determined. In fact, in many cases, PDF of noise is not Gaussian and it has unknown distribution. In this study, we introduce a generalized probability density function for modeling noise in images and propose a method to estimate its parameters. Because the generalized probability density function has multiple parameters, so use common parameter estimation techniques such as derivative method to maximize the likelihood function would be extremely difficult. In this study, we propose the use of evolutionary algorithms for global optimization. The results show that this method accurately estimates the probability density function parameters.

Non-parametric representation of pixel intensity distribution is quite effective to construct proper background model and to detect foreground objects accurately. However, from the viewpoint of practical application, the computation cost of the distribution estimation should be reduced. In this paper, we present fast estimation of the probability density function (PDF) of pixel value using Parzen density estimation and foreground object detection based on the estimated PDF. Here, the PDF is computed by partially updating the PDF estimated at the previous frame, and it greatly reduces the computation cost of the PDF estimation. Thus, the background model adapts quickly to changes in the scene and, therefore, foreground objects can be robustly detected. Several experiments show the effectiveness of our approach.

A function approximation based on an orthonormal wave function expansion in a complex space is derived. Although a probability density function (PDF) cannot always be expanded in an orthogonal series in a real space because a PDF is a positive real function, the function approximation can approximate an arbitrary PDF with high accuracy. It is applied to an actor-critic method of reinforcement learning to derive an optimal policy expressed by an arbitrary PDF in a continuous-action continuous-state environment. A chaos control problem and a PDF approximation problem are solved using the actor-critic method with the function approximation, and it is shown that the function approximation can approximate a PDF well and that the actor-critic method with the function approximation exhibits high performance.

In this paper, piecewise linear approximation of the probability density function is done. On the basis of the obtained approximated probability density function, a companding quantizer is designed. The signal at the entrance of the companding quantizer is modeled by Laplacian probability density function. Performances of the proposed companding quantizer are estimated on the basis of comparison of calculated values of signal to quantization noise ratio and approximation error, with the calculated values that correspond to the model of piecewise uniform scalar quantizer, that is also proposed in this paper.

The tabulated premixed conditional moment closure (T-PCMC) method has been shown to provide the capability to model turbulent, premixed methane flames with detailed chemistry and reasonable runtimes in Reynolds-averaged Navier–Stokes (RANS) environment by Martin et al. (2013, “Modeling an Enclosed, Turbulent Reacting Methane Jet With the Premixed Conditional Moment Closure Method,” ASME Paper No. GT2013-95092). Here, the premixed conditional moment closure (PCMC) method is extended to large eddy simulation (LES). The new model is validated with the turbulent, enclosed reacting methane backward facing step data from El Banhawy et al. (1983, “Premixed, Turbulent Combustion of a Sudden-Expansion Flow,” Combust. Flame, 50, pp. 153–165). The experimental data have a rectangular test section at atmospheric pressure and temperature with an inlet velocity of 10.5 m/s and an equivalence ratio of 0.9 for two different step heights. Contours of major species, velocity, and temperature are provided. The T-PCMC model falls into the class of table lookup turbulent combustion models in which the combustion model is solved offline over a range of conditions and stored in a table that is accessed by the computational fluid dynamic (CFD) code using three controlling variables: the reaction progress variable (RPV), variance, and local scalar dissipation rate. The local scalar dissipation rate is used to account for the affects of the small-scale mixing on the reaction rates. A presumed shape beta function probability density function (PDF) is used to account for the effects of subgrid scale (SGS) turbulence on the reactions. SGS models are incorporated for the scalar dissipation and variance. The open source CFD code OpenFOAM is used with the compressible Smagorinsky LES model. Velocity, temperature, and major species are compared to the experimental data. Once validated, this low “runtime” CFD turbulent combustion model will have great utility for designing the next generation of lean premixed (LPM) gas turbine combustors.

In order to assess the availability of wind potential energy in a given place, a fundamental part is to investigate the behavior of the wind during a certain period. A standard way to characterize wind data is by using statistical models such as the Weibull distribution. To define the parameters of the Weibull probability density function (PDF) which best describes the input data, the estimation of two parameters is needed. Rather than testing different estimation methods for the Weibull distribution parameters and choosing the one with greater fitting level, we propose a iterative method aiming to maximize the suitability of the PDF into the raw data histogram. The method is based on the Newton-Raphson algorithm and uses as initial guess the parameters yielded by a Maximum Likelihood estimation. We also assess the performance of the hybrid Weibull distribution proposed by Tackle and Brown for different types of data and compare it to the standard Weibull PDF. The results show that, for every set of data analyzed, the proposed method outperformed the state-of-the-art Maximum Likelihood method. Also the hybrid Weibull PDF model yielded better fit than the standard Weibull PDF in all cases.

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations