Analytical Probability Density Function Research Articles

Liouville models of particle-laden flow

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics. This strategy gives rise to an exact (deterministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. Analytical PDF solutions for canonical models of particle-laden flows serve to establish a relationship between the Langevin and Liouville approaches. Finally, our framework is used to derive a new analytical PDF model for fluidized homogeneous heating systems.

Reliability analysis for product package via probability density function of acceleration random response

Analytical probability density function of acceleration random response for cubic nonlinear product package has seldomly been investigated due to the difficulty of solving nonlinear random problem and less attention paid to the acceleration response. In this paper, the analytical solution for the probability density function of linear package acceleration response under random vibration is obtained, and on this basis, the equivalent linearization method is introduced to further explore the approximate analytical solution of acceleration response probability density function for cubic nonlinear product package under random vibration. Some suggestions on packaging optimal design are given by analyzing the sensitivity of acceleration random response to the external excitation and system parameters, and the first-passage failure probability for the acceleration response of cubic nonlinear package under random vibration is analyzed. The results show that the equivalent linearization method can precisely predict the acceleration response probability density function for the cubic nonlinear package under certain conditions, and the applicable range of the method is given by the equivalent stiffness formula. Nonlinear characteristic parameter γ of the cubic package activates acceleration response a completely different trend from the quasi-linear system. Properly increasing the system stiffness and decreasing the damping ratio can effectively reduce the acceleration response strength of the product package. As the vibration time increases and the product fragility decreases, the first-passage failure probability of the product package increases. The analysis provides theoretical foundation and guidance for packaging optimization design.

Ice rose diagrams for probabilistic characterization of the ice drift behavior in the Beaufort Sea

The purpose of the present paper is to investigate the viability of the so-called ice rose diagram, which is a graphic tool to provide a succinct view of the ice drift information such as speed and direction of ice drift events in terms of their relative distribution at a local observation station to support the dynamic ice-structure interaction in cold regions. The ice drift data is collected by using local subsurface measurements based on acoustic Doppler current profilers (ADCP) in the Beaufort Sea during 2006–2017. Probabilistic analyses are performed in order to characterize the time-series data from these ADCP measurements. Both the ice drift speed and direction are treated as random variables for which analytical probability density functions are fitted. It is found that sea ice in the Beaufort Sea tends to drift at a lower speed during the winter season (i.e. the growth season) than during the summer season (i.e. the melting season). The Weibull distribution provides the most appropriate fit to the data of ice drift speeds during the growth season. For the melting season, a mixed distribution provides the most appropriate fit for the drift speed. Regarding directionality of the ice drift, a mixed Von Mises distribution is applied in order to represent its statistical properties. In this work, the relationship between ice drift speed and wind speed is also studied. It is found that the magnitude of the ice drift speed is approximately 2.5% of the wind speed during the winter season.

On the Distribution of SINR for Widely Linear MMSE MIMO Systems With Rectilinear or Quasi-Rectilinear Signals

Although the widely linear least mean square error (WLMMSE) receiver has been an appealing option for multiple-input-multiple-output (MIMO) wireless systems, a statistical understanding on its pose-detection signal-to-interference-plus-noise ratio (SINR) in detail is still missing. To this end, we consider a WLMMSE MIMO transmission system with rectilinear or quasi-rectilinear (QR) signals over the uncorrelated Rayleigh fading channel and investigate the statistical properties of its SINR for an arbitrary antenna configuration with $N_t$ transmit antennas and $N_r$ receive ones. We first derive an analytic probability density function (PDF) of the SINR in terms of the confluent hypergeometric function of the second kind, for WLMMSE MIMO systems with an arbitrary $N_r$ and $N_t=2, 3$. For a more general case in practice, i.e., $N_t>3$, we resort to the moment generating function to obtain an approximate but closed form PDF under some mild conditions, which, as expected, is more Gaussian-like as $2N_r-N_t$ increases. The so-derived PDFs are able to provide key insights into the WLMMSE MIMO receiver in terms of the outage probability, the symbol error rate, and the diversity gain, all presented in closed form. In particular, its diversity gain and the gain improvement over the conventional LMMSE one are explicitly quantified as $N_r-(N_t-1)/2$ and $(N_t-1)/2$, respectively. Finally, Monte Carlo simulations support the analysis.

Transition of fluctuations from Gaussian state to turbulent state.

Variation of the statistical properties of an incompressible velocity, passive vector and passive scalar in isotropic turbulence was studied using direct numerical simulation. The structure functions of the gradients, and the moments of the dissipation rates, began to increase at about from the Gaussian state and grew rapidly at in the turbulent state. A contour map of the probability density functions (PDFs) indicated that PDF expansion of the gradients of the passive vector and passive scalar begins at around , whereas that of the longitudinal velocity gradient PDF is more gradual. The left tails of the dissipation rate PDF were found to follow a power law with an exponent of 3/2 for the incompressible velocity and passive vector dissipation rates, and 1/2 for the scalar dissipation rate and the enstrophy; they remained constant for all Reynolds numbers, indicating the universality of the left tail. The analytical PDFs of the dissipation rates and enstrophy of the Gaussian state were obtained and found to be the Gamma distribution. It was shown that the number of terms contributing to the dissipation rates and the enstrophy determines the decay rates of the two PDFs for low to moderate amplitudes.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

An Asymptotically MSE-Optimal Estimator Based on Gaussian Mixture Models

This paper investigates a channel estimator based on Gaussian mixture models (GMMs) in the context of linear inverse problems with additive Gaussian noise. We fit a GMM to given channel samples to obtain an analytic probability density function (PDF) which approximates the true channel PDF. Then, a conditional mean estimator (CME) corresponding to this approximating PDF is computed in closed form and used as an approximation of the optimal CME based on the true channel PDF. This optimal CME cannot be calculated analytically because the true channel PDF is generally unknown. We present mild conditions which allow us to prove the convergence of the GMM-based CME to the optimal CME as the number of GMM components is increased. Additionally, we investigate the estimator's computational complexity and present simplifications based on common model-based insights. Further, we study the estimator's behavior in numerical experiments including multiple-input multiple-output (MIMO) and wideband systems.

Analytical prediction of low-frequency fluctuations inside a one-dimensional shock

Linear instability of high-speed boundary layers is routinely examined assuming quiescent edge conditions, without reference to the internal structure of shocks or to instabilities potentially generated in them. Our recent work has shown that the kinetically modeled internal nonequilibrium zone of straight shocks away from solid boundaries exhibits low-frequency molecular fluctuations. The presence of the dominant low frequencies observed using the direct simulation Monte Carlo (DSMC) method has been explained as a consequence of the well-known bimodal probability density function (PDF) of the energy of particles inside a shock. Here, PDFs of particle energies are derived in the upstream and downstream equilibrium regions, as well as inside shocks, and it is shown for the first time that they have the form of the noncentral Chi-squared (NCCS) distributions. A linear correlation is proposed to relate the change in the shape of the analytical PDFs at a specified upstream number density and temperature as a function of Mach number, within the range \(3 \le M \le 10\), with the DSMC-derived average characteristic low-frequency of shocks, as computed in our earlier work. At a given Mach number \(M=7.2\) and upstream number density \(n_1=10^{22}\,\hbox {m}^{-3}\), it is shown that the variation in DSMC-derived low frequencies is correlated with the change in most-probable-speed inside shocks at the location of maximum bulk velocity gradient for upstream translational temperature in the range \(\sim 90 \le T_{tr,1}/(K) \le 1420\). Using the proposed linear functions, average low frequencies are estimated within the examined ranges of Mach number and input temperature and a semi-empirical relationship is derived to predict low-frequency oscillations in shocks. Our model can be used to provide realistic physics-based boundary conditions in receptivity and linear stability analysis studies of laminar-turbulent transition in high-speed flows.

An analytic probability density function for partially premixed flames with detailed chemistry

Laminar premixed flame profiles of methane/air free flames and strained flames at different fuel/air ratios and strain rates are analyzed using detailed chemistry with Lewis numbers equal to one. It is shown that the detailed chemistry flame profiles of progress variables CO2 + CO and H2O + H2 in canonically stretched coordinates can be fitted accurately by a slight generalization of recently proposed analytical presumed flame profiles over a wide range of fuel/air ratios through adaptation of a single model parameter. Strained flame profiles can be reproduced using an additional linear coordinate transformation, emulating the compression of the preheat zone by strain as predicted by premixed flame theory. The model parameter can alternatively be determined using only the laminar flame speeds and the fully burnt temperatures from the laminar flame calculations. The stretch factor of the coordinate transformation is proportional to cp/lambda, which drops by a factor up to 4 across the laminar flame. It is shown how the non-constant cp/lambda modifies the laminar flame probability density function (pdf) and a polynomial fit to cp/lambda as a function of the progress variable allows analytical results for the laminar flame pdf and the mean value of the progress variable and of the reaction source term. An analytic pdf for partially premixed flames is proposed based on Bayes's theorem as a combination of a beta pdf for the mixture fraction and the laminar flame pdf's evaluated at the respective fuel/air ratio.

On finding the zero‐shear‐rate viscosity of polymer melts

AbstractA significant fraction of the experimental works on the rheology of polymer melts include an attempt to find the zero‐shear‐rate viscosity η0. This is done for good reasons, because η0 is a limiting property that depends only on thermodynamic variables and, importantly, the molecular and supermolecular structure of the melt. As with all limiting properties, η0 is impossible to measure directly. Fortunately with many melts, it can be estimated from viscosity measurements at very low shear rates or frequencies, but still remains one of those properties that becomes in the limit very prone to error. The common approach is to use a set of frequency‐ or shear‐rate‐dependent data and extrapolate to find η0. As with any extrapolation, the major question is the function used for the extrapolation. This question is addressed in some detail in this article. The question of which function to use was discarded in favor of using a large sample of 20 equations of many functional forms. This sample of randomly chosen equations was used to generate a set of η0 values, and the statistics of this distribution were examined, in the usual fashion, by description with an analytical probability density function that gives a high probability of being a likely generator of the data. In addition, a weighted average was proposed, where the weighting factor takes into account the quality of the fit. For testing these ideas, the room temperature melts of poly(vinyl isobutyl ether), poly(isobutylene), and poly(dimethyl siloxane) were used. The η0 of the latter was reachable; for the other resins, a falling ball technique was attempted.

Estimation Performance for the Cubature Particle Filter under Nonlinear/Non-Gaussian Environments

This paper evaluates the state estimation performance for processing nonlinear/non-Gaussian systems using the cubature particle filter (CPF), which is an estimation algorithm that combines the cubature Kalman filter (CKF) and the particle filter (PF). The CPF is essentially a realization of PF where the third-degree cubature rule based on numerical integration method is adopted to approximate the proposal distribution. It is beneficial where the CKF is used to generate the importance density function in the PF framework for effectively resolving the nonlinear/non-Gaussian problems. Based on the spherical-radial transformation to generate an even number of equally weighted cubature points, the CKF uses cubature points with the same weights through the spherical-radial integration rule and employs an analytical probability density function (pdf) to capture the mean and covariance of the posterior distribution using the total probability theorem and subsequently uses the measurement to update with Bayes’ rule. It is capable of acquiring a maximum a posteriori probability estimate of the nonlinear system, and thus the importance density function can be used to approximate the true posterior density distribution. In Bayesian filtering, the nonlinear filter performs well when all conditional densities are assumed Gaussian. When applied to the nonlinear/non-Gaussian distribution systems, the CPF algorithm can remarkably improve the estimation accuracy as compared to the other particle filter-based approaches, such as the extended particle filter (EPF), and unscented particle filter (UPF), and also the Kalman filter (KF)-type approaches, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF) and CKF. Two illustrative examples are presented showing that the CPF achieves better performance as compared to the other approaches.

A model for describing phase-converted image intensity noise in digital fringe projection techniques

Digital fringe projection is a surface-profiling technique that is gaining popularity due to the increasing availability and quality of low-cost projection equipment and digital cameras. Noise in the pixel field of imaged targets induces error in the reconstructed phase and ultimately the surface profile measurement. In this paper, we present an approximate analytical probability density function for the estimated phase given an arbitrarily-correlated Gaussian pixel noise structure. This probability density function can be used to estimate the single point phase measurement uncertainty from easily obtainable pixel intensity noise statistics. We confirm the accuracy of the new model by comparing it to a Monte-Carlo simulation of the phase distribution. A complimentary graphics model is proposed which simulates the physical process of full-field phase measurement using a pin-hole camera model and three-dimensional point clouds of the measurement surface, allowing for another level of model verification.

Laplace approximation in sparse Bayesian learning for structural damage detection

The Bayesian theorem has been demonstrated as a rigorous method for uncertainty assessment and system identification. Given that damage usually occurs at limited positions in the preliminary stage of structural failure, the sparse Bayesian learning has been developed for solving the structural damage detection problem. However, in most cases an analytical posterior probability density function (PDF) of the damage index is not available due to the nonlinear relationship between the measured modal data and structural parameters. This study tackles the nonlinear problem using the Laplace approximation technique. By assuming that the item in the integration follows a Gaussian distribution, the asymptotic solution of the evidence is obtained. Consequently the most probable values of the damage index and hyper-parameters are expressed in a coupled closed form, and then solved sequentially through iterations. The effectiveness of the proposed algorithm is validated using a laboratory tested frame. As compared with other techniques, the present technique results in the analytical solutions of the damage index and hyper-parameters without using hierarchical models or numerical sampling. Consequently, the computation is more efficient.

Noise estimation for the velocity in MRI phase-contrast

The purpose of this study is to estimate the precision or statistical variability of the velocity measurements computed from MRI phase-contrast. From the analytical probability density function (PDF) of the phase in the signal we obtain the PDF of the velocity by means of an auto-convolution. This PDF allows the estimation of the precision of the velocity, important for the correct interpretation of the many parameters that are based on it. We show that for high Signal-to-Noise Ratio (SNR) voxels, the distribution is well approximated by a Gaussian distribution. On the other hand, this is not true for lower SNR voxels, where the distribution adopts a form in between the Gaussian and the uniform distributions. This was confirmed empirically. Also, knowing the PDF on a coil by coil basis it is possible to combine the data from multiple coils in an optimal way. We showed that the optimal combination reduces the resulting global variability of the velocity, in comparison with the commonly used Weighted Mean or with a SENSE reconstruction with R = 1.

A probabilistic perspective on thermodynamic parameter uncertainties: Understanding aqueous speciation of mercury

Abstract Speciation plays an important role in determining the fate and transport of metals in terrestrial surface and subsurface systems. Equilibrium speciation modeling in aqueous systems relies on thermodynamic constants (log K values) of complexes, which are subject to uncertainties. Here, using Monte Carlo (MC) simulations with Latin hypercube sampling (LHS) we systematically analyze the propagation of thermodynamic constant uncertainty through speciation modeling of an inorganic mercury-sulfide-chloride-water system. We find that seemingly small variances of the input log K normal distributions can lead to output species concentrations spanning multiple orders of magnitude, with highly skewed probability distributions. When equilibrium with mineral metacinnabar (β-HgS(s)) is neglected, the relative uncertainty of each output species is strongly positively correlated with the skewness of its concentration probability distribution, i.e., the lowest uncertainty occurs when the species concentration probability distribution is the most negatively skewed as its concentration approaches the total element concentration limit. The highest uncertainty in the identity of dominant species is located around species equivalence points. For cases where the mineral equilibrium is included, we derive analytical probability density functions for the log concentrations of all major species in the system. The mineral log K uncertainty is found to be an important contributor to all output concentration uncertainties. The analysis of combined effects of both pH and total sulfide concentration on output concentration uncertainties shows that high concentration uncertainties occur under highly sulfidic alkaline conditions and low uncertainties at low pH and sulfide concentrations. An analysis as presented here can distinguish between conditions that require a full uncertainty analysis and those for which the classical deterministic speciation modeling suffices.

Novel Methods for Generating Fox's $H$-Function Distributed Random Variables With Applications

We present two novel methods for the generation of Fox's H-function distributed random variables (RVs), which have been recently used to model fading in various wireless communication scenarios. The first proposed method is based on the use of standard normal RVs, whereas the second is based on the use of Gamma RVs. Using Monte Carlo simulations, the proposed methods are tested and are shown to provide indistinguishable results from the corresponding analytical H-function probability density functions (PDFs) for a diverse set of parameters. Moreover, as an application, simulated ergodic capacity and average bit error rate results obtained using the proposed methods are compared to analytical expressions previously obtained in the literature and perfect agreement is observed. Our results also provide valuable insights into what constraints on the H-function parameters are sufficient to guarantee valid H-function PDFs.

Statistics-Based Approach for Blind Post-Compensation of Modulator’s Imperfections and Power Amplifier Nonlinearity

Power amplifier (PA) nonlinearity and in-phase and quadrature-phase (I/Q) imbalance are major concerns for wireless transmitters. In this paper, we present a new closed-form expression for the probability density function (PDF) of I and Q components in the presence of transmitter’s impairments and propose a blind post-compensation approach for the mitigation of these impairments. These impairments include static PA nonlinearity and frequency-independent I/Q imbalance. The accuracy of the analytical PDF is evaluated using Kullback–Leibler divergence and Hellinger square distance. Simulation results show a reasonable correspondence between the derived PDF and non-parametric kernel density estimation-based PDF. After a closed-form PDF is obtained, higher order statistics-based method is used to estimate PA nonlinearity in the presence of I/Q impairments. Finally, a maximum-likelihood estimation of I/Q imbalance parameters is obtained using the analytical PDF. Simulation results show a normalized mean-squared error (NMSE) of around −40 dB and an adjacent channel power ratio of around −53 dBc, along with an error vector magnitude (EVM) of around 1%, for a 3-MHz local thermal equilibrium signal. Using laboratory measurements, an NMSE of around −35 dB and an EVM of 1.5% are achieved.

Stochastic averaging technique for SDOF strongly nonlinear systems with delayed feedback fractional-order PD controller

A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom (SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative (PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged Ito stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function (PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov (FPK) equation. Two examples of van der Pol oscillator and Rayleigh-Duffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation

Position-domain integrity risk-based ambiguity validation for the integer bootstrap estimator

Integrity monitoring for ambiguity resolution is of significance for utilizing the high-precision carrier phase differential positioning for safety–critical navigational applications. The integer bootstrap estimator can provide an analytical probability density function, which enables the precise evaluation of the integrity risk for ambiguity validation. In order to monitor the effect of unknown ambiguity bias on the integer bootstrap estimator, the position-domain integrity risk of the integer bootstrapped baseline is evaluated under the complete failure modes by using the worst-case protection principle. Furthermore, a partial ambiguity resolution method is developed in order to satisfy the predefined integrity risk requirement. Static and kinematic experiments are carried out to test the proposed method by comparing with the traditional ratio test method and the protection level-based method. The static experimental result has shown that the proposed method can achieve a significant global availability improvement by 51% at most. The kinematic result reveals that the proposed method obtains the best balance between the positioning accuracy and the continuity performance.

Analysis of wealth inequality with a random money transfer model

Increasing gap in wealth distribution is among the key issues that have been discussed worldwide in recent years. In this paper, we use the money transfer model to explain the formation of wealth distribution, by imposing two types of debt constraints, and the analytic function of wealth distribution is derived by adopting Boltzmann statistics. With a limit of individual debt, it is shown that the stationary distribution of wealth follows the exponential law, which is verified by many empirical studies. While the limit is imposed on the total amount of bank loan, the stationary distribution becomes an asymmetric Laplace one. Furthermore, an excellent agreement is found between these analytical probability density functions and numerical results by simulation at the steady state.

The best fit for the observed galaxy counts-in-cell distribution function

The Sloan Digital Sky Survey (SDSS) is the first dense redshift survey encompassing a volume large enough to find the best analytic probability density function that fits the galaxy Counts-in-Cells distribution $f_V(N)$, the frequency distribution of galaxy counts in a volume $V$. Different analytic functions have been previously proposed that can account for some of the observed features of the observed frequency counts, but fail to provide an overall good fit to this important statistical descriptor of the galaxy large-scale distribution. Our goal is to find the probability density function that better fits the observed Counts-in-Cells distribution $f_V(N)$. We have made a systematic study of this function applied to several samples drawn from the SDSS. We show the effective ways to deal with incompleteness of the sample (masked data) in the calculation of $f_V(N)$. We use LasDamas simulations to estimate the errors in the calculation. We test four different distribution functions to find the best fit: the Gravitational Quasi-Equilibrium distribution, the Negative Binomial Distribution, the Log Normal distribution and the Log Normal Distribution including a bias parameter. In the two latter cases, we apply a shot-noise correction to the distributions assuming the local Poisson model. We show that the best fit for the Counts-in-Cells distribution function is provided by the Negative Binomial distribution. In addition, at large scales the Log Normal distribution modified with the inclusion of the bias term also performs a satisfactory fit of the empirical values of $f_V(N)$. Our results demonstrate that the inclusion of a bias term in the Log Normal distribution is necessary to fit the observed galaxy Count-in-Cells distribution function.