Approximate Probability Density Research Articles

Non-Gaussian noise induced stochastic resonance in FitzHugh–Nagumo neural system with time delay

In this paper, non-Gaussian noise induced stochastic resonance for the FitzHugh–Nagumo neural system with a time delay is investigated. Through the path integral method, the non-Gaussian noise is approximated as a colored noise, and according to the unified colored noise theory and the method of probability density approximation, a stochastic differential equation with a Markovian property is obtained. Then, by applying the two-state theory, the expression of the signal-to-noise ratio (SNR) is derived. Finally, the effects of non-Gaussian noise and time delay parameters in the neural system on the SNR are discussed with the help of analytical results.

Bayesian parameter estimation for the SWIFT model of eye-movement control during reading

Process-oriented theories of cognition must be evaluated against time-ordered observations. Here we present a representative example for data assimilation of the SWIFT model, a dynamical model of the control of fixation positions and fixation durations during natural reading of single sentences. First, we develop and test an approximate likelihood function of the model, which is a combination of a spatial, pseudo-marginal likelihood and a temporal likelihood obtained by probability density approximation Second, we implement a Bayesian approach to parameter inference using an adaptive Markov chain Monte Carlo procedure. Our results indicate that model parameters can be estimated reliably for individual subjects. We conclude that approximative Bayesian inference represents a considerable step forward for computational models of eye-movement control, where modeling of individual data on the basis of process-based dynamic models has not been possible so far.

An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

Abstract In this study, a new method to obtain approximate probability density function (pdf) of random variable of solution of stochastic differential equations (SDEs) by using generalized entropy optimization methods (GEOM) is developed. By starting given statistical data and Euler–Maruyama (EM) method approximating SDE are constructed several trajectories of SDEs. The constructed trajectories allow to obtain random variable according to the fixed time. An application of the newly developed method includes SDE model fitting on weekly closing prices of Honda Motor Company stock data between 02 July 2018 and 25 March 2019.

RESEARCH OF DIAGNOSTIC PARAMETERS OF COMPOSITE MATERIALS USING JOHNSON DISTRIBUTION

In this paper, it was proposed to carry out a preliminary normalization of diagnostic parameters using the Johnson distribution, which with three basic distribution groups (SL, SB, SU), covers a wide class of empirical distributions. The mathematical description of the family allows us to find the approximating probability density function in an explicit form, to determine the distribution parameters for obtaining the corresponding function (curve), as well as the inverse function for finding the quantiles of the specified levels. To assess the accuracy of the obtained normalized data, they were compared with the data obtained by replacing the resulting law with a Gaussian one. Percentages of values were compared in the implementation under study, which concentrated in the limits of estimated quantiles. Implementations were obtained using the simulation method. By the same method, the correctness (relative systematic error) of determining the quantile values of the specified levels was evaluated. The error value δ was estimated between the conditionally true quantile value calculated from the generated pseudo-general complex and the value estimated using the methods considered in the paper. Obtained data show that the relative error in the calculation of quantiles using the Johnson distribution does not exceed 0.07% and decreases in two orders of magnitude than the currently accepted procedure for replacing sample laws with Gaussian.

Reliability of a class of nonlinear systems under switching random excitations

Systems subjected to switching random excitations are practically significant because they include many safety-critical systems such as power plants and communication networks. In this paper, the reliability of multi-degree, nonlinear, non-integrable Hamiltonian systems subjected to switching random excitations is investigated. Such a system is formulated as a continuous–discrete hybrid based upon the Markov jump theory. Stochastic averaging is applied to suppress the rapidly varying parameters of the Markov jump process in order to generate a probability-weighted diffusion equation. The associated backward Kolmogorov equation is then set up, from which the approximate reliability function and probability density of first passage time are obtained. The utility and accuracy of this approximate procedure are demonstrated by two examples.

Secrecy Analysis in DF Relay Over Generalized-$K$ Fading Channels

In this paper, we analyze the secrecy performance of the decode-and-forward (DF) relay system in generalized- $K$ fading channels. In a typical four-node communications model, a source ( $S$ ) sends confidential information to a destination ( $D$ ) via a relay ( $R$ ) using DF strategy in two time slots, while an eavesdropper ( $E$ ) wants to overhear the information from $S$ to $D$ over generalized- $K$ fading channels. To be more realistic, we assume that $E$ can receive the signals of two time slots, and there is no direct link between $S$ and $D$ because of heavy fading. Based on those assumptions, we derive closed-form expressions for the secrecy outage probability (SOP) and ergodic secrecy capacity (ESC) by using a tight approximate probability density function of the generalized- $K$ model. Furthermore, asymptotic expressions for the SOP and ESC are also derived in the high signal-to-noise ratio region, not only because we can get some insights about SOP and ESC, but also because expressions for SOP and ESC can be simplified significantly. The single relay system is subsequently extended into a multi-relay system, where the asymptotic SOP analysis of three proposed relay selection strategies is investigated. Moreover, the security-reliability tradeoff analysis in the multi-relay system is also presented given that $S$ adopts a constant code rate. Finally, the Monte-Carlo simulation is used to demonstrate the accuracy of the derived closed-form expressions.

New approximate and asymptotic closed-form expressions for the outage probability and the average BER of MIMO-FSO system with MRC diversity technique over Gamma–Gamma fading channels with generalized pointing errors

This paper studies the outage probability (OP) and the average bit-error rate (BER) of multiple-input multiple-output (MIMO) free-space optical (FSO) system with maximal ratio combining (MRC) diversity technique over Gamma–Gamma (GG) fading channels with generalized pointing errors modeled by the Beckmann distribution. A new approximate closed-form probability density function (PDF) at high signal-to-noise ratio (SNR) for MRC scheme is derived by inverse Laplace transformation method. Based on the developed PDF, new approximate and asymptotic closed-form expressions for the OP and the average BER are derived, respectively. Then, the OP and the average BER performance of the MIMO-FSO system with MRC scheme at high SNR are analyzed with different parameters, including number of transmitters and receivers, receiver aperture sizes, nonzero boresight, and different jitters. Finally, numerical results and Monte Carlo simulations corroborate the accuracy of the derived approximate and asymptotic expressions over a wide range of SNR.

Parallel probability density approximation.

Probability density approximation (PDA) is a nonparametric method of calculating probability densities. When integrated into Bayesian estimation, it allows researchers to fit psychological processes for which analytic probability functions are unavailable, significantly expanding the scope of theories that can be quantitatively tested. PDA is, however, computationally intensive, requiring large numbers of Monte Carlo simulations in order to attain good precision. We introduce Parallel PDA (pPDA), a highly efficient implementation of this method utilizing the Armadillo C++ and CUDA C libraries to conduct millions of model simulations simultaneously in graphics processing units (GPUs). This approach provides a practical solution for rapidly approximating probability densities with high precision. In addition to demonstrating this method, we fit a piecewise linear ballistic accumulator model (Holmes, Trueblood, & Heathcote, 2016) to empirical data. Finally, we conducted simulation studies to investigate various issues associated with PDA and provide guidelines for pPDA applications to other complex cognitive models.

Asymptotic capacity analysis under different adaptive transmission for TAS/MRC MIMO scheme subject to Weibull fading channels

This paper investigates a unified average channel capacity (ACC) analysis under different adaptive transmission of multiple-input multiple-output (MIMO) system, combining single transmit antenna selection (TAS) and maximal-ratio combining (MRC) receiver operating under independent flat Weibull multipath fading channels (WFCs). Based on a tight approximate probability density function expression of the signal-to-noise ratio (SNR) at the considered receiver output, we derive new accurate closed-form expressions of ACC for TAS/MRC system employing several rate adaptation techniques, namely optimal rate adaptation, optimal simultaneous power and rate adaptation, channel inversion with fixed rate (CIFR), truncated CIFR (TCIFR), and continuous power TCIFR (CTCIFR). Further, asymptotic expressions, in both high and low-SNR regimes, are investigated. The results show high accuracy for significant values of Lt×Lr TAS/MRC system. Indeed, the smaller Lt×Lr, the better the PDF’s accuracy, therefore, the better are the capacities. Numerical outcomes have been assessed by utilizing Mathematica and Matlab Software and corroborated via Monte-Carlo simulations.

Performance analysis of transmit antenna selection–maximal ratio combining relay systems over non‐identical channel estimation error

This study investigates the performance analysis of a transmit antenna selection–maximal ratio combining relay system over non-identical channel estimation error. The proposed system selects both relay and user terminal with a maximum signal-to-noise ratio (SNR) using feedback information. The exact closed-form expression of the proposed system for the outage probability and symbol error rate (SER) is derived in the presence of non-identical channel estimation error. Using the approximated probability density function, the authors quantify the diversity order for the SER and outage probability. From the above results, the diversity order of the proposed system is proportional to the number of relay nodes and user terminals. Thus, the diversity order of the proposed system is superior to that of the previous selection systems. The diversity order of the proposed system is independent of the number of receive antennas at the relay and user terminal and the channel correlation coefficient. Furthermore, the SNR gain of the proposed system is in proportion to the sum of the channel correlation coefficients and the number of receive antennas at the relay and user terminal.

Analysis of Massive MIMO With Low-Resolution ADC in Nakagami-${m}$ Fading

The uplink of multi-user massive multi-input-multi-output systems, in which the base station is equipped with a low-resolution analog-to-digital converter architecture, is considered in this letter. All the channels are assumed to undergo independent and nonidentically distributed Nakagami- $m$ fading. By means of moment matching, the signal-to-interference-plus-quantization-noise ratio at the output of maximum ratio combiner is approximated by a gamma random variable. Using this approximate probability density function, tight approximations for metrics, such as outage probability and rate, are derived. The gap between the simulation and the approximate result is negligible. In addition, for a large number of antennas, a decrease in the number of antennas should be accompanied by a proportionate decrease in the threshold to maintain the same outage probability.

Average SEP and channel capacity analysis over Generic/IG composite fading channels: A unified approach

This paper develops an analytical framework to analyse important performance metrics of the wireless communication system over generic fading channels characterized by α−μ/Inverse Gaussian (IG), η−μ/IG, and κ−μ/IG distributions. The important performance metrics, namely, the outage probability (OP), the amount of fading (AOF), the average symbol error probability (SEP) of coherent and non-coherent schemes, the channel capacity under optimal rate adaptation (ORA), channel inversion with fixed rate (CIFR), and truncated CIFR (TCIFR) policy are analysed. Further, low and high signal-to-noise-ratio (SNR) approximations for channel capacity are presented. Furthermore, for asymptotic analysis the approximate composite probability density function (PDF) at the origin was developed and utilized to derive the closed-form expressions for the average SEP of both coherent and non-coherent modulation schemes with maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC) diversity schemes. The proposed asymptotic solutions are simple and can be easily used to describe the system behaviour at high SNR. The correctness of the derived results has been verified by comparing them with Monte Carlo simulations.

Stochastic Constitutive Model of Isotropic Thin Fiber Networks Based on Stochastic Volume Elements.

Thin fiber networks are widely represented in nature and can be found in man-made materials such as paper and packaging. The strength of such materials is an intricate subject due to inherited randomness and size-dependencies. Direct fiber-level numerical simulations can provide insights into the role of the constitutive components of such networks, their morphology, and arrangements on the strength of the products made of them. However, direct mechanical simulation of randomly generated large and thin fiber networks is characterized by overwhelming computational costs. Herein, a stochastic constitutive model for predicting the random mechanical response of isotropic thin fiber networks of arbitrary size is presented. The model is based on stochastic volume elements (SVEs) with SVE size-specific deterministic and stochastic constitutive law parameters. The randomness in the network is described by the spatial fields of the uniaxial strain and strength to failure, formulated using multivariate kernel functions and approximate univariate probability density functions. The proposed stochastic continuum approach shows good agreement when compared to direct numerical simulation with respect to mechanical response. Furthermore, strain localization patterns matched the one observed in direct simulations, which suggests an accurate prediction of the failure location. This work demonstrates that the proposed stochastic constitutive model can be used to predict the response of random isotropic fiber networks of arbitrary size.

Novel approximate and asymptotic expressions of the outage probability and BER in gamma–gamma fading FSO links with generalized pointing errors

Abstract In this paper, a novel closed-form approximate probability density function (PDF) at high signal-to-noise ratio (SNR) for the composite gamma–gamma (GG) atmospheric turbulence with a generalized pointing error model using the Beckmann distribution is derived in terms of an appropriate gamma distribution by a moment generating function (MGF) matching approach. Based on the obtained PDF, novel closed-form approximate and asymptotic expressions for the outage probability (OP) and the average bit-error rate (BER) are derived. Then, the average BER performance and the outage performance of the considered FSO system at high SNR are investigated with different turbulence conditions, receiver aperture sizes, nonzero boresight, and different jitter variances. Numerical results and Monte Carlo simulation results verify the accuracy of our derived approximate and asymptotic expressions over a wider SNR range. This work can be further extended as well to diversity links.

Randomized mixture models for probability density approximation and estimation

Neural networks (NNs) with random weights are an interesting alternative to conventional NNs that are used more for data modeling. The random vector functional-link (RVFL) network is an established and theoretically well-grounded randomized learner. A key theoretical result for RVFL networks is that they provide universal approximation for continuous maps, in expectation, with respect to the square-integral norm. We specialize and modify this result, and show that RFVL networks can provide functional approximations that converge in Kullback–Leibler divergence, when the target function is a probability density function. Expanding on the approximation results, we demonstrate the RFVL networks lead to a simple randomized mixture model (MM) construction for density estimation from sample data. An expectation–maximization (EM) algorithm is derived for the maximum likelihood estimation of our randomized MM. The EM algorithm is proved to be globally convergent and the maximum likelihood estimator is proved to be consistent. A set of simulation studies is given to provide empirical evidence towards our approximation and density estimation results.

Bit Error Probability for MMSE Receiver in GFDM Systems

In this letter, we consider the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels. We derive an approximate probability density function for the signal-to-interference-plus-noise ratio. This expression allows us to obtain a new approximate, but rather accurate formulation for the bit error probability for a $\mathcal {M}$ -quadrature amplitude modulation scheme. Our results resort on the pivotal properties exhibited by eigenvalues of a circulant matrix. Since the entries of the channel matrix $\text {H}_{\text {ch}}$ are complex Gaussian distributed, and the eigenvalues are given as a weighted sum of its entries, the joint eigenvalue distribution is also Gaussian. Comparisons of the simulated and analytical results validate our formulation and allow a quick and efficient tool to compute the bit error rate for the GFDM system.

Probabilistic solution of the vibratory energy harvester excited by Gaussian white noise

This paper aims to investigate the statistical characteristics of strongly nonlinear vibratory energy harvesters under Gaussian white noise excitation. The high-dimensional Fokker–Planck–Kolmogorov (FPK) equation of the coupled electromechanical system is reduced to a low-dimensional equation by using the state-space-split method. The conditional moment given by the equivalent linearization method is employed to decouple the FPK equations of coupled system, and then obtained an equivalent nonlinear uncoupled subsystem. The exact stationary solution of the reduced FPK equation of the subsystem is established. The mean output power is derived by the second order conditional moment from the associated approximate probability density function of mechanical subsystem. The procedure is applied to mono- and bi-stable energy harvesters. Effectiveness of the probability density function of the proposed approach is examined via comparison with equivalent linearization method and Monte Carlo simulation. The effects of the system parameters on the mean-square displacement and the mean output power are discussed. The approximate analytical outcomes are qualitatively and quantitatively supported by the numerical simulations.

Performance Analysis of Dual Selection With Maximal Ratio Combining Over Nonidentical Imperfect Channel Estimation

This paper investigates the performance analysis of dual selection (i.e., combining user selection and transmit antenna selection) with maximal ratio combining (MRC) over nonidentical imperfect channel estimation in a multiple-input multiple-output system. This paper derives the exact and approximate cumulative density function and probability density function (PDF) for the dual selection with MRC over nonidentical imperfect channel estimation. Using the distribution, the exact and approximate closed-form expressions of the outage probability are derived with quantification of diversity order. Further, this paper derived the exact ergodic capacity and approximate outage capacity of the the dual selection via Gaussian and Gamma approximations. The numerical results show that the capacity of the dual selection with MRC is enhanced by the number of receive antennas as well as the number of user terminals and sum of channel correlation coefficients, while the order of the receive spatial diversity for outage probability is eliminated by the nonidentical channel estimation error.

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation.

We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method.

Applying Hankel’s Expansion for Performance Analysis in Double-Nakagami (Generalized-K) Fading Channels

The double-scattering fading models have found applications in many transmission scenarios. In this paper, approximate probability density functions (PDFs) are proposed for double-Nakagami fading and Gamma–Gamma fading, based on Hankel’s expansion of the modified Bessel function of the second kind. The proposed PDF is not only consistent with the exact PDF when the difference between the two shape parameters is equal to an integer plus one half, but also accurately approximates the distribution when the shape parameters are both greater than or equal to 2. The proposed approximate PDF has been compared with the power series expansion and the mixture Gamma model. It is found that the proposed PDF is in simple form and no parameters need to be tabulated. Moreover, the proposed PDF does not require an increasing number of terms to maintain precision with increasing diversity order of the reception scheme. Closed-form outage probability, ergodic capacity, and symbol error rate expressions have been derived using the proposed approximate PDFs for equal-gain combining, selection combining, and two-stage reception schemes. The proposed approximations are also leveraged for performance analysis of space-time block codes and free-space optical links. The precision of the proposed approximations is validated by comparing with the simulation results.