Marginal Probability Density Research Articles

Unbiased density computation for stochastic resetting *

We establish a practical means for unbiased computation of the marginal probability density function of the dynamics under stochastic resetting. In contrast to conventional dynamics devoid of resetting, the marginal probability density function under resetting may exhibit cusps and, in multi-dimensions, explosions at reset positions, arising from the compelled displacement of mass. Standard numerical techniques, such as kernel density estimation, fall short in accurately reproducing those characteristics due to their inherent smoothing effects. The proposed unbiased estimation formulas are derived using advanced stochastic calculus in their general formulations, yet their implementation in specific problem settings involves only elementary numerical operations, requiring minimal mathematical expertise and marking the very first instance of a numerical method free from bias in this context. We present numerical results throughout to validate the derived estimation formulas and, more broadly, to demonstrate the effectiveness of our approach in accurately capturing the irregularities of the marginal probability density function.

Novel gradient-enhanced Bayesian neural networks for uncertainty propagation

Uncertainty propagation (UP) is crucial for assessing the impact of input uncertainty on structural responses, holding significant importance in engineering applications. However, achieving accurate and efficient UP remains challenging, especially for highly nonlinear structures. Bayesian neural networks (BNN) have gained attention for addressing UP issues, yet current BNN models only utilize input samples and corresponding structural responses for training. However, incorporating gradients of structural responses with respect to input samples provides valuable information. This study proposes a novel approach called gradient-enhanced Bayesian neural networks (GEBNN) to tackle UP tasks. In the GEBNN, a modified evidence lower bound (MELBO) loss is developed to consider both structural responses and gradient information simultaneously. This includes disparities between actual and predicted responses, as well as disparities between actual and predicted derivatives. Additionally, a gradient screening strategy based on the marginal probability density functions (PDFs) of input samples is established to identify significant derivative data for GEBNN training. Once the GEBNN is configured, it is utilized to replace the computationally intensive finite element model to efficiently provide UP results. Various applications, including nonlinear numerical examples, and mechanical, civil, and aeronautical structures, are presented to demonstrate the effectiveness of the GEBNN. The results show that the GEBNN significantly enhances the computational accuracy of the BNN in solving UP tasks.

A Gaussian mixture multiple-model belief propagation filter for multisensor-multitarget tracking

This paper presents a novel Gaussian mixture multi-model belief propagation (GMM-BP) filter for maneuvering multitarget tracking with multiple sensors. The filter is built upon the BP-based multisensor-multitarget tracking scheme, enabling accurate estimation of target numbers and states. It assumes linear Gaussian target motion, birth process, and sensor measurement models and utilizes the Gaussian mixture model to represent the target's marginal posterior probability density function. Additionally, multiple models are incorporated for target motion, enhancing the maneuvering target tracking capability. Closed-form recursions for means, covariances, and weights of Gaussian components in the filter are derived. To enhance scalability with the number of sensors, a simplified multi-sensor data fusion process is proposed, thereby preserving a linear relationship between Gaussian components and the number of sensors. The performance of the filter was evaluated in a three-dimensional (3D) simulated environment. Compared to the particle-implemented BP filter and the labeled multi-Bernoulli (LMB) filter, the proposed GMM-BP filter exhibited a significant improvement in computational efficiency while maintaining a high level of tracking accuracy.

Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems

This paper extends the recently developed method of separable Gaussian neural networks (SGNN) to obtain solutions of the Fokker–Planck–Kolmogorov (FPK) equation in high-dimensional state space. Several challenges when extending SGNN to high-dimensional state space are addressed including proper definition of domain for placing Gaussian neurons and region for data sampling, and numerical integration issue of evaluating marginal probability density functions. Three benchmark nonlinear dynamic systems with increasing complexity and dimension are examined with the SGNN method. In particular, the steady-state probability density of the response is obtained with the SGNN method and compared with the results of extensive Monte Carlo simulations. It should be pointed out that some solutions of high-dimensional FPK equations for nonlinear dynamic systems would be very difficult to obtain without SGNN.

Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations

This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

Periodicity-assist double delay-controlled stochastic resonance for the fault detection of bearings

This paper analyses the stochastic resonance of a double delay-controlled nonlinear system based on the high-order-degradation potential function. It provides a theoretical derivation of the stationary probability density, mean first-passage time (MFPT), and signal-to-noise ratio (SNR), while the marginal probability density is examined via Monte-Carlo simulation. In the field of fault detection for faulty bearings, the stochastic resonance of the nonlinear system encounters a challenge where the exact fault period must be known beforehand. Unfortunately, it is not available for the real engineering application. Consequently, this paper proposes the periodicity-assisted stochastic resonance method, which uses the blind deconvolution technique. The proposed method allows for adaptive determination of the fault frequency from the vibration signal and successfully enhances the power of the signal component corresponding to the determined fault frequency. Compared with the state-of-the-art methods, the proposed method shows better performance, which is validated by the simulated and experimental data. Overall, the proposed method is advantageous for enhancing the weak fault characteristic which is interfered by the strong noise and random impulses.

Multipaths’ statistics for scatterers with inverted elliptic–parabolic spatial density around the mobile

The joint and marginal probability densities of multipaths’ angles-of-arrival (AOA) and times-of-arrival (TOA) at the cellular base station are developed in closed form in this paper. Unlike the general simplification assumption in the open literature in which the scatterers are assumed to be located in a circular region for non-uniform spatial densities, the scatterers in this paper are assumed to be located in an elliptical region to properly model the elliptical footprint around the mobile station from the elevated base station with directional antenna. The inverted elliptic–parabolic spatial density was adopted to model the non-uniform distribution of the scatterers around the mobile. The uplink’s AOA–TOA joint distributions, AOA and TOA marginal distributions were analytically derived in closed form. How the eccentricity of the elliptical scatterer region affects the probability density functions is discussed. Furthermore, the derived AOA marginal distribution is compared to that of the elliptic conic and inverted parabolic models. The proposed model is shown to have better least-squares fit to some empirical AOA data compared to the elliptic conic and inverted parabolic models.

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time . If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density . The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location and the amount of time spent at the origin. The continuum limit involves rescaling by a factor , where is the lattice spacing. In the limit , the rescaled functional becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

Encounter-based reaction-subdiffusion model II: partially absorbing traps and the occupation time propagator

In this paper we develop an encounter-based model of reaction-subdiffusion in a domain Ω with a partially absorbing interior trap . We assume that the particle can freely enter and exit , but is only absorbed within . We take the probability of absorption to depend on the amount of time a particle spends within the trap, which is specified by a Brownian functional known as the occupation time A(t). The first passage time (FPT) for absorption is identified with the point at which the occupation time crosses a random threshold with probability density . Non-Markovian models of absorption can then be incorporated by taking to be non-exponential. The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the occupation time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. However, care must be taken when combining fractional diffusion with chemical reactions in the same medium. Therefore, we derive the occupation time propagator equation from first principles by taking the continuum limit of a heavy-tailed continuous-time random walk. We then use the solution of the propagator equation to investigate conditions under which the mean FPT for absorption within a trap is finite. We show that this depends on the choice of threshold density and the subdiffusivity. Hence, as previously found for evanescent reaction-subdiffusion models, the processes of subdiffusion and absorption are intermingled.

Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting

The Ornstein–Uhlenbeck (O-U) process with resetting is considered as the anomalous transport taking place on a three-dimensional comb. The three-dimensional comb is a comb inside a comb structure, consisting of backbones and fingers in the following geometrical correspondence x–backbone →y–fingers–backbone →z–fingers. Realisation of the O-U process on the three-dimensional comb leads to anomalous (non-Markovian) diffusion. This specific anomalous transport in the presence of resets results in non-equilibrium stationary states. Explicit analytical expressions for the mean values and the mean squared displacements along all three directions of the comb are obtained and verified numerically. The marginal probability density functions for each direction are obtained numerically by Monte Carlo simulation of a random transport described by a system of coupled Langevin equations for the comb geometry.

Noise-induced bistability in a simple mutual inhibition system.

In this study, we study noise-induced bistability in a simple bivariate mutual inhibition system with slow fluctuating responses to external signals. We give a general condition that the marginal stationary probability density of one of the two variables experiences a transition from a unimodal shape to a bimodal one. We show that the transition occurs even when the stationary probability density of the response to external signals is monotone. The mechanism for the transition is investigated in terms of the calculation of the mean first passage time. We also discuss the genericity of the transition mechanism.

Polynomial Propagation of Moments in Stochastic Differential Equations

We address the problem of approximating the moments of the solution, , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid . In particular, we assume an explicit numerical scheme for the generation of sample paths and then obtain recursive equations that yield any desired noncentral moment of as a function of the initial condition . The core of the methodology is the decomposition of the numerical solution into a “central part” and an “effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., for some deterministic and known ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

Non-parametric simulation of random field samples from incomplete measurements using generative adversarial networks

ABSTRACT Random field theory is an effective tool for modelling spatial or temporal variability and uncertainty in natural phenomena, and it has been widely applied in many areas such as structural dynamics, geology, hydrology, and meteorology. Conventional methods of random field simulations are often parametric, in which explicit function forms for trend function, auto-correlation function, and marginal probability density function (PDF) should be selected, together with their corresponding parameters estimated from measurements. Complete random field measurements with many data points are indispensable for selection of the appropriate function forms and accurate estimates of their corresponding parameters. However, the measurements in practice are often incomplete. Without sufficient measurement data, the random field samples (RFSs) generated by conventional methods might contain unexpected uncertainty and might be misleading. To tackle this problem, a purely non-parametric random field simulation method is developed in this study that generates RFSs directly from incomplete measurement data using generative adversarial networks (GAN). Statistical analysis is performed to estimate statistical properties from the generated RFSs, including mean, standard deviation (SD), autocovariance function (AF), and cumulative density function (CDF). The results show that the proposed method properly simulates RFSs from incomplete measurement data in a purely data-driven manner.

Artificial Bee Colony Based Iterative Heuristic Signal Detection Technique for Optical Wireless Communications

In this paper, we consider a problem of optical signal detection and present a novel artificial bee colony (ABC) based heuristic approach for optical spectrum sensing in a free-space optical wireless network that overcomes channel impairment due to noise uncertainty. In particular, we propose a generic ABC-based optimization framework in conjunction with eigenvalue-based detection (EVD). The uniqueness of the proposed algorithm is that it is executed iteratively by comparing various local solutions until an optimum solution is obtained. The channel is characterized as correlated multivariate fading and the received signal is quantified as a continuous waveform wideband detector. We derive the joint statistic of the correlated multivariate channel fading in the multiple-input multiple-output (MIMO) optical network. Subsequently, the marginal probability density function of the channel coefficients is obtained. For comparison purposes with existing optical spectrum sensing techniques, noise uncertainty is considered. It is demonstrated from simulations that the proposed ABC-EVD spectrum sensing yields performance improvement and the analytical results derived are verified. From the comparative study, it is also found that the proposed scheme outperforms existing optical spectrum sensing techniques, particularly in a regime of low signal-to-noise ratio. When compared to EVD, it is shown that, at a received signal-to-noise ratio of -15 dB, the performance gain achieved in terms of the probability of detection is 15% higher over the entire range of target probability of false alarm. When compared with the ED technique, the proposed technique outperforms it with approximately 50% higher probability of detection at the noise uncertainty of 1 dB and received SNR of -10 dB. The proposed algorithm offers fast convergence, strong robustness, and high flexibility.

A statistical moment-based spectral approach to the chance-constrained stochastic optimal control of epidemic models

This paper presents a spectral approach to the uncertainty management in epidemic models through the formulation of chance-constrained stochastic optimal control problems. Specifically, a statistical moment-based polynomial expansion is used to calculate surrogate models of the stochastic state variables of the problem that allow for the efficient computation of their main statistics as well as their marginal and joint probability density functions at each time instant, which enable the uncertainty management in the epidemic model. Moreover, the surrogate models are employed to perform the corresponding sensitivity and risk analyses. The proposed methodology provides the designers of the optimal control policies with the capability to increase the predictability of the outcomes by adding suitable chance constraints to the epidemic model and formulating a proper cost functional. The chance-constrained optimal control of a COVID-19 epidemic model is considered in order to illustrate the practical application of the proposed methodology.

Effects of Mixture Distribution on the Structure and Propagation of Turbulent Stratified Slot-Jet Flames

The influence of mixture stratification on the development of turbulent flames in a slot-jet configuration has been analysed using Direct Numerical Simulation data. Mixture stratification was imposed at the inlet by varying the equivalence ratio between 0.6 and 1.0 with different alignments to the reaction progress variable gradient: aligned gradients (back-supported), opposed gradients (front-supported) and misaligned gradients. An additional premixed case with a global equivalence ratio of 0.8 was simulated for comparison. The flame is shortest for the front-supported case, followed by the premixed flame, with the back-supported and misaligned gradient flames being the tallest and of comparable size. This behaviour has been explained in terms of the variations of the mean equivalence ratio within the flame and the volume-integrated reaction rate in the streamwise direction. The difference in mixture composition for these cases results in significant variations in the burning rate, flame area, flame wrinkling and flame brush thickness in the streamwise direction. The globally front-supported case has the highest volume-integrated burning rate and flame area, while the back-supported case has the lowest. The misaligned scalar gradient case exhibits qualitatively similar behaviour to that of the globally back-supported case. The burning intensity is unity for a major part of the flame length but assumes values greater than unity towards the flame tip where the effects of flame curvature become strong. All cases predominantly exhibit the premixed mode of combustion within the flamelet regime, so flamelet assumption-based reaction rate closures, originally proposed for premixed combustion, were evaluated using a priori analysis. The terms which require improved closures have been identified and existing closures have been improved where necessary. It was found that the global nature of mixture stratification does not influence the performance of the mean reaction rate closures or the parameterisation of marginal probability density functions of scalars in turbulent stratified mixture combustion.

Performance enhancement of snap-through vibration energy harvester with displacement amplifier

Vibration energy harvesters (VEH) based on bistable and snap-through mechanisms are being widely employed to harvest energy from different vibration sources. However, under weak ambient excitation, the performance of these systems is marginal. Multi-stability and enhancement techniques, such as displacement amplifier, are being used with bistable VEH to improve its performance. Similar to this, multi-stable snap-through VEH has also been developed; however, the usage of a displacement amplifier in the context of snap-through VEH has not yet been investigated. In this novel study, a displacement amplifier/ displacement amplification mechanism (DAM) is integrated into a snap-through VEH, and the enhanced performance under weak excitation is investigated. A generalized transformation is applied to develop a reduced-order model of the proposed system, and its dynamics and performance under harmonic and random excitation are investigated analytically and numerically. The frequency–amplitude relationship is derived under harmonic excitation, and a detailed investigation of the influence of system parameters on the frequency and force responses is performed. It is observed that by adjusting the mass and stiffness ratio between the VEH and DAM, it is possible to harvest more energy under harmonic excitation of low intensity. In the case of Gaussian white noise excitation, effective potential, and stochastic averaging methods are used to obtain marginal, joint, and stationary probability density functions and mean square power. The influence of noise intensity and other system parameters on the aforementioned measures is thoroughly investigated. It is observed that for specific parameter values, the proposed VEH can harvest 3.3 times more power under harmonic excitation and four times more power under random excitation compared to the snap-through VEH without DAM.

Application of wastewater-based surveillance and copula time-series model for COVID-19 forecasts

The objective of this study was to develop a novel copula-based time series (CTS) model to forecast COVID-19 cases and trends based on wastewater SARS-CoV-2 viral load and clinical variables. Wastewater samples were collected from wastewater pumping stations in five sewersheds in the City of Chesapeake VA. Wastewater SARS-CoV-2 viral load was measured using reverse transcription droplet digital PCR (RT-ddPCR). The clinical dataset included daily COVID-19 reported cases, hospitalization cases, and death cases. The CTS model development included two steps: an autoregressive moving average (ARMA) model for time series analysis (step I), and an integration of ARMA and a copula function for marginal regression analysis (step II). Poisson and negative binomial marginal probability densities for copula functions were used to determine the forecasting capacity of the CTS model for COVID-19 forecasts in the same geographical area. The dynamic trends predicted by the CTS model were well suited to the trend of the reported cases as the forecasted cases from the CTS model fell within the 99 % confidence interval of the reported cases. Wastewater SARS CoV-2 viral load served as a reliable predictor for forecasting COVID-19 cases. The CTS model provided robust modeling to predict COVID-19 cases.

Chaotic diffusion of the fundamental frequencies in the Solar System (Corrigendum)

The long-term variations in the orbit of the Earth govern the insolation on its surface and hence its climate. The use of the astronomical signal, whose imprint has been recovered in the geological records, has revolutionized the determination of the geological timescales. However, the orbital variations beyond 60 Myr cannot be reliably predicted because of the chaotic dynamics of the planetary orbits in the Solar System. Taking this dynamical uncertainty into account is necessary for a complete astronomical calibration of geological records. Our work addresses this problem with a statistical analysis of 120 000 orbital solutions of the secular model of the Solar System ranging from 500 Myr to 5 Gyr. We obtain the marginal probability density functions of the fundamental secular frequencies using kernel density estimation. The uncertainty of the density estimation is also obtained here in the form of confidence intervals determined by the moving block bootstrap method. The results of the secular model are shown to be in good agreement with those of the direct integrations of a comprehensive model of the Solar System. Application of our work is illustrated on two geological data sets: the Newark-Hartford records and the Libsack core.

A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data

The primary objective of this article was to introduce a new probabilistic model for the discussion and analysis of random covariates. The introduced model was derived based on the Marshall–Olkin shock model. After proposing the mathematical form of the new bivariate model, some of its distributional properties, including joint probability distribution, joint reliability distribution, joint reversed (hazard) rate distribution, marginal probability density function, conditional probability density function, moments, and distributions for both Y=max{X1,X2} and W=min{X1,X2}, were investigated. This novel model can be applied to discuss and evaluate symmetric and asymmetric data under various kinds of dispersion. Moreover, it can be used as a probability approach to analyze different shapes of hazard rates. The maximum likelihood approach was utilized for estimating the parameters of the bivariate model. A simulation study was carried out to assess the performance of the parameters, and it was noted that the maximum likelihood technique can be used to generate consistent estimators. Finally, two real datasets were analyzed to illustrate the notability of the novel bivariate distribution, and it was found that the suggested distribution provided a better fit than the competitive bivariate models.