Marginal Probability Density Research Articles

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.

Statistical property analysis for a stochastic chemostat model with degenerate diffusion

<abstract><p>By considering the fact that the growth of microorganisms in a chemostat is subject to white noise, we construct a stochastic chemostat model with degenerate diffusion by using a discrete Markov chain. By solving the corresponding Fokker-Planck equation, we derive the explicit expression of the stationary joint probability density, which peaks near the deterministic equilibrium. Next, we simulate the the marginal probability density functions for different noise intensities and further discuss the relationship of the marginal probability density function and noise intensities. For the statistical properties of the stochastic model, we mainly investigate the effect of white noise on the variance and skewness of the concentration of microorganisms.</p></abstract>

Analysis of a stochastic SIB cholera model with saturation recovery rate and Ornstein-Uhlenbeck process.

In this paper, a stochastic SIB(Susceptible-Infected-Vibrios) cholera model with saturation recovery rate and Ornstein-Uhlenbeck process is investigated. It is proved that there is a unique global solution for any initial value of the model. Furthermore, the sufficient criterion of the stationary distribution of the model is obtained by constructing a suitable Lyapunov function, and the expression of probability density function is calculated by the same condition. The correctness of the theoretical results is verified by numerical simulation, and the specific expression of the marginal probability density function is obtained.

A stochastic model of grain boundary dynamics: A Fokker–Planck perspective

Many technologically useful materials are polycrystals composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. The energetics and connectivities of the grain boundaries play an essential role in defining the effective properties of materials across multiple scales. In this paper we derive a Fokker–Planck model for the evolution of the planar grain boundary network. The proposed model considers anisotropic grain boundary energy which depends on lattice misorientation and takes into account mobility of the triple junctions, as well as independent dynamics of the misorientations. We establish long time asymptotics of the Fokker–Planck solution, namely the joint probability density function of misorientations and triple junctions, and closely related the marginal probability density of misorientations. Moreover, for an equilibrium configuration of a boundary network, we derive explicit local algebraic relations, a generalized Herring Condition formula, as well as formula that connects grain boundary energy density with the geometry of the grain boundaries that share a triple junction. Although the stochastic model neglects the explicit interactions and correlations among triple junctions, the considered specific form of the noise, under the fluctuation–dissipation assumption, provides partial information about evolution of a grain boundary network, and is consistent with presented results of extensive grain growth simulations.

Cooperative Localization Based on Augmented State Belief Propagation for Mobile Agent Networks

Belief propagation (BP) is widely used to solve the cooperative localization problem due to its excellent performance and natural distributed structure of implementation. For a mobile agent network, its factor graph inevitably encounters loops. In this case, the BP algorithm becomes iterative and can only provide an approximate marginal probability density function of the estimate with finite iterations. We propose an augmented-state BP algorithm for mobile agent networks to alleviate the effect of loops. By performing state augmentation, the messages in the factor graph will actually be allowed to be backward propagated, which reduces the number of loops in the factor graph, increases the available information of agents, and thus, benefits the localization. Experimental results demonstrate the better performance of the proposed algorithm over the original BP method.

Tempered stable autoregressive models

In this article, we introduce and study a one sided tempered stable first order autoregressive model called TAR(1). Under the assumption of stationarity of the model, the marginal probability density function of the error term is found. It is shown that the distribution of the error term is infinitely divisible. Parameter estimation of the introduced TAR(1) process is done by adopting the conditional least square and method of moments based approach and the performance of the proposed methods are evaluated on simulated data. Also, we study an autoregressive model of order one with tempered stable innovations. Using appropriate test statistic it is shown that the model fits very well on real and simulated data. Our models generalize the inverse Gaussian and one-sided stable autoregressive models existing in the literature.

Unsupervised deep learning method for bridge condition assessment based on intra-and inter-class probabilistic correlations of quasi-static responses

Data-driven methods for structural condition assessment have been extensively investigated using deep learning (DL). However, studies on quasi-static response data-based structural health diagnoses are relatively insufficient. The difficulty is that quasi-static response data contain coupled effects of structural parameters and external loads. Considering that the correlation between quasi-static responses subjected to identical external loads is only a function of structural parameters and independent from the external loads, the correlation can therefore be employed as an indicator of the structural condition. This study proposes a condition assessment approach for cable-stayed bridges based on correlation modeling between the deflection of girders and tension in cables. The correlation is modeled by an unsupervised DL network comprising two variational autoencoders (AE) and two generative adversarial networks (GANs). The input and output are marginal probability density functions (PDFs). The DL network is trained as the reconstruction and translation processes to model the intra-class and inter-class correlations. Assumptions of shared latent space and cycle consistency are taken to ensure mutual modeling capacity. The Wasserstein distance between the predicted and ground-truth PDFs of tension in cables is used as an indicator of the structural condition. Using probabilistic correlation of quasi-static responses only requires the PDF of external loads to be identical and does not need the external loads to be precisely identical at any moment, thus relieving time-synchronization restrictions for different sensors. The results show that the predicted PDFs agree well with the ground-truth values under normal conditions. Furthermore, the Wasserstein distance is sensitive to damage and shows noticeable variations when the damage of the stay cable occurs.

Hybrid nested sampling method for identifying the uncertainty of the high-dimensional updating parameters in Bayesian structural model updating

Bayesian inference is a practical and straightforward approach to quantifying the uncertainty of the model parameters in structural finite element model updating. Sampling methods are frequently used to estimate the uncertainty of the selected updating parameters in a statistically principled way. Generally, uncertainty can be described by global optimum, local optimum, expectation, variance, and marginal probability density function (PDF). However, it is rare to see model updating methods focusing on studying the high-dimensional distribution of the updating parameters due to the computational cost. This paper develops a hybrid nested sampling method to identify the global optimum and high-dimensional confidence interval simultaneously. The proposed method samples the posterior PDF by shrinking the range of the live sample set layer by layer and achieves the global optimum guided by a hybrid search strategy. Finally, the performance of the proposed method was investigated by numerical simulations and an actual shear-type structure’s model test. After analysis and comparison, the results show that the proposed method performs very well in accuracy and robustness.

Stochastic Bifurcation and Energy Transfer in an Inerter-Based Pendulum Vibration Absorber

Abstract In this theoretical study, the vibration suppression, energy transfer, and bifurcation characteristics, as a function of dimensionless parameters, are investigated for an inerter-based pendulum vibration absorber (IPVA) attached to a linear single-degree-of-freedom spring-mass-damper system (primary structure), subject to white noise excitation. A perturbation method is introduced to detect and track the bifurcation points of the system. It is shown that the marginal probability density function of the pendulum angular displacement undergoes a P-bifurcation at critical parameter values, transitioning from monomodality to bi-modality. A cumulant-neglect technique is used to predict the mean squares of the system, which are compared to the response of a linear system without the IPVA to quantify the vibration suppression. It is shown that the IPVA leads to effective vibration mitigation of the structure in the neighborhood of the P-bifurcation, which is achieved by transferring the kinetic energy of the structure to the pendulum. The results are validated by a Monte Carlo simulation that is used to numerically approximate the marginal probability density function for the pendulum angle as well as the mean squares.

Loopy sum-product algorithm based joint detection, tracking and classification of extended objects with analytic implementations

This paper addresses the problem of joint detection, tracking and classification (JDTC) of multiple extended objects (EOs), where each EO is restricted to be an ellipse modeled as a random matrix, and the classification of an EO is obtained on the basis of its size information. In order to reduce the computational complexity of data association and calculating the marginal posterior probability density function (PDF), the loopy sum-product algorithm (LSPA) is carried out based on a suitably devised factor graph. The goal of LSPA-based JDTC is achieved by iteratively computing the belief of each EO which is used to approximate the marginal posterior. Furthermore, in order to get a closed-form implementation, the belief of each EO is modeled via a sum of belief of each class, which is further modeled as the class probability multiplied by a mixture Gamma-Gaussian-inverse-Wishart (GGIW) distribution conditional on the class. The details of implementation are discussed in this paper and, finally, the effectiveness of the proposed approach is assessed via simulation experiments.

Effect of Normal Transformation Methods on Performance of Multivariate Normal Distribution

Multivariate normal distribution is used widely to characterize the uncertainties and correlations for correlated geotechnical data. The success of constructing a multivariate normal distribution depends on the reliable estimation of the marginal probability density functions (PDFs) and the correlation matrix. This paper focused on the normal transformation which is related to the fitted marginal PDFs and investigated its effect on the performance of the constructed multivariate normal distributions, i.e., the normality of the multivariate normal distribution, the fitness of the simulated data with the original data, the rationality of the derived point estimate equations, and validation of the equations based on validation data sets. Three normal transformation methods with different types of fitted marginal PDF, namely Johnson transformation, three-parameter lognormal transformation, and Box–Cox transformation, were compared based on their application to a real soil database. It was found that all the three normal transformation methods are applicable in the framework of multivariate normal distribution, although the transformed variables do not follow the multivariate normal distribution. The consistence of normality of the transformed variables with the performance of the constructed multivariate normal distribution in estimating the unknown parameters using Bayesian updating technique was verified. The Johnson transformation method is the recommended method for constructing the multivariate normal distribution for the real databases due to its robustness and superiority in normal transformation.

Event-by-event cumulants of azimuthal angles

We further develop the recently proposed event-by-event cumulants of azimuthal angles. The role of reflection symmetry, permutation symmetry, frame independence, and relabeling of particle indices in the cumulant expansion is discussed in detail. We argue that mathematical and statistical properties of cumulants are preserved if cumulants of azimuthal angles are defined event-by-event in terms of single-event averages of azimuthal angles, while they are violated in the traditional approach in which cumulants are defined in terms of all-event averages. We derive for the first time the example analytic solutions for the contribution of combinatorial background in the measured two- and three-particle correlations. We demonstrate that these solutions for the combinatorial background are universal, as they can be written generically in terms of multiplicity-dependent combinatorial weights and marginal probability density functions of starting multivariate distribution. The new general results between multiparticle azimuthal correlators and flow amplitudes and symmetry planes are presented.

Running vacuum model versus ΛCDM – a Bayesian analysis

ABSTRACT We study the significance of the running vacuum model in which the vacuum energy density depends on the square of Hubble parameter, in comparison with the ΛCDM model. The Bayesian inference method is employed to appraise the relative significance of the running vacuum model, using the combined data sets, SN1a+CMB+BAO and SN1a+CMB+BAO+OHD. The model parameters and the corresponding errors are estimated from the marginal probability density function of the model parameters. The parameter that distinguish the running vacuum model from the ΛCDM model is ν. With the SN1a+CMB+BAO data set, we have found that the parameter ν is different from zero at ∼2.7σ. With the second data set, SN1a+CMB+BAO+OHD, the significance improved considerably to 3.4σ. Marginalizing over all model parameters with suitable prior, we have obtained the Bayes factor as the ratio of Bayesian evidence of our model and the ΛCDM model. The analysis based on Jeffrey’s scale of bayesian inference shows that the evidence of our model against the ΛCDM model is weak for the data set SN1a+CMB+BAO. We have obtained a definite evidence of running vacuum model for SN1a+CMB+BAO + OHD data set. This indicates that the dark energy could be dynamical.

Large deviations for the empirical measure of the zig-zag process

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng–Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker–Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker–Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

Chaotic diffusion of the fundamental frequencies in the Solar System

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 to 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.

Toward Predicting Human Performance Outcomes From Wearable Technologies: A Computational Modeling Approach.

Wearable technologies for measuring digital and chemical physiology are pervading the consumer market and hold potential to reliably classify states of relevance to human performance including stress, sleep deprivation, and physical exertion. The ability to efficiently and accurately classify physiological states based on wearable devices is improving. However, the inherent variability of human behavior within and across individuals makes it challenging to predict how identified states influence human performance outcomes of relevance to military operations and other high-stakes domains. We describe a computational modeling approach to address this challenge, seeking to translate user states obtained from a variety of sources including wearable devices into relevant and actionable insights across the cognitive and physical domains. Three status predictors were considered: stress level, sleep status, and extent of physical exertion; these independent variables were used to predict three human performance outcomes: reaction time, executive function, and perceptuo-motor control. The approach provides a complete, conditional probabilistic model of the performance variables given the status predictors. Construction of the model leverages diverse raw data sources to estimate marginal probability density functions for each of six independent and dependent variables of interest using parametric modeling and maximum likelihood estimation. The joint distributions among variables were optimized using an adaptive LASSO approach based on the strength and directionality of conditional relationships (effect sizes) derived from meta-analyses of extant research. The model optimization process converged on solutions that maintain the integrity of the original marginal distributions and the directionality and robustness of conditional relationships. The modeling framework described provides a flexible and extensible solution for human performance prediction, affording efficient expansion with additional independent and dependent variables of interest, ingestion of new raw data, and extension to two- and three-way interactions among independent variables. Continuing work includes model expansion to multiple independent and dependent variables, real-time model stimulation by wearable devices, individualized and small-group prediction, and laboratory and field validation.

Angle-of-arrival and time-of-arrival statistics of multipaths arising from scatterers on an elliptical disc with conic spatial density around the mobile

This paper presents the angle-of-arrival (AOA) and time-of-arrival (TOA) statistics of multipaths arriving from scatterers with conic spatial density and located on an elliptical disc around the mobile. For simplification of analysis, a common practice in geometric channel modeling of macrocell networks in the open literature is to assume the scatterers to be located on a circular disc around the mobile for non-uniform scatterers’ spatial densities. This paper extends the analysis for scatterers with conic spatial density on a circular disc around the mobile to that on an elliptical disc around the mobile. The AOA-TOA joint probability densities and AOA marginal probability densities from the base station and mobile station sides were derived analytically in closed form. The TOA marginal probability densities for both the base-station and mobile-station sides were also studied. The effect of the eccentricity of the elliptical disc on the derived probability densities was discussed. In most cases, increase in the eccentricity results to reduction in the AOA support region.

Time-squeezing and time-expanding transformations in harmonic force fields.

A variety of real life phenomena exhibit complex time-inhomogeneous nonlinear diffusive motion in the presence of an external harmonic force. In capturing the characteristics of such dynamics, the class of Ornstein-Uhlenbeck processes, with its physical time appropriately modulated, has long been regarded as the most relevant model on the basis of its mean reversion property. In this paper, we contrast two similar, yet definitely different, time-changing mechanisms in harmonic force fields by systematically deriving and presenting a variety of key properties all at once, that is, whether or not and how those time-changing mechanisms affect the characteristics of mean-reverting diffusion through sample path properties, the marginal probability density function, the asymptotic degeneracy of increments, the stationary law, the second-order structure, and the ensemble- and time-averaged mean square displacements. Some of those properties turn out rather counter-intuitive due to, or irrespective of, possible degeneracy of time-changing mechanisms in the long run. In light of those illustrative comparisons, we examine whether such time-changing operations are worth the additional operational cost, relative to physically relevant characteristics induced, and deduce practical implications and precautions from modeling and inference perspectives, for instance, on the experimental setup involving those anomalous diffusion processes, such as the observation start time and stepsize when measuring mean square displacements, so as to preclude potentially misleading results and paradoxical interpretations.

Non-parametric modelling and simulation of spatiotemporally varying geo-data

ABSTRACT Modelling and simulation of spatially or temporally varying geo-data play a pivotal role in the development of digital twins of civil infrastructures and smart cities. Measurements on geo-data are however often sparse, and it is challenging to model or simulate the spatiotemporally varying geo-data directly from sparse measurements. Non-parametric methods are appealing to tackle this challenge because they bypass the difficulty in the selection of suitable parametric models or function forms and offer great flexibility for mimicking complicated characteristics of geo-data in a data-driven manner. This paper provides a state-of-the-art review of non-parametric modelling and simulation of spatiotemporally varying geo-data under the framework of spectral representation or compressive sensing/sampling (CS). Similarity and differences between the spectral representation-based methods and the CS-based methods are discussed, including modelling of unknown trend function, marginal probability density function (PDF), and spatial or temporal autocovariance structure. Advantages of the CS-based methods are highlighted, such as superior performance for sparse measurements (i.e. capable of dealing with a sampling frequency lower than Nyquist frequency) and incorporation of the uncertainty associated with the interpretation of sparse measurements. Numerical examples are presented to demonstrate both spectral representation-based methods and CS-based methods.

Bayesian Geophysical Inversion Using Invertible Neural Networks

AbstractConstraining geophysical models with observed data usually involves solving nonlinear and nonunique inverse problems. Neural mixture density networks (MDNs) provide an efficient way to estimate Bayesian posterior marginal probability density functions (pdf's) that represent the nonunique solution. However, it is difficult to infer correlations between parameters using MDNs, and in turn to draw samples from the posterior pdf. We introduce an alternative to resolve these issues: invertible neural networks (INNs). These are simultaneously trained to represent uncertain forward functions and to solve Bayesian inverse problems. In its usual form, the method does not account for uncertainty caused by data noise and becomes less effective in high dimensionality. To overcome these issues, in this study, we include data uncertainties as additional model parameters, and train the network by maximizing the likelihood of the data used for training. We apply the method to two types of imaging problems: One‐dimensional surface wave dispersion inversion and two‐dimensional travel time tomography, and we compare the results to those obtained using Monte Carlo and MDNs. Results show that INNs provide comparable posterior pdfs to those obtained using Monte Carlo, including correlations between parameters, and provide more accurate marginal distributions than MDNs. After training, INNs estimate posterior pdfs in seconds on a typical desktop computer. Hence they can be used to provide efficient solutions for repeated inverse problems using different data sets. Also even accounting for training time, our results show that INNs can be more efficient than Monte Carlo methods for solving inverse problems only once.