non-Gaussian Probability Density Function Research Articles

Application of maximum statistical entropy in formulating a non-gaussian probability density function in flow uncertainty analysis with prior measurement knowledge

In mechanical, civil and chemical engineering systems the accuracies of flow measurement instruments is conventionally specified by certified measurement capabilities (CMCs) that are symmetric, however it is physically possible for some flow instruments and equipment to exhibit asymmetric non-Gaussian behaviour. In this paper the influence of non-Gaussian uncertainties is investigated using direct Monte Carlo simulations to construct a probability density function (PDF) using representative non-Gaussian surface roughness data for a commercial steel pipe friction factor. Actual PDF results are compared and contrasted with a symmetric Gaussian PDF, and reveal inconsistencies in the statistical distributions that cannot be neglected in high accuracy flow measurements. The non-Gaussian PDF is visualized with a kernel density estimate (KDE) scheme to infer an initial qualitative shape of the actual PDF using the approximate locations of the normalized peaks as a initial metrologist estimate of the measurement density. This is then utilized as inputs in a maximum statistical entropy functional to optimize the actual non-Gaussian PDF using a nonlinear optimization of Lagrange multipliers for a mathematically unique PDE. Novelties in the present study is that a new methodology has been developed for statistical sampling from non-monotonic non-Gaussian distributions with accompanying Python and Matlab/GNU Octave computer codes, and a new methodology for utilizing metrologist's expert prior knowledge of PDF peaks and locations for constructing an a priori estimate of the shape of unknown density have been incorporated into the maximum statistical entropy nonlinear optimization problem for a faster and more efficient approach for generating statistical information and insights in constructing high accuracy non-Gaussian PDFs of real world messy engineering measurements.

A physics-informed data-driven algorithm for ensemble forecast of complex turbulent systems

A new ensemble forecast algorithm, named the physics-informed data-driven algorithm with conditional Gaussian statistics (PIDD-CG), is developed to predict the probability density functions (PDFs) of complex turbulent systems with partial observations. The PIDD-CG algorithm integrates a unique multiscale statistical closure modeling strategy with a highly efficient nonlinear data assimilation scheme to create a mixture of conditional statistics. An effective data-driven modeling method is integrated with the dominant conditional statistics to serve as the forecast ensemble members that can significantly reduce the high computational cost of recovering high-dimensional PDFs. The multiscale features in the time evolution of these conditional statistics ensembles are effectively predicted by an appropriate combination of physics-informed analytic formulae and recurrent neural networks. An information metric is adopted as the loss function for the latter to more accurately capture the desirable turbulent features. The proposed algorithm displays effective forecasting performance in both the transient and statistical equilibrium non-Gaussian PDFs of strongly turbulent systems with intermittency, regime switching, and extreme events. It also facilitates the development of efficient statistical reduced-order models in recovering and predicting the large-scale coherent structures of a large group of multiscale complex systems.

Spatial evolution of the turbulent/turbulent interface geometry in a cylinder wake

This study aims to examine the spatial evolution of the geometrical features of the turbulent/turbulent interface (TTI) in a cylinder wake. The wake is exposed to various turbulent backgrounds in which the turbulence intensity and the integral length scale are varied independently, and comparisons to a turbulent/non-turbulent interface (TNTI) are drawn. The turbulent wake was marked with a high Schmidt number ( $Sc$ ) scalar, and a planar laser induced fluorescence experiment was carried out to capture the interface between the wake and the ambient flow from $x/d = 5$ to 40, where $x$ is the streamwise coordinate from the centre of the cylinder, and $d$ is the cylinder's diameter. It is found that the TTI generally spreads faster towards the ambient flow than the TNTI. A transition region of the interfaces’ spreading is found at $x/d \approx 15$ , after which the interfaces propagate at a slower rate than previously (upstream), and the mean interface positions of both the TNTI and TTI scale with the local wake half-width. The locations of both the TNTI and TTI have non-Gaussian probability density functions (PDFs) in the near wake because of the influence of the large-scale coherent motions present within the flow. Further downstream, after the large-scale coherent motions have dissipated, the TNTI position PDF does become Gaussian. For the first time, we explore the spatial variation of the ‘roughness’ of the TTI, quantified via the fractal dimension, from near field to far field. The length scale in the background flow has a profound effect on the TTI fractal dimension in the near wake, whilst the turbulence intensity becomes important only for the fractal dimension farther downstream.

A novel variational robust filter with Gaussian mixture model for unknown non-Gaussian noises

In numerous practical applications, the presence of unpredictable abnormal behaviors often leads to random measurement outliers, causing the random measurement noises are no longer Gaussian but unknown non-Gaussian distributions, and it is intractable to obtain accurate a priori information about the noises. To address this challenge and enable nonlinear state estimation under such measurement noises, we propose a novel robust filter based on Gaussian mixture model (GMM). Firstly, a GMM is introduced to model the unknown non-Gaussian measurement likelihood probability density function (ML-PDF). Without relying on any prior noise statistics, the GMM can accurately model the ML-PDF utilizing a set of model parameters adaptively learned by the variational Bayesian (VB) method. Then, considering the high computational burden associated with the existing variational iterative process, we develop an improved VB method, which independently calculates the posterior distributions of state and unknown model parameters, thereby improving the computational efficiency. Finally, a new robust filter is derived by integrating the GMM with the improved VB approach, The effectiveness and superiority of the filter are demonstrated through both numerical simulations and real-world data analysis.

Data-driven statistical reduced-order modeling and quantification of polycrystal mechanics leading to porosity-based ductile damage

Predicting the process of porosity-based ductile damage in polycrystalline metallic materials is an essential practical topic. Ductile damage and its precursors are represented by extreme values in stress and material state quantities, the spatial probability density function (PDF) of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts utilizing sophisticated continuum-based physical models generally lack in representing the statistics of structural evolution during material deformation. Computational tools which do represent complex structural evolution are typically expensive. The inevitable model error and the lack of uncertainty quantification may also induce significant forecast biases, especially in predicting the extreme events associated with ductile damage. In this paper, a data-driven statistical reduced-order modeling framework is developed to provide a probabilistic forecast of the deformation process of a polycrystal aggregate leading to porosity-based ductile damage with uncertainty quantification. The framework starts with computing the time evolution of the leading few moments of specific state variables from the spatiotemporal solution of full-field polycrystal simulations. Then a sparse model identification algorithm based on causation entropy, including essential physical constraints, is utilized to discover the governing equations of these moments. An approximate solution of the time evolution of the PDF is obtained from the predicted moments exploiting the maximum entropy principle. Numerical experiments based on polycrystal realizations of a representative body-centered cubic (BCC) tantalum illustrate a skillful reduced-order model in characterizing the time evolution of the non-Gaussian PDF of the von Mises stress and quantifying the probability of extreme events. The learning process also reveals that the mean stress is not simply an additive forcing to drive the higher-order moments and extreme events. Instead, it interacts with the latter in a strongly nonlinear and multiplicative fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast when applied to the realizations outside the training data set, indicating the robustness of the model and the skill for extrapolation. Finally, an information-based measurement is employed to quantitatively justify that the leading four moments are sufficient to characterize the crucial highly non-Gaussian features throughout the entire deformation history considered.

Characterizing the Anchoring Effects of Official Forecasts on Private Expectations

The paper proposes a method for simultaneously estimating the treatment effects of a change in a policy variable on a numerable set of interrelated outcome variables (different moments from the same probability density function). Firstly, it defines a non-Gaussian probability density function as the outcome variable. Secondly, it uses a functional regression to explain the density in terms of a set of scalar variables. From both the observed and the fitted probability density functions, two sets of interrelated moments are then obtained by simulation. Finally, a set of difference-in-difference estimators can be defined from the available pairs of moments in the sample. A stylized application provides a 29-moment characterization of the direct treatment effects of the Peruvian Central Bank’s forecasts on two sequences of Peruvian firms’ probability densities of expectations (for inflation -π- and real growth -g-) during 2004-2015.

Adaptive diagnosis and non-fragile predictive control for HFV engine with non-Gaussian uncertain output

This study presents an adaptive fault estimation and model predictive fault-tolerant control (FTC) for the hypersonic vehicle engine with non-Gaussian uncertain output jet plume. The non-Gaussian probability density function (PDF) describing the plume velocity in a stochastic system is approximated by Type II fuzzy radial basis functions. In the fault observer, the novel prey adaptive estimation method is designed for the valve faults with different amplitudes. Ultimately, this fuzzy-prey fusion adaptive observer shields perturbation and accurately estimates complex faults including incipient and intermittent faults. A predictive tolerant controller compensates for all the faults effects and non-fragile compensation factors of FTC eliminate the perturbation interference, thence the output PDF matches with the expected value. Finally, the more refined internal structure of plume output distribution replaces the local and rough valve position variables, helping this study realizes the hypersonic vehicle engine integrated refine control. Simulation result verifies the effectiveness of the proposed approaches and the superiority compared with the existing method.

Moving horizon estimator for nonlinear and non-Gaussian stochastic disturbances

Over the last two decades, moving horizon estimation (MHE) has increasingly been used by researchers and industrial practitioners of nonlinear model predictive control as an alternative to recursive Bayesian estimation schemes. The MHE formulations available in the literature assume that uncertainties in the state dynamics can be modelled as additive and Gaussian stochastic processes. In practice, the unmeasured disturbances affect the system dynamics in a much more complex way and Gaussian assumption may prove inadequate to represent their behaviour. However, unlike the recursive Bayesian estimation area, handling of non-additive and non-Gaussian state disturbances has not received much attention in the MHE literature. In this work, a novel formulation of MHE is proposed in the Bayesian framework to solve state estimation problems associated with systems subjected to nonlinear non-Gaussian stochastic disturbances. To begin with, a Bayesian formulation that maximizes the joint posterior density function of the initial state and stochastic inputs is developed for a batch of data, which is further extended to the moving horizon framework. The proposed formulation associates the random fluctuations affecting the system dynamics with the unmeasured inputs entering the systems. The probabilistic formulation of MHE together with modelling of the state disturbances arising from physical sources enables us to work with any (non-Gaussian or Gaussian) probability density function (PDF) and systematically handle the stochastic inputs affecting the dynamics in a nonlinear manner without making any simplifying assumptions. We proceed to show that the disturbances/states with constraints can also be incorporated in our proposed formulation and can be interpreted as following truncated probability density functions. Efficacy of the proposed MHE approach is demonstrated using three benchmark simulation case studies and an experimental case study. Simulation studies show that the proposed MHE framework is able to perform similar to EnKF, a sampling based sequential Bayesian estimation approach which is capable of handling nonlinear non-Gaussian disturbances. It is further shown that the proposed MHE performs much better than the conventional MHE approach which assumes additive Gaussian noise in the state dynamics. Thus, the proposed MHE scheme provides an optimization based alternative to sampling based estimation schemes, such as Ensemble Kalman filtering, which can handle state estimation problems when state and measurement densities are non-Gaussian.

Statistics-based Bayesian modeling framework for uncertainty quantification and propagation

A new Bayesian modeling framework is proposed to account for the uncertainty in the model parameters arising from model and measurements errors, as well as experimental, operational, environmental and manufacturing variabilities. Uncertainty is embedded in the model parameters using a single level hierarchy where the uncertainties are quantified by Normal distributions with the mean and the covariance treated as hyperparameters. Unlike existing hierarchical Bayesian modelling frameworks, the likelihood function for each observed quantity is built based on the Kullback–Leibler divergence used to quantify the discrepancy between the probability density functions (PDFs) of the model predictions and measurements. The likelihood function is constructed assuming that this discrepancy for each measured quantity follows a truncated normal distribution. For Gaussian PDFs of measurements and response predictions, the posterior PDF of the model parameters depends on the lower two moments of the respective PDFs. This representation of the posterior is also used for non-Gaussian PDFs of measurements and model predictions to approximate the uncertainty in the model parameters. The proposed framework can tackle the situation where only PDFs or statistical characteristics are available for measurements. The propagation of uncertainties is accomplished through sampling. Two applications demonstrate the use and effectiveness of the proposed framework. In the first one, structural model parameter inference is considered using simulated statistics for the modal frequencies and mode shapes. In the second one, uncertainties in the parameters of the probabilistic S-N curves used in fatigue are quantified based on experimental data.

The large-scale structure from non-Gaussian primordial perturbations

ABSTRACT The late-time effect of primordial non-Gaussianity offers a window into the physics of inflation and the very early Universe. In this work, we study the consequences of a particular class of primordial non-Gaussianity that is fully characterized by initial density fluctuations drawn from a non-Gaussian probability density function, rather than by construction of a particular form for the primordial bispectrum. We numerically generate multiple realizations of cosmological structure and use the late-time matter power spectrum, bispectrum, and trispectrum to determine the effect of these modified initial conditions. We show that the initial non-Gaussianity has only a small imprint on the first three polyspectra, when compared to a standard Gaussian cosmology. Furthermore, some of our models present an interesting scale-dependent deviation from the Gaussian case in the bispectrum and trispectrum, although the signal is at most at the per cent level. The majority of our models are consistent with cosmic microwave background constraints on the trispectrum, while the others are only marginally excluded. Finally, we discuss further possible extensions of our study.

Modified Particle Filters for Detection of False Data Injection Attacks and State Estimation in Networked Nonlinear Systems

Networked control systems which transfer data over communication networks may suffer from malicious cyber attacks by injecting false data to the transferred information. Such attacks can cause performance degradation of the closed-loop system and the filtering problem. The sequential importance sampling (SIS) particle filtering (PF) methods employ the sequential Monte Carlo approach to estimate the generally non-Gaussian posterior probability density function (pdf) for Bayesian estimation of generally non-linear non-Gaussain systems. In this paper, it is firstly shown that with the normal SIS PF, the injected false data to the networked systems remains stealthy and therefore it is not possible to reduce the degrading effect of the attack on the estimation. However, with a modification in the proposal pdf, a modified SIS PF is then proposed which guarantees the attack detectability where the attacked measurements are incorporated in the particle generation process and thus the particles are updated and make the attack detectable. Using the derived thresholds and under small enough measurement noises, it is also proved that no false alarm occurs. After estimation of the attack value, the posterior pdf conditioned on truly detected attack leads to an estimation equivalent to the attack free SIS PF in terms of estimation bias and estimation covariance error. Finally, the accuracy of the presented concepts is demonstrated for a networked interconnected four-tank system.

Statistics and Dynamics of Instantaneous Leading Point in Nonpremixed Bluff-Body Flames

The subtle flow–flame interaction is essential for flame stabilization in propulsion and power systems. In this study, the flamelet/progress variable approach combined with large-eddy simulation is employed to simulate the Sydney nonpremixed bluff-body flame HM1E, for which the leading point concept is employed to reveal the complex flow–flame interaction. The Eulerian data as well as the leading point tracked data are analyzed. The Eulerian data show that compositional structure at axial location ranging from 0.5 to 1.0 has two branches, the unburnt branch and the burning branch. To reveal the underlying physics of the compositional structure, the statistics of leading point are analyzed and show that there are Gaussian-like and non-Gaussian probability density function (PDF) distributions upstream and downstream of about . Further analysis of the flow–leading point interaction shows that the region of ranging from 0.25 and 0.5 is where the instability of the outer shear layer occurs. Two different interaction patterns exist upstream and downstream of corresponding to Gaussian-like and non-Gaussian PDFs, respectively. The core and braid structures developed in the shear layer downstream of are responsible for the non-Gaussian PDF. In addition, the leading point migration in streamwise direction leads to the exhibition of two-branch compositional structure for Eulerian data.

Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments: Coupled translational and rotational motions.

We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction ϕ and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on ϕ, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing ϕ. We also analyze the relative motion in the dimers, finding that larger ϕ suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(ϕ) of translational and rotational motion of the dumbbells an exponential decay with ϕ for weak and a power-law variation D(ϕ)∝(ϕ-ϕ^{★})^{2.4} for strong crowding is found. A comparison of simulation results with theoretical predictions for D(ϕ) is discussed and some relevant experimental systems are overviewed.

Analysis of bilinear hysteretic structures with nonlinear fluid viscous dampers using modified stochastic linearization technique

Viscous dampers are increasingly used to design seismic resistant structures in modern construction due to their ability to reduce structural and non-structural damage by absorbing a large portion of the earthquake input energy. To facilitate the design of such complex nonlinear systems, this study aims to propose new non-Gaussian probability density functions (PDFs) to improve the accuracy in the determination of the seismic response of bilinear hysteretic structures equipped with linearized nonlinear viscous dampers. The employed linearization method is based on a stochastic linearization technique using stationary response velocity PDFs, while the difference between the governing nonlinear differential equation and the equivalent linear equation is minimized. To generate suitable functions, a Genetic Algorithm (GA) optimization method is adopted for both hard and soft soil-based structures equipped with nonlinear viscous dampers using three different power coefficients. The computational error of the employed modified stochastic linearization technique is then evaluated for single-degree-of-freedom (SDOF) systems under a set of five thousand randomly generated earthquake excitations with peak ground accelerations (PGAs) of 0.2 g to 0.6 g. The results indicate that the proposed method can significantly reduce (up to 80%) the computational errors. Subsequently, the lateral roof displacements of 5-story structures equipped with nonlinear viscous dampers are obtained under a set of natural ground motions using different linearization methods. It is shown that the proposed functions lead to the least error for both hard and soft soil-based structures compared to the accurate results.

BAMCAFE: A Bayesian machine learning advanced forecast ensemble method for complex turbulent systems with partial observations.

Ensemble forecast based on physics-informed models is one of the most widely used forecast algorithms for complex turbulent systems. A major difficulty in such a method is the model error that is ubiquitous in practice. Data-driven machine learning (ML) forecasts can reduce the model error, but they often suffer from partial and noisy observations. In this article, a simple but effective Bayesian machine learning advanced forecast ensemble (BAMCAFE) method is developed, which combines an available imperfect physics-informed model with data assimilation (DA) to facilitate the ML ensemble forecast. In the BAMCAFE framework, a Bayesian ensemble DA is applied to create the training data of the ML model, which reduces the intrinsic error in the imperfect physics-informed model simulations and provides the training data of the unobserved variables. Then a generalized DA is employed for the initialization of the ML ensemble forecast. In addition to forecasting the optimal point-wise value, the BAMCAFE also provides an effective approach of quantifying the forecast uncertainty utilizing a non-Gaussian probability density function that characterizes the intermittency and extreme events. It is shown using a two-layer Lorenz 96 model that the BAMCAFE method can significantly improve the forecasting skill compared to the typical reduced-order imperfect models with bare truncation or stochastic parameterization for both the observed and unobserved large-scale variables. It is also shown via a nonlinear conceptual model that the BAMCAFE leads to a comparable non-Gaussian forecast uncertainty as the perfect model while the associated imperfect physics-informed model suffers from large forecast biases.

Uncertainty Estimation for Ensemble Particle Image Velocimetry

We present a novel approach to estimate the uncertainty in ensemble particle image velocimetry (PIV) measurements. Ensemble PIV is widely used when the cross-correlation signal-to-noise ratio (SNR) is insufficient to perform a reliable instantaneous velocity measurement. Despite the utility of ensemble PIV, uncertainty quantification for this type of measurement has not been studied. The existing uncertainty quantification algorithms for PIV are developed and used only for instantaneous PIV measurement and do not account for the improved SNR in ensemble PIV. Existing instantaneous uncertainty quantification methods can be divided into direct and indirect categories. Indirect methods require calibration based on the effect of various image parameters (such as noise, particle size, density, velocity gradient, etc.) on the correlation SNR. Indirect methods have not been calibrated for error sources relevant in an Ensemble PIV measurement. Also, they have lower sensitivity to the error sources compared to direct approaches. Direct methods, such as the moment of correlation (MC) and Image Matching (IM), find the uncertainty based on the images and correlation planes without any calibration and are more reliable (Bhattacharya et al., 2018; Sciacchitano et al., 2013). Ensemble PIV is based on ensemble correlations; therefore, MC, which uses the generalized cross-correlation (GCC) plane as a measure of uncertainty, is the most suitable method to be modified to be applicable for the ensemble PIV. The GCC plane is the inverse Fourier transform of the phase correlation and represents the probability density function (PDF) of particles’ displacements (Bhattacharya et al., 2018; Eckstein and Vlachos, 2009). We replaced instantaneous GCC with ensemble GCC and modified MC’s normalization factor to account for the number of ensembles. The MC’s primary limitation is that it assumes a Gaussian shape for the PDF of displacements and estimate the standard deviation of the underlying PDF using a fitted Gaussian. However, the PDF deviates from Gaussian distribution due to velocity gradient or non-Gaussian random displacements. Therefore, MC’s reliability and applicability are reduced for flow fields with non-Gaussian PDFs. Also, our analysis shows that ensemble MC consistently underestimates the uncertainty. So, a generalized and reliable method for uncertainty quantification for ensemble PIV is needed.

Mixture of robust Gaussian processes and its hard-cut EM algorithm with variational bounding approximation

The Gaussian process is a powerful statistical learning model and has been applied widely in nonlinear regression and classification. However, it fails to model multi-modal data from a non-stationary source since a prior Gaussian process is generally stationary. Based on the idea of the mixture of experts, the mixture of Gaussian processes was established to increase the model flexibility. On the other hand, the Gaussian process is also sensitive to outliers and thus robust Gaussian processes have been suggested to own the heavy-tailed property. In practical applications, the datasets may be multi-modal and contain outliers at the same time. In order to overcome these two difficulties together, we propose a mixture of robust Gaussian processes (MRGP) model and establish a precise hard-cut EM algorithm for learning its parameters. Since the exact solving process is intractable due to the fact that non-Gaussian probability density functions of the noises are adopted into the likelihood of the proposed model on the dataset, we employ a variational bounding method to approximate the marginal likelihood functions so that the hard-cut EM algorithm can be implemented effectively. Moreover, we conduct various experiments on both synthetic and real-world datasets to evaluate and compare our proposed MRGP method with several competitive nonlinear regression methods. The experimental results demonstrate that our MRGP model with the hard-cut EM algorithm is much more effective and robust than the competitive nonlinear regression models.

Theoretical determination of asymmetric rolling amplitude in irregular beam seas

A methodology for predicting the probability density function of roll motion for irregular beam seas is developed in the author’s previous research. In this paper, two methods for the prediction of the probability density function (PDF) of the rolling amplitude are examined. One of these methods is an approach based on a non-Gaussian PDF with the use of the equivalent linearization and the moment method, which has been used by Maki in the field of naval architecture. In the framework of this method, the instantaneous joint PDF of the roll and roll rate can be calculated. Thereby, the transformation from the joint PDF to the PDF of the total energy H is necessary. In this paper, the transformation from the joint PDF of instantaneous roll and rate to the PDF of H was conducted. The other method is the energy-based stochastic averaging method, whereby the PDF of the total energy H is determined from which the roll amplitude is calculated. It is noteworthy that the proposed theory aims to predict the PDF of the amplitude of the vessel due to wind or flooding. For such conditions, the restoring curve (GZ curve) has asymmetricity. To take into account this asymmetricity, a new theory for the transform of the PDF of the total energy H to the PDFof amplitude is proposed. The obtained theoretical results show almost complete agreement, and both of the two theoretical methods i.e., the energy-based stochastic averaging method and the non-Gaussian PDF method with the use of the equivalent linearization and moment method, show equivalent prediction performance even in the asymmetric condition.

Robust particle filtering with enhanced outlier resilience and real-time disturbance compensation

This paper is concerned with the state estimation problem for nonlinear/non-Gaussian systems suffered from both time-varying disturbances (TVD) and measurement outliers. Conventional particle filtering (PF) approach can be used to track the non-Gaussian probability density functions, but its sampling efficiency is degraded in the presence TVD. To address this problem, we propose a disturbance observer based PF (DOBPF) method where the knowledge on the dynamics of TVD are fully exploited and real-time disturbance compensation is achieved. Furthermore, to enhance the resilience of our method against outliers, we adopt the skew-t distribution to characterize both skewness and heavy-tailedness of the measurement noise. On this basis, the variational Bayes approach is incorporated into the DOBPF under the marginalization PF framework to infer the noise statistics. Compared with conventional PF approaches, the proposed outlier-resilient DOBPF method exhibits improved resilience against measurement outliers and increased sampling efficiency in the presence of TVD. Simulation and experimental results confirm the effectiveness of the proposed algorithm.

Likely oscillatory motions of stochastic hyperelastic spherical shells and tubes

We examine theoretically the dynamic inflation and finite amplitude oscillatory motion of inhomogeneous spherical shells and cylindrical tubes of stochastic hyperelastic material. These bodies are deformed by radially symmetric uniform inflation, and are subjected to either a surface dead load or an impulse traction, uniformly applied in the radial direction. We consider composite shells and tubes with two concentric stochastic homogeneous neo-Hookean phases, and inhomogeneous bodies of stochastic neo-Hookean material with constitutive parameters varying continuously in the radial direction. For the homogeneous materials, we define the elastic parameters as spatially-independent random variables, while for the radially inhomogeneous bodies, we take the parameters as spatially-dependent random fields, described by non-Gaussian probability density functions. Under radially symmetric dynamic deformation treated as quasi-equilibrated motion, we show that the bodies oscillate, i.e., the radius increases up to a point, then decreases, then increases again, and so on, and the amplitude and period of the oscillations are characterised by probability distributions, depending on the initial conditions, the geometry, and the probabilistic material properties.