non-Gaussian Processes Research Articles

Notice of Violation of IEEE Publication Principles: Fault Detection Using Industrial Video Information

Notice of Violation of IEEE Publication Principles Fault Detection Using Industrial Video Information by Y. Zhang, Z. Wang, Y. Fu and X. Li, in the IEEE Transactions on Industrial Electronics, vol. PP, no. 99 After careful and considered review of the content of this paper and associated multimedia content by a duly constituted expert committee, this work has been found to be in violation of IEEE's Publication Principles. This paper and associated multimedia files contain fraudulent data and images. Due to the nature of this violation, reasonable effort should be made to remove all past references to this paper and refrain from future references. In this study, a new data-modelling algorithm, referred to as relaxed independent component analysis, is proposed for the purpose of industrial video process monitoring. The contributions are as follows. (1) Physical variables and image data are unified in a framework to propose fault detection method of industrial process for the first time. (2) We find that independent components are sensitive to extract features of image and physical variables. In original KICA method, KPCA+ICA are used to extract nonlinear independent components. Since ICA is linear method, the original method only extracts semi-nonlinear independent components. In our method, the complete nonlinear independent components are extracted using the independence criterion. (3) A process monitoring approach of relaxed independent component analysis is presented. When applied to the monitoring of an electrical fused magnesia furnace, it can effectively improve the accuracy of forecasting the non-Gaussian process and reduce the ratio of false alarms compared to kernel principle component analysis and traditional kernel independent component analysis.

Binary spatial random field reconstruction from non-Gaussian inhomogeneous time-series observations

We develop a new model for spatial random field reconstruction of a binary-valued spatial phenomenon. In our model, sensors are deployed in a wireless sensor network across a large geographical region. Each sensor measures a non-Gaussian inhomogeneous temporal process which depends on the spatial phenomenon. Two types of sensors are employed: one collects point observations at specific time points, while the other collects integral observations over time intervals. Subsequently, the sensors transmit these time-series observations to a Fusion Center (FC), and the FC infers the spatial phenomenon from these observations. We show that the resulting posterior predictive distribution is intractable and develop a tractable two-step procedure to perform inference. Firstly, we develop algorithms to perform approximate Likelihood Ratio Tests on the time-series observations, compressing them to a single bit for both point sensors and integral sensors. Secondly, once the compressed observations are transmitted to the FC, we utilize a Spatial Best Linear Unbiased Estimator (S-BLUE) to reconstruct the binary spatial random field at any desired spatial location. The performance of the proposed approach is studied using simulation. We further illustrate the effectiveness of our method using a weather dataset from the National Environment Agency (NEA) of Singapore with fields including temperature and relative humidity.

TELETRAFFIC FORECASTING IN MEDIA SERVICE SYSTEMS

Context. The development of information and communication technologies has led to an increase in the volume of information sent over the network. Media service platforms play an important role in the creation and processing of bitrate in the information network. Therefore, there is a need to develop a methodology for predicting bitrate in various media service platforms by creating an effective algorithm that minimizes the forecast error.
 Objective. The aim of the work is to synthesize in analytical form the state transition matrix of the Kalman filter for nonstationary self-similar processes when predicting the bitrate in telecommunication networks.
 Method. A methodology has been developed for predicting teletraffic in media service platforms, based on a modification of the Kalman filter for non-Gaussian processes. This methodology uses an original procedure for calculating statistics, which makes it possible to reduce the filtering and forecast error that arises due to the uncertainty of the analytical model of the process under study. The methodology does not require knowledge of the analytical model of the process, as well as strict restrictions on its stochastic characteristics.
 Results. A methodology for estimating and forecasting bitrate in telecommunication systems is proposed. This methodology was used to study teletraffic processes in the media service platforms Google Meet, Zoom, Microsoft Teams. The passage of real bitrate through the specified media service platforms was studied. A comparison of real teletraffic with predicted teletraffic was carried out. The influence of the order of the state transition matrix of the Kalman filter on the error of estimation and prediction has been studied. It has been established that even a low (second) order of the state transition matrix allows one to obtain satisfactory forecast results. It is shown that the use of the proposed methodology makes it possible to predict traffic with a relative error of the order of 3– 4%.
 Conclusions. An original algorithm for assessing and forecasting the characteristics of media traffic has been developed. Recommendations for improving the technology for building media service platforms are formulated. It is shown that the bitrates generated by various media service platforms, in the case of applying the proposed estimation and forecasting methodology, are invariant with respect to the type of stochastic processes being processed.

Gamma mixed fractional Lévy Ornstein–Uhlenbeck process

In this article, a non-Gaussian long memory process is constructed by the aggregation of independent copies of a fractional Lévy Ornstein–Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.

Nearest-Neighbor Mixture Models for Non-Gaussian Spatial Processes

Nearest-Neighbor Mixture Models for Non-Gaussian Spatial Processes

Controlling the Flexibility of Non-Gaussian Processes Through Shrinkage Priors

The normal inverse Gaussian (NIG) and generalized asymmetric Laplace (GAL) distributions can be seen as skewed and semi-heavy-tailed extensions of the Gaussian distribution. Models driven by these more flexible noise distributions are then regarded as flexible extensions of simpler Gaussian models. Inferential procedures tend to overestimate the degree of non-Gaussianity in the data and therefore we propose controlling the flexibility of these non-Gaussian models by adding sensible priors in the inferential framework that contract the model towards Gaussianity. In our venture to derive sensible priors, we also propose a new intuitive parameterization of the non-Gaussian models and discuss how to implement them efficiently in Stan. The methods are derived for a generic class of non-Gaussian models that include spatial Matérn fields, autoregressive models for time series, and simultaneous autoregressive models for aerial data. The results are illustrated with a simulation study and geostatistics application, where priors that penalize model complexity were shown to lead to more robust estimation and give preference to the Gaussian model, while at the same time allowing for non-Gaussianity if there is sufficient evidence in the data.

A hierarchical Bayesian modeling framework for identification of Non-Gaussian processes

Non-Gaussian processes are frequently encountered in engineering problems, posing a challenge when it comes to identification. The main challenge in the identification arises from the fact that a non-Gaussian process can be treated as a collection of infinite dimensional non-Gaussian variables. The application of the hierarchical Bayesian modeling (HBM) framework is constrained due to the inherent complexity of dimensionality and non-Gaussian characteristics associated with these variables. To tackle the issue of dimensionality, the improved orthogonal series expansion (iOSE) representing a non-Gaussian process by time functions with non-Gaussian coefficients, which are readily obtained from discretizing the process at some specific time points, is introduced within the HBM framework. In particular, the iOSE is embedded to convert the identification of a non-Gaussian process into a finite number of non-Gaussian coefficients. Regarding their non-Gaussian characteristics, polynomial chaos expansion (PCE) is used to quantify the uncertainty of the non-Gaussian coefficients with parameters in PCE treated as hyper parameters to be estimated by the HBM framework. The proposed framework is applicable to the identification of both stationary and nonstationary non-Gaussian processes. The effectiveness of non-Gaussian process quantification by the proposed framework is demonstrated using simulated data of a non-stationary extreme value process. Theapplicability of this approach for non-Gaussian process identification is validated by accurately identifying a stochastic load in a structural dynamic problem. Furthermore, it is successfully applied to the reconstruction of random mode shapes of a building arising from different environmental conditions.

Neural network-aided simulation of non-Gaussian stochastic processes

Simulation of non-Gaussian stochastic processes is of paramount importance to dynamic reliability assessment of structures driven by non-Gaussian loads. In this paper, a novel translation model based on neural network is proposed to convert the target non-Gaussian power spectrum to the underlying Gaussian power spectrum for simulating non-Gaussian stochastic processes. First, a valid neural network model, whose structure is similar to the transformation from a non-Gaussian power spectrum to its underlying Gaussian power spectrum, is established. Then, starting with an initial underlying Gaussian power spectrum, the non-Gaussian power spectrum corresponding to the target non-Gaussian distribution is obtained through the Wiener-Khintchine transform and translation process. After that, the discrete data of non-Gaussian power spectrum is employed as the input layer in the neural network above, while the discrete data of Gaussian power spectrum serves as the output layer. With the trained neural network, the required underlying Gaussian power spectrum can be directly obtained by inputting the target non-Gaussian power spectrum. Finally, Gaussian samples can be generated by using the spectral representation method, which are then converted into non-Gaussian samples via the translation process. The effectiveness, accuracy, and applicability of the proposed method are demonstrated through two numerical examples.

Solving the inverse cosmological calibration problem of gamma-ray bursts

ABSTRACT We have received a new physical characteristics fitting based on actual observational data from the Swift mission’s long-duration gamma-ray bursts (LGRBs). We considered such characteristics as the Amati parameters for linear correlation (Eiso–Ep,i) and the k-correction for gravitational lensing and Malmquist bias (GLMB) effect. We used the Pantheon SN Ia catalogue and the standard Lambda cold dark matter model with a fixed Hubble constant of H0 = 70 km s−1 Mpc−1 as the baseline for the Hubble function μ(z). In our paper, we formulated the inverse cosmological calibration problem (ICCP) in the non-parametric statistics framework. The ICCP involves fitting non-observable physical characteristics while assuming a fixed cosmological model. To solve this problem, we developed a new method that is resistant to non-Gaussian processes. This method is based on error propagation through the Monte Carlo method and the Theil–Sen method for linear regression estimate. We have demonstrated the stability and robustness of this assessment method. The parameter estimates are as follows: $a=0.92^{+0.12}_{-0.12}$, $b=50.32^{+0.33}_{-0.32}$ without considering the GLMB effect, and $a=0.63^{+0.13}_{-0.14}$, $b=50.12^{+0.33}_{-0.31}$, and $k=1.98^{+0.25}_{-0.24}$ with the effect included. The proposed method can be applied to any other calibration sample of known standard candles, a calibrated sample of LGRBs, and the Hubble function μ(z). In the future, the ICCP idea can be used as an alternative cosmological test for estimating cosmological parameters, including the GLMB effect, or even for the selection of models, providing new information about the Universe. This can be done by analysing the residual values of observational data within the Bayesian statistics paradigm.

Statistical features of systems driven by non-Gaussian processes: theory & practice

Nowadays many tools, e.g. fluctuation relations, are available to characterize the statistical properties of non-equilibrium systems. However, most of these tools rely on the assumption that the driving noise is normally distributed. Here we consider a class of Markov processes described by Langevin equations driven by a mixture of Gaussian and Poissonian noises, focusing on their non-equilibrium properties. In particular, we prove that detailed balance does not hold even when correlation functions are symmetric under time reversal. In such cases, a breakdown of the time reversal symmetry can be highlighted by considering higher order correlation functions. Furthermore, the entropy production may be different from zero even for vanishing currents. We provide analytical expressions for the average entropy production rate in several cases. We also introduce a scale dependent estimate for entropy production, suitable for inference from experimental signals. The empirical entropy production allows us to discuss the role of spatial and temporal resolutions in characterizing non-equilibrium features. Finally, we revisit the Brownian gyrator introducing an additional Poissonian noise showing that it behaves as a two dimensional linear ratchet. It has also the property that when Onsager relations are satisfied its entropy production is positive although it is minimal. We conclude discussing estimates of entropy production for partially accessible systems, comparing our results with the lower bound provided by the thermodynamic uncertainty relations.

Identification of hybrid energy harvesting systems with non-Gaussian process

Identification of hybrid energy harvesting systems with non-Gaussian process

Non-Gaussian buffeting analysis of large structures by means of a Proper Orthogonal Decomposition

Proper Orthogonal Decomposition is implemented in a Bispectral Stochastic Analysis of large MDOF structures. Although Higher Order Stochastic Analysis has been introduced few decades ago, its establishment has found place only in a theoretical way, or applied to very small systems. Its heavy computational cost as well as resource consumption burden has let it aside for practical civil engineering applications. Nevertheless, in wind engineering, the community is increasingly considering the importance of non-Gaussian nature of wind induced vibrations. This is of interest for the extreme value analysis, for which many theoretical and empirical models exist for non-Gaussian random processes. In this context, a (bi-)spectral approach is realized in a 2-D frequency space where the bispectrum is computed then the 3rd statistical moment is obtained by means of a twofold integration. These operations are quite heavy as soon as the dimension of the problem increases. A novel algorithm implementation is proposed. It hinges on (i) the use of Proper Orthogonal Decomposition and the (ii) development of an optimized numerical method. The proposed algorithmic arrangement minimizes the number of operations, and consequently saves time and memory, while conserving precision in estimating 3rd order statistical moments of structural responses.

Cooperative positioning method of underwater autonomous vehicle formation based on extended Kalman filter

Autonomous Underwater Vehicles (UAVs) have shown significant potential application value in marine environmental surveillance, development, and utilization of resources. However, due to the complex underwater acoustic channel environment and the relative motion between AUVs, this may lead to heavy-tailed non-Gaussian process noise and measurement noise, leading to increased error and the Extended Kalman Filter (EKF) may fail. In prior studies, we propose a multi-AUV formation hierarchical target location algorithm based on the EKF, which can realize multi-AUV self-localization and stationary target localization. The AUV formation consists of one high-precision piloting AUV and several low-precision following AUVs. The following AUVs are divided into two levels, the reference AUVs and the AUVs to be tested. According to the designed positioning period, the reference AUV receives the position parameters from a high-precision piloting AUV and transmits its own position parameters to the AUV to be measured. Then use the EKF to complete the cooperative position correction of the AUV cluster. In this paper, we will study the influence of heavy-tailed noise on this method and study the EKF based on Huber estimation to improve the antijamming capability.

Assessment of train running safety on railway bridges based on velocity-related indices under random near-fault ground motions

Random analysis has emerged as the dominant approach for analyzing train-bridge coupled (TBC) systems, replacing deterministic analysis. This shift is driven by considerations of system economy and randomness. However, the seismic train running safety assessment with multiple non-Gaussian distribution parameters faces the double dilemma of low computational efficiency and inaccuracy for TBC systems. In this study, an approach is applied by the combination of the new point estimate method and moment expansion approximation (NPEM-MEA) to address this problem. Furthermore, the application of this approach is illustrated by the random dynamic responses of a three-dimensional TBC system subjected to random near-fault ground motions. Validation of the NPEM-MEA’s feasibility is conducted using the Monte Carlo method, and comparative analysis confirms the accuracy and efficiency of the method. Additionally, the influences of random near-fault ground motions are discussed based on the random dynamic responses and train running safety indices in the TBC system. The simulation results underscore the necessity of simultaneously considering the randomness of multiple parameters in seismic train running safety assessment. This methodology holds potential for safety assessments involving non-Gaussian random processes in complex systems of a similar nature.

Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.

Gaussian mixture model sample selection strategy–based active semi-supervised soft sensor for industrial processes

Soft sensors have become reliable tools for estimating difficult-to-measure target variables in modern industrial processes. In order to make full use of labeled and unlabeled samples, an active semi-supervised soft sensor modeling method is proposed, which combines active learning and semi-supervised learning to maximize model performance and minimize the laboratory analysis cost of expanding the labeled sample data set. First, manifold regularization is introduced into the deep extreme learning machine (DELM) algorithm to form a semi-supervised DELM that improves the performance of a model trained with unlabeled samples. Then, considering non-Gaussian processes and the error information between the predicted and true values, an active sample selection strategy based on error Gaussian mixture model is developed. Using this strategy, the most uncertain and representative unlabeled samples are selected for labeling, and thereby expanding the labeled sample data set. Finally, the effectiveness of the proposed method is verified using industrial debutanizer process data.

Multiple discrete orthonormal S-transforms and its application in analyzing, modelling, and simulating random process and field

We examine the derivation and characteristics of the basis function employed in the discrete orthonormal S-transform (DOST). The examination is aimed at clarifying the interpretation of the power characteristics associated with the partitions and basis functions employed in DOST. Based on this examination, it is suggested that a one-half shift to the time index of the original DOST basis functions could be considered to facilitate the interpretation of the power distribution in the time–frequency domain. Although the non-redundant DOST is a highly efficient transform, its resolution in the time–frequency space is relatively coarse as compared to some of the continuous time–frequency decomposition techniques. To improve the resolution, and without significantly increasing the computational time, we propose the multiple DOSTs (MDOSTs) approach that uses M DOSTs, each associated with a different partition scheme in the time–frequency domain and a set of corresponding orthogonal basis functions. The redundancy factor of MDOSTs is simply equal to M. In addition, we show that the MDOSTs can be used to model and simulate nonstationary non-Gaussian processes and fields within the framework of the iterative power and amplitude correction algorithm. The use of MDOSTs for analyzing, modelling, and simulating seismic ground motion and corrosion surface is illustrated by numerical examples. The results show that using MDOSTs can effectively improve the time–frequency resolutions while maintaining high computational efficiency.

Peak factor estimation of non-Gaussian wind pressures based on a novel piecewise Johnson transformation model

In present study, a novel piecewise Johnson transformation model (PJTM) is proposed by conducting separation-mirror-expansion on a non-Gaussian process, aiming to enhance the accuracy and efficiency of the estimation of non-Gaussian peak factor. The parameters of the proposed PJTM are strictly derived from the analytical formula of the unbounded system (SU system) and the semi-analytical formula of the bounded system (SB system). Based on the proposed PJTM and the crossing rate theory, the algorithm of peak factor estimation, as well as the overall calculation procedure, is thus established with explicit formulas and applied to estimating the peak factors of sufficient long-time non-Gaussian wind pressures on a long-span roof. It is noted that the peak factors of the non-Gaussian wind pressures are also calculated by the commonly used Johnson transform model (JTM) and the traditional Hermite polynomial model (HPM). The results show that the peak factors of the non-Gaussian wind pressures estimated by the proposed PJTM agree well with the real values based on a sufficient long-time of data. In addition, the proposed PJTM has comparable accuracy but a broader applicability range than the HPM, and significantly superior accuracy and efficiency than those of JTM. The present study can provide valuable guidance for the wind-resistance design of long-span roofs.

A novel shock life model for BGA solder joints based on evolutionary power spectral density

Multiple shock damage of Ball grid array (BGA) solder joint is one of the important reasons for space mission failure. A novel shock life model is proposed for BGA solder joints to evaluate its life under shock loads based on the evolutionary power spectral density (EPSD). Considering the bandwidth and non-Gaussian features of the stress signals during different stages, the shock stress response was treated as a non-stationary and non-Gaussian stochastic process. EPSD of the shock stress response is derived in matrix form based on wavelet transforms. The weight coefficient and non-Gaussian damage factor are proposed to describe the influence of the bandwidth and non-Gaussian characteristics of shock response on fatigue damage, respectively. The damage values of different stress stages were calculated by weight coefficient and corrected using non-Gaussian damage factor. Then the shock life model is proposed based on damage accumulation theory. Tests for the BGA solder joints shock life were conducted, and the result shows that the proposed model is feasible. Therefore, the developed method in the paper is expected to provide reference for the prediction of BGA solder joint shock life in aerospace engineering.

Modeling and simulation of financial returns under non-Gaussian distributions

It is well known that the probability distribution of high-frequency financial returns is characterized by a leptokurtic, heavy-tailed shape. This behavior undermines the typical assumption of Gaussian log-returns behind the standard approach to risk management and option pricing. Yet, there is no consensus on what class of probability distributions should be adopted to describe financial returns and different models used in the literature have demonstrated, to varying extent, an ability to reproduce empirically observed stylized facts. In order to provide some clarity, in this paper we perform a thorough study of the most popular models of return distributions as obtained in the empirical analyses of high-frequency financial data. We compare the statistical properties and simulate the dynamics of non-Gaussian financial fluctuations by means of Monte Carlo sampling from the different models in terms of realistic tail exponents. Our findings show a noticeable consistency between the considered return distributions in the modeling of the scaling properties of large price changes. We also discuss the convergence rate to the asymptotic distributions of the non-Gaussian stochastic processes and we study, as a first example of possible applications, the impact of our results on option pricing in comparison with the standard Black and Scholes approach.