Stochastic Properties Research Articles

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.

Stochastic properties of generalized finite α-mixtures

Most of the real-life populations are heterogeneous and homogeneity is often just a simplifying assumption for the relevant statistical analysis. Mixtures of lifetime distributions that correspond to homogeneous subpopulations were intensively studied in the literature. Various distributional and stochastic properties of finite and continuous mixtures were discussed. In this paper, following recent publications, we develop further a mixture concept in the form of the generalized α-mixtures that include all mixture models that are widely explored in the literature. We study some main stochastic properties of the suggested mixture model, that is, aging and appropriate stochastic comparisons. Some relevant examples and counterexamples are given to illustrate our findings.

Memory order decomposition of symbolic sequences.

We introduce a general method for the study of memory in symbolic sequences based on higher-order Markov analysis. The Markov process that best represents a sequence is expressed as a mixture of matrices of minimal orders, enabling the definition of the so-called memory profile, which unambiguously reflects the true order of correlations. The method is validated by recovering the memory profiles of tunable synthetic sequences. Finally, we scan real data and showcase with practical examples how our protocol can be used to extract relevant stochastic properties of symbolic sequences.

Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

Kinematic dynamo in incompressible isotropic turbulent flows with high magnetic Prandtl number is considered. The approach interpreting an arbitrary magnetic field distribution as a superposition of localized perturbations (blobs) is developed. We derive a general relation between stochastic properties of an isolated blob and a stochastically homogenous distribution of magnetic field advected by the same stochastic flow. This relation allows us to investigate the evolution of a localized blob at a late stage when its size exceeds the viscous scale. It is shown that in three-dimensional flows, the average magnetic field of the blob increases exponentially in the inertial range of turbulence, as opposed to the late-batchelor stage when it decreases. Our approach reveals the mechanism of dynamo generation in the inertial range both for blobs and homogenous contributions. It explains the absence of dynamo in the two-dimensional case and its efficiency in three dimensions. We propose a way to observe the mechanism in numerical simulations.

Common and Idiosyncratic Conditional Volatility Factors: Theory and Empirical Evidence

We propose a multiplicative dynamic factor structure for the conditional modelling of the variances of an N-dimensional vector of financial returns. We identify common and idiosyncratic conditional volatility factors. The econometric framework is based on an observation-driven time series model that is simple and parsimonious. The common factor is modeled by a normal density and is robust to fat-tailed returns as it averages information over the cross-section of the observed N-dimensional vector of returns. The idiosyncratic factors are designed to capture the erratic shocks in returns and therefore rely on fat-tailed densities. Our model is potentially of a high-dimension, is parsimonious and it does not necessarily suffer from the curse of dimensionality. The relatively simple structure of the model leads to simple computations for the estimation of parameters and signal extraction of factors. We derive the stochastic properties of our proposed dynamic factor model, including bounded moments, stationarity, ergodicity, and filter invertibility. We further establish consistency and asymptotic normality of the maximum likelihood estimator. The finite sample properties of the estimator and the reliability of our method to track the common conditional volatility factor are investigated by means of a Monte Carlo study. Finally, we illustrate our approach with two empirical studies. The first study is for a panel of financial returns from ten stocks of the S&P100. The second study is for the panel of returns from all S&P100 stocks.

An overview on recent progress of extended state observers for uncertain systems: Methods, theory, and applications

AbstractExtended state observer (ESO) is featured with online estimating both the system's states and the total effect of external disturbance as well as nonlinear uncertain dynamics. The original ESO design, proposed for the cascade‐of‐integrator system without measurement noise, has been well studied with theoretical foundation and many successful applications. In the last years, multifarious modified ESOs have also been developed to handle many engineering systems captured by the models with general nonlinear structure, time‐delay, the noises with stochastic properties, and etc. This paper aims to comprehensively investigate the representative modified ESOs in recent years. Firstly, the ESO's design and tuning law are illustrated for general uncertain systems with multiple control inputs, multiple measured outputs and various uncertainties. The quantitative results regarding the estimation errors of ESO are also presented. Next, the extended state‐based Kalman‐typed filters to handle state estimation problem against both stochastic noises and uncertain dynamics are introduced. The consistency, stability, and asymptotic optimization of the proposed filters are rigorously shown. Furthermore, some novel designs of ESO to deal with the estimation problem of uncertain nonlinear systems with time‐delay and measurement biases are discussed. We believe this overview will be helpful for practitioners in applications of recent modified ESOs.

The persistence of inequality across Indian states: A time series approach

AbstractThere is a growing literature on the importance of persistent regional inequalities in developing countries, with evidence that the lack of convergence in incomes across regions is associated with a rise in conflict and social unrest. However, there is a relative paucity of literature on the stochastic characteristics of gross domestic product at the sub‐national level. In this paper we identify new stochastic properties of Indian states' gross domestic product between 1960 and 2019 using a fractional stochastic convergence approach. We test for fractional stochastic convergence in relative incomes in order to identify high persistence and mean reversion. Interval estimates of the largest autoregressive coefficient for the relative incomes of Indian states are wide, thus including many alternatives that are persistent. Interval estimates of the half‐life of relative income shocks suggest that in several cases shocks die out within 0–10 years. Finally, we estimate a fractionally integrated model and obtain mixed evidence of mean reversion and non‐stationarity, with six out of 16 states experiencing mean reversion. These results are highly encouraging and contradict earlier studies which show long‐term divergence and polarization of regional incomes across Indian states, and are thus of great relevance to policy‐makers.

Entropy and stochastic properties in catalysis at nanoscale

This work approaches the Michaelis-Menten model for enzymatic reactions at a nanoscale, where we focus on the quasi-stationary state of the process. The entropy and the kinetics of the stochastic fluctuations are studied to obtain new understanding about the catalytic reaction. The treatment of this problem begins with a state space describing an initial amount of substrate and enzyme-substrate complex molecules. Using the van Kampen expansion, this state space is split into a deterministic one for the mean concentrations involved, and a stochastic one for the fluctuations of these concentrations. The probability density in the fluctuation space displays a behavior that can be described as a rotation, which can be better understood using the formalism of stochastic velocities. The key idea is to consider an ensemble of physical systems that can be handled as if they were a purely conceptual gas in the fluctuation space. The entropy of the system increases when the reaction starts and slightly diminishes once it is over, suggesting: 1. The existence of a rearrangement process during the reaction. 2. According to the second law of thermodynamics, the presence of an external energy source that causes the vibrations of the structure of the enzyme to vibrate, helping the catalytic process. For the sake of completeness and for a uniform notation throughout this work and the ones referenced, the initial sections are dedicated to a short examination of the master equation and the van Kampen method for the separation of the problem into a deterministic and stochastic parts. A Fokker-Planck equation (FPE) is obtained in the latter part, which is then used as grounds to discuss the formalism of stochastic velocities and the entropy of the system. The results are discussed based on the references cited in this work.

ON THE ELL FAMILY OF DISTRIBUTIONS WITH ACTUARIAL APPLICATIONS

AbstractIn this paper, a new three-parameter discrete family of distributions, the $r{\mathcal B}ell$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

Influence of Wind Power on Modeling of Bidding Strategy in a Promising Power Market with a Modified Gravitational Search Algorithm

It is expected that large-scale producers of wind energy will become dominant players in the future electricity market. However, wind power output is irregular in nature and it is subjected to numerous fluctuations. Due to the effect on the production of wind power, producing a detailed bidding strategy is becoming more complicated in the industry. Therefore, in view of these uncertainties, a competitive bidding approach in a pool-based day-ahead energy marketplace is formulated in this paper for traditional generation with wind power utilities. The profit of the generating utility is optimized by the modified gravitational search algorithm, and the Weibull distribution function is employed to represent the stochastic properties of wind speed profile. The method proposed is being investigated and simplified for the IEEE-30 and IEEE-57 frameworks. The results were compared with the results obtained with other optimization methods to validate the approach.

Information criteria and cross validation for Bayesian inference in regular and singular cases

In data science, an unknown information source is estimated by a predictive distribution defined from a statistical model and a prior. In an older Bayesian framework, it was explained that the Bayesian predictive distribution should be the best on the assumption that a statistical model is convinced to be correct and a prior is given by a subjective belief in a small world. However, such a restricted treatment of Bayesian inference cannot be applied to highly complicated statistical models and learning machines in a large world. In 1980, a new scientific paradigm of Bayesian inference was proposed by Akaike, in which both a model and a prior are candidate systems and they had better be designed by mathematical procedures so that the predictive distribution is the better approximation of unknown information source. Nowadays, Akaike’s proposal is widely accepted in statistics, data science, and machine learning. In this paper, in order to establish a mathematical foundation for developing a measure of a statistical model and a prior, we show the relation among the generalization loss, the information criteria, and the cross-validation loss, then compare them from three different points of view. First, their performances are compared in singular problems where the posterior distribution is far from any normal distribution. Second, they are studied in the case when a leverage sample point is contained in data. And last, their stochastic properties are clarified when they are used for the prior optimization problem. The mathematical and experimental comparison shows the equivalence and the difference among them, which we expect useful in practical applications.

A note on comparison theorems for graphs

In the present note we are concerned with the study of curvature-based comparison theorems on graphs related to main stochastic properties, such as the Feller property and stochastic completeness. We show that, under our main hypothesis, whilst previous results concerning stochastic properties are improved, it is not possible to obtain comparison theorems concerning volume growth. Finally, we prove an analogue of the Bishop-Gromov's relative volume comparison theorem and present a series of examples related to various possible notions of curvature.

Monitoring mean changes in persistent multivariate time series

ABSTRACT We focus on sequential (online) monitoring of changes in the mean vector of high-dimensional persistent VARMA time series by using multivariate control charts. Applying either modified or residual control charts to the original process or to the process of VARMA residuals could be problematic when the degree of autoregressive persistence is high. To overcome these difficulties, we suggest to monitor the process of the vector first differences of the VARMA series. We derive the stochastic properties of this difference vector process which are required for the design of the multivariate control charts. Then, we illustrate the performance of our approach in a Monte Carlo simulation study for detection of various types of mean changes.

A CHAOTIC MULTILAYER LIF SCHEME TO MODEL THE PRIMARY VISUAL CORTEX

Precise mathematical modeling of the primary visual cortex (V1) is still a challenging problem. Due to the high similarity of visual system of cat and human, in this paper, we present a hybrid model to track the electrical responses of neurons that are measured by a multi-electrode array implanted in cat V1. The proposed model combines a stochastic phenomenological model with a multilayer leaky integrate-and-fire (LIF) model to predict V1 responses. Since all the existing visual cortex models do not capture the stochastic properties of synaptic changes, the proposed phenomenological model provides input currents for V1 by utilizing continuous chaotic neural equations with a quantization rule. Then a multilayer LIF model is presented to mimic the functions of lateral geniculate nucleus (LGN) and V1 neurons by their corresponding differential equations. The input current in these models is from the presynaptic neurons, which are computed using the LIF model. The LGN-V1 neuronal connections are adopted from previous studies, where the receptive fields (RFs) of LGN neurons converge onto elongated spatial structures that denote RFs of V1 neurons. The main purpose of this paper is to develop a short-term plasticity model that is more consistent with the nature of the LGN and V1 responses compared to state-of-the-art models. Previous studies have not incorporated the stochastic and quantized behaviors of neurons that in the recorded data of implemented electrodes. The experimental results show the ability of the proposed model to accurately predict spikes recorded experimentally, indicating the model outperforms the state-of-the-art method.

Advanced InSAR Tropospheric Corrections From Global Atmospheric Models that Incorporate Spatial Stochastic Properties of the Troposphere

AbstractTropospheric delays are still the main error source of satellite‐based Interferometric Synthetic Aperture Radar (InSAR) mapping of Earth’s surface movements. Recent studies have demonstrated the potential of global atmospheric models (GAMs) in reducing InSAR tropospheric delays. However, the importance of appropriate interpolation and weighting strategies in GAM corrections has largely been overlooked. Here we present a new GAM‐based tropospheric correction method that incorporates spatial stochastic models of the troposphere into the weighting strategy of the correction. The method determines the correlation between a pixel of interest and neighboring GAM grid locations (3D) according to the spatial variability of the tropospheric random field, instead of subjectively using an inverse distance method, a local spline function, or other standard interpolation scheme. Also, our new method considers horizontal heterogeneities of the tropospheric field by estimating the integral of the tropospheric delays along the satellite line‐of‐sight (LOS) direction, instead of calculating projected zenith‐delays. The method can be used with any GAM, but we here implement it with the latest ECMWF (European Center for Medium‐Range Weather Forecasts) ERA5 reanalysis products. We validate the new method with hundreds of Sentinel‐1 images from 2015 to 2020 over the island of Hawaii, a location with variable topography, surface conditions, local climate, and deformation, and explore the tropospheric corrections for both interferograms and time‐series analysis products (deformation velocities and time‐series solutions). Compared with other GAM corrections (PyAPS, d‐LOS, and GACOS), our new method yields a larger reduction of the average standard deviation of the corrected interferograms, i.e., from 2.55 to 1.91 cm, instead of 2.47 cm (PyAPS), 2.44 cm (d‐LOS), and 2.10 cm (GACOS). Also, a larger fraction of 87% of the interferograms (243 out of 280) is improved, compared with 52%, 53%, and 66% for the other GAM corrections, respectively. These results demonstrate the importance of considering (1) tropospheric stochastic models in GAM corrections, (2) horizontal heterogeneities when estimating the LOS delays, and (3) tropospheric delays when mapping long‐wavelength or small‐magnitude deformations using InSAR.

Plasticity and fracture of cast and SLM AlSi10Mg: High-throughput testing and modeling

Both additively-manufactured and cast metals are known to exhibit stochastic mechanical properties at the macroscopic level. Using a robot-assisted mechanical testing system, more than 360 experiments are performed on flat specimens extracted from AlSi10Mg components that are either cast or made through laser powder bed fusion (LPBF), commonly known as selective laser melting (SLM). Aside from basic EBSD analysis of the respective microstructures, micro-computed tomography is performed revealing a significantly higher porosity and pore size for the cast material. The results from uniaxial tension experiments reveal a 10% higher yield strength (on average) and an about 20% higher ultimate tensile strength for the SLM made AlSi10Mg. The tracking of the specimen origin within the SLM component shows a clear location dependence of the observed hardening response on the build height. The shear and tension fracture experiments revealed a strong stress-state dependence and significantly higher fracture strains for the cast material as compared to its SLM-made counterpart. To facilitate the computer aided engineering of structures with additively-manufactured AlSiMg alloys, a build height dependent hardening model is proposed along with a probabilistic plasticity and fracture modeling framework. • Performed about 360 plasticity and fracture experiments using a robotic system. • Employed micro-CT and EBSD to characterize microstructural features. • SLM-made AlSi10Mg is stronger, but also more brittle than its cast counterpart. • Property variations are due to porosity in castings and build height effect in the SLM. • Stochastic plasticity and fracture models are determined.

Nucleation of frictional sliding by coalescence of microslip

The onset of frictional motion is mediated by the dynamic propagation of a rupture front, analogous to a shear crack. The rupture front nucleates quasi-statically in a localized region of the frictional interface and slowly increases in size. When it reaches a critical nucleation length it becomes unstable, propagates dynamically and eventually breaks the entire interface, leading to macroscopic sliding. The nucleation process is particularly important because it determines the stress level at which the frictional interface fails, and therefore, the macroscopic friction strength. However, the mechanisms governing nucleation of frictional rupture fronts are still not well understood. Specifically, our knowledge of the nucleation process along a heterogeneous interface remains incomplete. Here, we study the nucleation of localized slip patches on linear slip-weakening interfaces with deterministic and stochastic heterogeneous friction properties. Using numerical simulations, we analyze the process leading to a slip patch of critical size for systems with varying correlation lengths of the local friction strength. Our deterministic interface model reveals that the growth of the critical nucleation patch at interfaces with small correlation lengths is non smooth due to the coalescence of neighboring slip patches. Existing analytical solutions do not account for this effect, which leads to an overestimation of global interface strength. Conversely, when the correlation length is large, the growth of the slip patch is continuous and our simulations match the analytical solution. Furthermore, nucleation by coalescence is also observed on stochastic interfaces with small correlation length. In this case, the applied load for a given slip patch size is a random variable. We show that its expectation follows a logistic function, which allows us to predict the strength of the interface well before failure occurs. Our model and observations provide new understanding of the nucleation process and its effect on the static frictional strength.

A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation

Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion.

Matrix Fisher–Gaussian Distribution on $\mathrm{SO}(3)\times \mathbb {R}^{ n }$ and Bayesian Attitude Estimation

In this article, a new probability distribution, referred to as the matrix Fisher–Gaussian distribution, is proposed on the product manifold of three-dimensional special orthogonal group and Euclidean space. It is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space into the manifold, while imposing a certain geometric constraint on the correlation term to avoid over parameterization. The unique feature is that it may represent large uncertainties in attitudes, linear variables of an arbitrary dimension, and angular–linear correlations between them in a global fashion without singularities. Various stochastic properties and an approximate maximum likelihood estimator are developed. Furthermore, two methods are developed to propagate uncertainties through a stochastic differential equation representing attitude kinematics. Based on these, a Bayesian estimator is proposed to estimate the attitude and time-varying gyro bias concurrently. Numerical studies indicate that the proposed estimator provides more accurate estimates against the multiplicative extended Kalman filter and unscented Kalman filter for challenging cases.

Stochastic investigation of daily air temperature extremes from a global ground station network

Near-surface air temperature is one of the most widely studied hydroclimatic variables, as both its regular and extremal behaviors are of paramount importance to human life. Following the global warming observed in the past decades and the advent of the anthropogenic climate change debate, interest in temperature’s variability and extremes has been rising. It has since become clear that it is imperative not only to identify the exact shape of the temperature’s distribution tails, but also to understand their temporal evolution. Here, we investigate the stochastic behavior of near-surface air temperature using the newly developed estimation tool of Knowable (K-)moments. K-moments, because of their property to substitute higher-order deviations from the mean with the distribution function, enable reliable estimation and an effective alternative to order statistics and, particularly for the outliers-prone distribution tails. We compile a large set of daily timeseries (30–200 years) of average, maximum and minimum air temperature, which we standardize with respect to the monthly variability of each record. Our focus is placed on the maximum and minimum temperatures, because they are more reliably measured than the average, yet very rarely analyzed in the literature. We examine segments of each timeseries using consecutive rolling 30-year periods, from which we extract extreme values corresponding to specific return period levels. Results suggest that the average and minimum temperature tend to increase, while overall the maximum temperature is slightly decreasing. Furthermore, we model the temperature timeseries as a filtered Hurst-Kolmogorov process and use Monte Carlo simulation to produce synthetic records with similar stochastic properties through the explicit Symmetric Moving Average scheme. We subsequently evaluate how the patterns observed in the longest records can be reproduced by the synthetic series.