A simulation method for stationary Gaussian random functions based on the sampling theorem

A unified method is developed for simulating realizations of real-valued stationary Gaussian processes, vector processes, fields, and vector fields. The method is based on parametric random models consisting of superpositions of deterministic functions of time or space with random amplitudes. The parametric models are based on the sampling theorem for random processes and generalizations of it for vector processes and random fields. The proposed simulation method is efficient and uses algorithms for generating realizations of random processes and fields that are similar to simulation techniques based on ARMA models. Several examples are presented to demonstrate the proposed simulation method and evaluate its efficiency and accuracy.     

Solution techniques for pulse problems in non-linear stochastic dynamics

Advantages and disadvantages of available solution techniques for pulse problems in non-linear stochastic dynamics are discussed. First, random pulse problems, both, those which do and do not lead to Markov theory, are presented. Next, the analytical and analytically numerical techniques suitable for Markov response problems such as moment equations, Petrov–Galerkin and cell-to-cell mapping techniques are briefly discussed. Usefulness of these techniques is limited by the fact that effectiveness of each of them depends on the mean rate of impulses. Another limitation is the size of the problem, i.e. the number of state variables of the dynamical system. In contrast, the applicability of the simulation techniques is not limited to Markov problems, nor is it dependent on the mean rate of impulses. Moreover their use is straightforward for a large class of point processes, at least for renewal processes.     

Karhunen–Loève expansion for multi-correlated stochastic processes

We propose two different approaches generalizing the Karhunen–Loève series expansion to model and simulate multi-correlated non-stationary stochastic processes. The first approach (muKL) is based on the spectral analysis of a suitable assembled stochastic process and yields series expansions in terms of an identical set of uncorrelated random variables. The second approach (mcKL) relies on expansions in terms of correlated sets of random variables reflecting the cross-covariance structure of the processes. The effectiveness and the computational efficiency of both muKL and mcKL is demonstrated through numerical examples involving Gaussian processes with exponential and Gaussian covariances as well as fractional Brownian motion and Brownian bridge processes. In particular, we study accuracy and convergence rates of our series expansions and compare the results against other statistical techniques such as mixtures of probabilistic principal component analysis. We found that muKL and mcKL provide an effective representation of the multi-correlated process that can be readily employed in stochastic simulation and dimension reduction data-driven problems.     

Fatigue life prediction under wide band random loading

In this paper, a method for the high‐cycle fatigue life prediction of components subjected to gaussian, stationary, wide band random loading is presented. It allows the user to evaluate the fatigue life of components subjected to uniaxial stress states directly from the stress power spectral density (PSD), avoiding onerous simulations in time domain. The proposed method can be applied to random stress processes having PSD of any shape, and the fatigue life predictions obtained are more accurate than that provided by most of the frequency domain techniques proposed in literature.     

非线性系统响应计算的一种数字模拟方法

目前,对单自由度非线性振子受随机激励的响应计算已经发展到了多种方法。本文从白噪音与泊松过程的联系出发,把系统受白噪音或调制白噪音激励的响应计算转换为对泊松随机脉冲激励的响应计算,从而发展了一种数字模拟方法。此法与其它模拟计算方法所得结果有良好的一致性,并具有较高的计算效率。     

Drilling intrusion probabilities for use in performance assessment for radioactive waste disposal

Product integral techniques are used to derive computational procedures to determine probabilities for scenarios resulting from drilling intrusions at geologic disposal facilities for radioactive waste. For these derivations, the occurence of individual drilling intrusions is assumed to be random in time and space, although the drilling rate is not assumed to be constant or even continuous through time. The use of product integral techniques allows the probability model for drilling intrusions to be initially cast in a very general form involving interval functions and then specialized to nonhomogeneous Poisson processes and ultimately to homogeneous Poisson processes. The resultant computational procedures are illustrated with results for the Waste Isolation Pilot Plant (WIPP) in southeastern New Mexico using the maximum drilling rate specified by the US Environmental Protection Agency (EPA) in its standard for the geologic disposal of radioactive waste (40 CFR 191).     

脉动风速过程模拟的正交展开-随机函数方法

在随机过程的正交展开基础上,采用随机函数的思想,提出了脉动风速随机过程模拟的正交展开-随机函数方法。通过将展开式中的一组标准正交随机变量表达为基本随机变量的正交函数形式,实现了用一个基本随机变量来表达原随机过程的目的。应用正交展开-随机函数方法,对脉动风速随机过程进行模拟分析。结果表明,在二阶数值统计意义上,仅需用一个基本随机变量即可对脉动风速随机过程进行精确模拟,进而体现了正交展开-随机函数方法的有效性和优越性。     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Uncertainty in analyses of one-dimensional steady-state seepage through random porous media

The aim of this paper is to assess the propagation of uncertainty in analyses of one-dimensional steady-state flow through random porous media. It is considered that uncertainty originates from lack of precise knowledge of soil hydraulic conductivity. This uncertainty is modelled by means of random variables. Taking into account experimental evidences, it is accepted that hydraulic conductivity has a log-normal probability distribution. The paper focuses on propagation of uncertainty from hydraulic conductivity to the computed flow rate through homogeneous and stratified random materials. The most common techniques available to evaluate propagation of uncertainty are briefly reviewed. The applicability and limitations of these techniques are assessed. Parametric studies to gauge the effect of uncertainty on soil hydraulic conductivity on the output statistics of flow rate are performed. Attenuation of uncertainty on flow rate as the number of materials in stratified soils increases is evaluated. Conclusions are presented regarding this attenuation and the usefulness of the different stochastic techniques employed, which proved to be more complementary than antagonistic.     

Time-Domain Measurements of Microwave Components

Recent advances in microwave component measurements using time-domain techniques are described. After reviewing the basic elements of a time-domain system, a substitution procedure is applied to determine the insertion loss of wide-band attenuators. Comparison of these measurements with frequency calibrations shows agreement to within 0.1 dB in 10 dB for attenuators between 10-50 dB, over the frequency range 0.4-8 GHz. Error sources are resolved by experiments designed to isolate and evaluate various contributions including: random errors due to noise and drifts; systematic errors caused by substitution attenuator inaccuracies, line mismatch, deflection nonlinearities, and inaccurate time window widths; time-to-frequency translation errors of aliasing and truncation; and mechanical errors due to connect-disconnect cycles. Results show that random processes are responsible for most of the observed error. The reported measurements establish the calibration capabilities and the expected magnitude of individual system errors for the particular system tested.     

Combination of primary load effects in ship structures

The models currently used for representing the still-water and the wave-induced load effects in ship structures are discussed. Models based both on random variables and on stochastic processes are considered. Load combination solutions are obtained based on the Ferry Borges-Castanheta model for random variables and also on an upcrossing formulation for stochastic processes. Load combination factors appropriate for code purposes are suggested and their values are derived from comparison with the load combination results.     

Nonstationary response of nonlinear systems using equivalent linearization with a compact analytical form of the excitation process

A new method based on equivalent linearization approaches is presented for estimating the nonstationary response of a class of nonlinear multi-degree-of-freedom systems subjected to nonstationary excitations. The highly efficient method is based on creating a compact analytical approximation of measured nonstationary excitation process data through use of a two-stage decomposition procedure. The analytic data condensation of the excitation process is performed in two stages; (1) by performing the Karhunen–Loeve spectral decomposition on the covariance matrix of the input random process to obtain the dominant eigenvectors, and (2) by fitting these eigenvectors with orthogonal polynomials to produce a truncated series of analytically approximated eigenvectors. The efficiency and accuracy of the method is demonstrated through simulation with synthetically generated excitation data as well as measured data from a real-world physical process. Although the decomposition procedure used can characterize very general input processes, because the equivalent linearization technique requires the Gaussian assumption of the response process, the constraint on applying this approach is similar to the constraints on all other equivalent linearization techniques. However, the additional freedom gained from being able to work with data-based nonstationary random processes is a significant addition to this area of research.     

Translation vectors with non-identically distributed components

A model for non-Gaussian random vectors is presented that relies on a modification of the standard translation transformation which has previously been used to model stationary non-Gaussian processes and non-Gaussian random vectors with identically distributed components. The translation model has the ability to exactly match target marginal distributions and a broad variety of correlation matrices. Joint distributions of the new class of translation vectors are derived, as are upper and lower bounds on the target correlation that depend on the target marginal distributions. Examples are presented that demonstrate the applicability of the approach to the modelling of heterogeneous material properties, and also illustrate the possible shortcomings of using second moment characterizations for such random vectors. Lastly, an outline is given of a method under development for extending the model to non-stationary, non-Gaussian random processes.     

Simulations of turbulence-induced phase and log-amplitude distortions

A new method, to our knowledge, allowing one to simulate correlated random processes is suggested. Structure (or correlation) functions of the processes under simulation are assumed to be given. The method is based on the generation of random wave vectors that allows one to simulate processes for a wide class of structure functions. The validity of the method proposed is illustrated by simulations of the turbulence-induced log-amplitude and phase distortions.     

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide (i) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, (ii) simple solutions for linear random vibration problems with non-Gaussian input, and (iii) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.     

Change‐point Detection in Phase I for Profiles with Binary Data and Random Predictors

In some applications, the quality of a process must be characterized by a profile, which describes the relationship between the response variable and explanatory variables. Moreover, for some processes, especially service processes, categorical response variables are common, making statistical process control techniques for profiles with categorical response data a must. We study Phase I analysis of profiles with binary data and random explanatory variables to identify the presence of change‐points in the reference profile dataset. The change‐point detection method based on logistic regression models is proposed. The method exploits directional shift information and integrates change‐point algorithm with the generalized likelihood ratio. A diagnostic scheme for identifying the change‐point location and the shift direction is also suggested. Numerical simulations are conducted to demonstrate the detection effectiveness and the diagnostic accuracy. A real example is used to illustrate the implementation of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Random walk models in biology.

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.     

Reliability analysis for multi-state systems subject to distinct random shocks

This paper analyzes the competing and dependent failure processes for multi-state systems suffering from four typical random shocks. Reliability analysis for discrete degradation is conducted by explicitly modeling the state transition characteristics. Semi-Markov model is employed to explore how the system vulnerability and potential transition gap affect the state residence time. The failure dependence is specified as that random shocks can not only lead to different abrupt failures but also cause sudden changes on the state transition probabilities, making it easier for the system to stay at the degraded states. Reliability functions for all the exposed failure processes are presented based on the corresponding mechanisms. Interactions between different failure processes are also taken into account to evaluate the actual reliability levels in the context of degradation and distinct random shocks. An illustrative example of a multi-state air conditioning system is studied to demonstrate how the proposed method can be applied to the engineering practice.     

Digital simulation of an arbitrary stationary stochastic process by spectral representation

In this paper we present a straightforward, efficient, and computationally fast method for creating a large number of discrete samples with an arbitrary given probability density function and a specified spectral content. The method relies on initially transforming a white noise sample set of random Gaussian distributed numbers into a corresponding set with the desired spectral distribution, after which this colored Gaussian probability distribution is transformed via an inverse transform into the desired probability distribution. In contrast to previous work, where the analyses were limited to auto regressive and or iterative techniques to obtain satisfactory results, we find that a single application of the inverse transform method yields satisfactory results for a wide class of arbitrary probability distributions. Although a single application of the inverse transform technique does not conserve the power spectra exactly, it yields highly accurate numerical results for a wide range of probability distributions and target power spectra that are sufficient for system simulation purposes and can thus be regarded as an accurate engineering approximation, which can be used for wide range of practical applications. A sufficiency condition is presented regarding the range of parameter values where a single application of the inverse transform method yields satisfactory agreement between the simulated and target power spectra, and a series of examples relevant for the optics community are presented and discussed. Outside this parameter range the agreement gracefully degrades but does not distort in shape. Although we demonstrate the method here focusing on stationary random processes, we see no reason why the method could not be extended to simulate non-stationary random processes.     

Machine Learning-Informed Predictive Design and Analysis of Electrohydrodynamic Printing Systems

Electrohydrodynamic (EHD) processes are promising techniques for manufacturing nanoscopic products with different shapes (such as thin films, nanofibers, 2D/3D nanostructures, and nanoparticles) and materials at a low cost using simple equipment. A key challenge in their adoption by nonexperts is the requirement of enormous time and resources in identifying the optimum design/process parameters for the underlying material and EHD system. Machine learning (ML) has made exciting advancements in predictive modeling of different processes, provided it is trained on high-quality datasets at appropriate volumes. This article extends the suitability of such ML-enabled approaches to a new technological domain of EHD spraying and drop-on-demand printing. Different ML models like ridge regression, random forest regression, support vector regression, gradient boosting regression, and multilayer perceptron are trained and their performance using evaluation metrics like RMSE and R2_score is examined. Tree-based algorithms like gradient boosting regression are found to be the most suitable technique for modeling EHD processes. The trained ML models show substantially higher accuracy (average error < 5%) in replicating these nonlinear processes as compared to previously reported scaling laws (average error ≈ 42%) and are well suited for predictive modeling/analysis of the underlying EHD system and process.