Nonstationary Stochastic Processes Research Articles

Time–frequency varying response functions of non-classically damped linear structures under fully non-stationary stochastic excitations

In order to reproduce typical characteristics of real earthquake ground-motion time histories, the seismic excitation should be modelled as a fully non-stationary stochastic process. For non-stationary input the time–frequency varying response (TFR) function plays a central role in the evaluation of the spectral characteristics of non-stationary response processes. In fact, by means of this function, it is possible to evaluate in explicit form the evolutionary power spectral density (EPSD) matrix function of the response and consequently the non-geometric spectral moments (NGSM s), which are required in the prediction of the safety of structural systems subjected to non-stationary random excitations.It has been recently recognized that by using supplemental devices it is possible to enhance the performance of structural systems exposed to seismic hazard. However, the introduction of supplemental devices into structures modifies their physical characteristics so that the common assumption of classical or proportional damping is inadequate. It follows that these modified structures must be correctly analysed as non-classically damped ones. To do this the complex modal analysis, which performs the modal superposition in terms of complex eigenvalues and eigenvectors, must be adopted.In this paper a method to evaluate in explicit closed-form solution the TFR vector function as well as the EPSD matrix function of the response of linear non-classically damped structural systems under fully non-stationary excitations is proposed.

Time-delayed feedback control of diffusion in random walkers.

Time delay in general leads to instability in some systems, while specific feedback with delay can control fluctuated motion in nonlinear deterministic systems to a stable state. In this paper, we consider a stochastic process, i.e., a random walk, and observe its diffusion phenomenon with time-delayed feedback. As a result, the diffusion coefficient decreases with increasing delay time. We analytically illustrate this suppression of diffusion by using stochastic delay differential equations and justify the feasibility of this suppression by applying time-delayed feedback to a molecular dynamics model.

Consistency of detrended fluctuation analysis.

The scaling function F(s) in detrended fluctuation analysis (DFA) scales as F(s)∼s^{H} for stochastic processes with Hurst exponent H. This scaling law is proven for stationary stochastic processes with 0<H<1 and nonstationary stochastic processes with 1<H<2. For H<0.5, it is observed that the asymptotic (power-law) autocorrelation function (ACF) scales as ∼s^{1/2}. It is also demonstrated that the fluctuation function in DFA is equal in expectation to (i) a weighted sum of the ACF and (ii) a weighted sum of the second-order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling and to employ DFA in a meaningful sense for signals that do not exhibit power-law statistics. The usefulness is illustrated by examples where it is demonstrated that a previous suggested modified DFA will increase the bias for signals with Hurst exponents 1<H≤1.5. As a final application of these developments, an estimator F[over ̂](s) is proposed. This estimator can handle missing data in regularly sampled time series without the need of interpolation schemes. Under mild regularity conditions, F[over ̂](s) is equal in expectation to the fluctuation function F(s) in the gap-free case.

Selective harvesting over networks

Active search (AS) on graphs focuses on collecting certain labeled nodes (targets) given global knowledge of the network topology and its edge weights under a query budget. However, in most networks, nodes, topology and edge weights are all initially unknown. We introduce selective harvesting, a variant of AS where the next node to be queried must be chosen among the neighbors of the current queried node set; the available training data for deciding which node to query is restricted to the subgraph induced by the queried set (and their node attributes) and their neighbors (without any node or edge attributes). Therefore, selective harvesting is a sequential decision problem, where we must decide which node to query at each step. A classifier trained in this scenario suffers from a tunnel vision effect: without recourse to independent sampling, the urge to query promising nodes forces classifiers to gather increasingly biased training data, which we show significantly hurts the performance of AS methods and standard classifiers. We find that it is possible to collect a much larger set of targets by using multiple classifiers, not by combining their predictions as an ensemble, but switching between classifiers used at each step, as a way to ease the tunnel vision effect. We discover that switching classifiers collects more targets by (a) diversifying the training data and (b) broadening the choices of nodes that can be queried next. This highlights an exploration, exploitation, and diversification trade-off in our problem that goes beyond the exploration and exploitation duality found in classic sequential decision problems. From these observations we propose D3TS, a method based on multi-armed bandits for non-stationary stochastic processes that enforces classifier diversity, matching or exceeding the performance of competing methods on seven real network datasets in our evaluation.

A note on general quadratic forms of nonstationary stochastic processes

ABSTRACTIn this paper, general quadratic forms of nonstationary, α-mixing time series are considered. Under mixing and moment assumptions, asymptotically normality of these forms are derived. These results do not assume that the variance of the generalized quadratic form has a limit, thus allowing for general types of nonstationarity. However, without well-defined limits, it is not possible to understand the differences in sampling properties of quadratic forms of nonstationary and stationary processes. To understand these differences, the nonstationary process is placed within the locally stationary framework. Under the assumption that the nonstationary process is locally stationary the asymptotic expectation and variance of the weighted sample covariance of the discrete Fourier transforms (an important class of quadratic forms) is derived and shown to be very different to its stationary counterpart.

A stochastic harmonic function representation for non-stationary stochastic processes

The time-domain representation of non-stationary stochastic processes is of paramount importance, in particular for response analysis and reliability evaluation of nonlinear structures. In the present paper a stochastic harmonic function (SHF) representation originally developed for stationary processes is extended to evolutionary non-stationary processes. Utilizing the new scheme, the time-domain representation of non-stationary stochastic processes is expressed as the linear combination of a series of stochastic harmonic components. Different from the classical spectral representation (SR), not only the phase angles but also the frequencies and their associated amplitudes, are treated as random variables. The proposed method could also be regarded as an extension of the classical spectral representation method. However, it is rigorously proved that the new scheme well accommodates the target evolutionary power spectral density function. Compared to the classical spectral representation method, moreover, the new scheme needs much fewer terms to be retained. The first four moments and the distribution properties, e.g., the asymptotical Gaussianity, of the simulated stochastic process via SHF representation are studied. Numerical examples are addressed for illustrative purposes, showing the effectiveness of the proposed scheme.

The non-stationary stochastic multi-armed bandit problem

We consider a variant of the stochastic multi-armed bandit with K arms where the rewards are not assumed to be identically distributed, but are generated by a non-stationary stochastic process. We first study the unique best arm setting when there exists one unique best arm. Second, we study the general switching best arm setting when a best arm switches at some unknown steps. For both settings, we target problem-dependent bounds, instead of the more conservative problem-free bounds. We consider two classical problems: (1) identify a best arm with high probability (best arm identification), for which the performance measure by the sample complexity (number of samples before finding a near-optimal arm). To this end, we naturally extend the definition of sample complexity so that it makes sense in the switching best arm setting, which may be of independent interest. (2) Achieve the smallest cumulative regret (regret minimization) where the regret is measured with respect to the strategy pulling an arm with the best instantaneous mean at each step.

Influence of Post-Yield Stiffness on Inelastic Seismic Response: A Stochastic Analysis

The main objective of this study is to evaluate inelastic response spectra conditioned on constant displacement ductility demand for a single degree of freedom system, which has positive post-yield stiffness. To develop this analysis, the nonlinear Bouc–Wen model is used and it is subject to a seismic action modeled by a nonstationary stochastic process. The study utilises stochastic linearisation technique and peak theory to obtain inelastic response spectra. More in detail, inelastic acceleration spectra, response modification factor spectra, strength reduction factor spectra, yield strength spectra, and inelastic displacement ratio spectra are obtained for constant displacement ductility demand. The novelty of the present study with respect to the previous ones in this field is to obtain inelastic response spectra in stochastic terms, starting from a stochastic model of the seismic motion. Results suggest that the effect of post-yield stiffness must be considered when determining inelastic response.

Online detection of continuous changes in stochastic processes

We are concerned with detecting continuous changes in stochastic processes. In conventional studies on non-stationary stochastic processes, it is often assumed that changes occur abruptly. By contrast, we assume that they take place continuously. The proposed scheme consists of an efficient algorithm and rigorous theoretical analysis under the assumption of continuity. The contribution of this paper is as follows: We first propose a novel characterization of processes for continuous changes. We also present a time- and space-efficient online estimator of the characteristics. Then, employing the proposed estimate, we propose a method for detecting changes together with a criterion for tuning its hyper-parameter. Finally, the proposed methods are shown to be effective through experimentation involving real-life data from markets, servers, and industrial machines.

A stochastic model for directional changes of swimming bacteria.

In this work we introduce a stochastic model to describe directional changes in the movement of swimming bacteria. We use the probability density function (PDF) of turn angles, measured on tumbling wild-type E. coli, to build a Langevin equation for the deflection of the bacterial body swimming in isotropic media. We have solved this equation analytically by means of the Green function method and shown that three parameters are sufficient to describe the movement: the characteristic time, the steady-state solution and the control parameter. We conclude that the tumble motion, which is manifested as abrupt turns, is primarily caused by the rotational boost generated by the flagellar motor and complementarily by the rotational diffusion introduced by noise. We show that in the tumble motion the deflection is a non-stationary stochastic process during times at which the tumbling occurs. By tuning the control parameter our model is able to explain small turns of the bacteria around their centres of mass along the run. We show that the deflection during the run is an Ornstein-Uhlenbeck process, which for typical run times is stationary. We conclude that, along the run, the rotational boosts do not exist and that only the rotational diffusion remains. Thus we have a single model to explain the turns of the bacterium during the run or tumble movements, through a control parameter that can be tuned through a critical value that can explain the transition between the two turn behaviours. This model is also able to explain in a very satisfactory way all available statistical experimental data, such as PDFs and average values of turning angles times, of both run and tumble motions.

A first passage based model for probabilistic fracture of polycrystalline silicon MEMS structures

Experiments have shown that the failure loads of Microelectromechanical Systems (MEMS) devices usually exhibit a considerable level of variability, which is believed to be caused by the random material strength and the geometry-induced random stress field. Understanding the strength statistics of MEMS devices is of paramount importance for the device design guarding against a tolerable failure risk. In this study, we develop a continuum-based probabilistic model for polycrystalline silicon (poly-Si) MEMS structures within the framework of first passage analysis. The failure of poly-Si MEMS structures is considered to be triggered by fracture initiation from the sidewalls governed by a nonlocal failure criterion. The model takes into account an autocorrelated random field of material tensile strength. The nonlocal random stress field is obtained by stochastic finite element simulations based on the information of the uncertainties of the sidewall geometry. The model is formulated within the contexts of both stationary and non-stationary stochastic processes for MEMS structures of various geometries and under different loading configurations. It is shown that the model agrees well with the experimentally measured strength distributions of uniaxial tensile poly-Si MEMS specimens of different gauge lengths. The model is further used to predict the strength distribution of poly-Si MEMS beams under three-point bending, and the result is compared with the Monte Carlo simulation. The present model predicts strong size effects on both the strength distribution and the mean structural strength. It is shown that the mean size effect curve consists of three power-law asymptotes in the small, intermediate, and large-size regimes. By matching these three asymptotes, an approximate size effect equation is proposed. The present model is shown to be a generalization of the classical weakest-link statistical model, and it provides a physical interpretation of the material length scale used in the weakest-link model.

Generalization bounds for non-stationary mixing processes

This paper presents the first generalization bounds for time series prediction with a non-stationary mixing stochastic process. We prove Rademacher complexity learning bounds for both average-path generalization with non-stationary $$\beta $$β-mixing processes and path-dependent generalization with non-stationary $$\phi $$ź-mixing processes. Our guarantees are expressed in terms of $$\beta $$β- or $$\phi $$ź-mixing coefficients and a natural measure of discrepancy between training and target distributions. They admit as special cases previous Rademacher complexity bounds for non-i.i.d. stationary distributions, for independent but not identically distributed random variables, or for the i.i.d. case. We show that, using a new sub-sample selection technique we introduce, our bounds can be tightened under the natural assumption of asymptotically stationary stochastic processes. We also prove that fast learning rates can be achieved by extending existing local Rademacher complexity analyses to the non-i.i.d. setting. We conclude the paper by providing generalization bounds for learning with unbounded losses and non-i.i.d. data.

Combined models for pre- and post-treatment longitudinal biomarker data: an application to CD4 counts in HIV-patients.

BackgroundThere has been some debate in the literature as to whether baseline values of a measurement of interest at treatment initiation should be treated as an outcome variable as part of a model for longitudinal change or instead used as a predictive variable with respect to the response to treatment. We develop a new approach that involves a combined statistical model for all pre- and post-treatment observations of the biomarker of interest, in which the characteristics of response to treatment are treated as a function of the ‘true’ value of the biomarker at treatment initiation.MethodsThe modelling strategy developed is applied to a dataset of CD4 counts from patients in the UK Register of HIV Seroconverters (UKR) cohort who initiated highly active antiretroviral therapy (HAART). The post-HAART recovery in CD4 counts for each individual is modelled as following an asymptotic curve in which the speed of response to treatment and long-term maximum are functions of the ‘true’ underlying CD4 count at initiation of HAART and the time elapsed since seroconversion. Following previous research in this field, the models developed incorporate non-stationary stochastic process components, and the possibility of between-patient differences in variability over time was also considered.ResultsA variety of novel models were successfully fitted to the UKR dataset. These provide reinforcing evidence for findings that have previously been reported in the literature, in particular that there is a strong positive relationship between CD4 count at initiation of HAART and the long-term maximum in each patient, but also reveal potentially important features of the data that would not have been easily identified by other methods of analysis.ConclusionOur proposed methodology provides a unified framework for the analysis of pre- and post-treatment longitudinal biomarker data that will be useful for epidemiological investigations and simulations in this context. The approach developed allows use of all relevant data from observational cohorts in which many patients are missing pre-treatment measurements and in which the timing and number of observations vary widely between patients.Electronic supplementary materialThe online version of this article (doi:10.1186/s12874-016-0187-2) contains supplementary material, which is available to authorized users.

Analysis of the Effect of Spatial Uncertainties on the Dynamic Behavior of Electrostatic Microactuators

AbstractThis paper examines the effect of spatial roughness on the dynamical behaviour of electrostatic microactuators. We develop a comprehensive physical model that comprises a nonlinear electrostatic actuation force aswell as a squeeze-film damping term to accurately simulate the dynamical behavior of a cantilever beam actuator. Spatial roughness is modeled as a nonstationary stochastic process whose parameters can be estimated from profilometric measurements. We propagate the stochastic model through the physical system and examine the resulting uncertainty in the dynamical behavior that manifests as a variation in the quality factor of the device. We identify two distinct, yet coupled, modes of uncertainty propagation in the system, that result from the roughness causing variation in the electrostatic actuation force and the damping pressure, respectively. By artificially turning off each of these modes of propagation in sequence, we demonstrate that the variation in the damping pressure has a greater effect on the damping ratio than that arising from the electrostatic force. Comparison with similar simulations performed using a simplified mass-spring-damper model show that the coupling between these two mechanisms can be captured only when the physical model includes the primary nonlinear interactions along with a proper treatment of spatial variations. We also highlight the difference between nonstationary and stationary covariance formulations by showing that the stationary model is unable to properly capture the full range of variation as compared to its nonstationary counterpart.

STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

Incremental Dynamic Analysis: A Nonlinear Stochastic Dynamics Perspective

AbstractAn efficient stochastic incremental dynamic analysis (IDA) methodology for nonlinear/hysteretic oscillators is developed by resorting to nonlinear stochastic dynamics concepts and tools such as stochastic averaging and statistical linearization. Specifically, modeling the excitation as a nonstationary stochastic process possessing an evolutionary power spectrum (EPS), an approximate closed-form expression is derived for the parameterized oscillator response amplitude probability density function (PDF). In this regard, an IDA surface is determined providing the PDF of the engineering demand parameter (EDP) for a given intensity measure (IM) value. In contrast to a computationally expensive Monte Carlo simulation (MCS) based determination of the IDA surface, the methodology developed herein determines the EDP PDF at minimal computational cost. Further, an approximate closed-form expression is derived for the parameterized nonlinear oscillator response EPS as well; thus, a conceptually novel IDA surf...

Random function based spectral representation of stationary and non-stationary stochastic processes

In conjunction with the formulation of random functions, a family of renewed spectral representation schemes is proposed. The selected random function serves as a random constraint correlating the random variables included in the spectral representation schemes. The objective stochastic process can thus be completely represented by a dimension-reduced spectral model with just few elementary random variables, through defining the high-dimensional random variables of conventional spectral representation schemes (usually hundreds of random variables) into the low-dimensional orthogonal random functions. To highlight the advantages of this scheme, orthogonal trigonometric functions with one and two random variables are constructed. Representative-point set of the dimension-reduced spectral model is derived by employing the probability-space partition techniques. The complete set with assigned probabilities of points gains a low-number-sample stochastic process. For illustrative purposes, the stochastic modeling of seismic acceleration processes is proceeded, of which the stationary and non-stationary cases are investigated. It is shown that the spectral acceleration of simulated processes matches well with the target spectrum. Stochastic seismic response analysis, moreover, and reliability assessment of a framed structure with Bouc-Wen behaviors are carried out using the probability density evolution method. Numerical results reveal the applicability and efficiency of the proposed simulation technique.

Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

Nous developpons et generalisons la theorie des valeurs extremes pour des processus stochastiques non-stationnaires, en affaiblissant la condition de melange uniforme qui avait ete utilisee auparavant. Nous appliquons nos resultats a des systemes dynamiques non autonomes, en particulier aux systemes dynamiques sequentiels engendres par des applications dilatantes et a une large classe de systemes dynamiques aleatoires. Quelques exemples sont presentes et calcules en detail.

Fractional Brownian motion and multivariate-t models for longitudinal biomedical data, with application to CD4 counts in HIV-positive patients.

Longitudinal data are widely analysed using linear mixed models, with ‘random slopes’ models particularly common. However, when modelling, for example, longitudinal pre‐treatment CD4 cell counts in HIV‐positive patients, the incorporation of non‐stationary stochastic processes such as Brownian motion has been shown to lead to a more biologically plausible model and a substantial improvement in model fit. In this article, we propose two further extensions. Firstly, we propose the addition of a fractional Brownian motion component, and secondly, we generalise the model to follow a multivariate‐t distribution. These extensions are biologically plausible, and each demonstrated substantially improved fit on application to example data from the Concerted Action on SeroConversion to AIDS and Death in Europe study. We also propose novel procedures for residual diagnostic plots that allow such models to be assessed. Cohorts of patients were simulated from the previously reported and newly developed models in order to evaluate differences in predictions made for the timing of treatment initiation under different clinical management strategies. A further simulation study was performed to demonstrate the substantial biases in parameter estimates of the mean slope of CD4 decline with time that can occur when random slopes models are applied in the presence of censoring because of treatment initiation, with the degree of bias found to depend strongly on the treatment initiation rule applied. Our findings indicate that researchers should consider more complex and flexible models for the analysis of longitudinal biomarker data, particularly when there are substantial missing data, and that the parameter estimates from random slopes models must be interpreted with caution. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Automatic algorithm to decompose discrete paths of fractional Brownian motion into self-similar intrinsic components

Fractional Brownian motion (fBm) is a nonstationary self-similar continuous stochastic process used to model many natural phenomena. A realization of the fBm can be numerically approximated by discrete paths which do not entirely preserve the self-similarity. We investigate the self-similarity at different time scales by decomposing the discrete paths of fBm into intrinsic components. The decomposition is realized by an automatic numerical algorithm based on successive smoothings stopped when the maximum monotonic variation of the averaged time series is reached. The spectral properties of the intrinsic components are analyzed through the monotony spectrum defined as the graph of the amplitudes of the monotonic segments with respect to their lengths (characteristic times). We show that, at intermediate time scales, the mean amplitude of the intrinsic components of discrete fBms scales with the mean characteristic time as a power law identical to that of the corresponding continuous fBm. As an application we consider hydrological time series of the transverse component of the transport process generated as a superposition of diffusive movements on advective transport in random velocity fields. We found that the transverse component has a rich structure of scales, which is not revealed by the analysis of the global variance, and that its intrinsic components may be self-similar only in particular cases.