Wiener Process Model Research Articles

On mimicry among sequential sampling models

Sequential sampling models are widely used in modeling the empirical data obtained from different decision making experiments. Since 1960s, several instantiations of these models have been proposed. A common assumption among these models is that the subject accumulates noisy information during the time course of a decision. The decision is made when the accumulated information favoring one of the responses reaches a decision boundary. Different models, however, make different assumptions about the information accumulation process and the implementation of the decision boundaries. Comparison among these models has proven to be challenging. In this paper we investigate the relationship between several of these models using a theoretical framework called the inverse first passage time problem. This framework has been used in the literature of applied probability theory in investigating the range of the first passage time distributions that can be produced by a stochastic process. In this paper, we use this framework to prove that any Wiener process model with two time-constant boundaries can be mimicked by an independent race model with time-varying boundaries. We also examine the numerical computation of the mimicking boundaries. We show that the mimicking boundaries of the race model are not symmetric. We then propose an equivalent race model in which the boundaries are symmetric and time-constant but the drift coefficients are time-varying.

Remaining Useful Life Prediction for a Nonlinear Heterogeneous Wiener Process Model With an Adaptive Drift

Nonlinear degradation trajectories are encountered frequently, and not all of them evolve homogeneously in practical systems. To take nonlinearity, heterogeneity, and the entire historical degradation data into account, we propose a nonlinear heterogeneous Wiener process model with an adaptive drift to characterize degradation trajectories. A state-space based method is employed to delineate our model. Due to the introduction of the adaptive drift, it is difficult to directly apply Kalman filter methods to update the distribution of the estimated degradation drift. To address this issue, we develop an online filtering algorithm based on Bayes' theorem. The expectation-maximization (EM) algorithm, as well as a novel Bayes'-theorem-based smoother, are adopted to estimate the unknown parameters in our model. Moreover, the distribution of the predicted remaining useful life (RUL) incorporating the complete distribution of the estimated degradation drift is achieved analytically. Finally, a simulation, and a case study are provided to validate the proposed approach.

A new class of Wiener process models for degradation analysis

For many products, it is not uncommon to see that a unit with a higher degradation rate has a more volatile degradation path. Motivated by this observation, we propose a new class of random effects model for the Wiener process model. We express the Wiener process in a special form and allow one of the parameters to be random across the product population so that a unit with a high degradation rate would also possess high volatility. Statistical inference of the model is discussed. By the same token, we introduce a stress–acceleration relation for the Wiener process so that both the degradation rate and the volatility of the product are increasing in the stress level. The proposed models are demonstrated by analyzing a dataset of fatigue crack growth and a dataset of head wears of hard disk drives. The applications suggest that our models perform better than existing models that ignore the positive correlation between the drift rate and the volatility.

Reliability Demonstration for Long-Life Products Based on Degradation Testing and a Wiener Process Model

<?Pub Dtl=""?> In this paper, the degradation based reliability demonstration test (RDT) plan design problems for long life products under a small sample circumstance are studied. Fixed sample method, sequential probability ratio test (SPRT) method, and sequential Bayesian decision method are provided based on univariate degradation testing. The simulation examples show the superiority of degradation based RDT methods compared with the traditional failure based methods, and the sequential-type methods have more test power than their fixed sample counterparts. The test power can be further improved by combining the test data of a reliability indicator with the data of its marker, based on which the bivariate fixed sample method and the sequential Bayesian decision method are defined. The simulation study shows the benefit from the combination. The degradation based RDT plan optimization model, and the corresponding searching-based solution algorithm using some heuristic rules discovered in the paper, are also presented. The case study of Rubidium Atomic Frequency Standard with a RDT plan design demonstrates the effectiveness of our methods on overcoming the difficulties of small samples in reliability demonstration of long life products.

An Additive Wiener Process-Based Prognostic Model for Hybrid Deteriorating Systems

Hybrid deteriorating systems, which are made up of both linear and nonlinear degradation parts, are often encountered in engineering practice, such as gyroscopes which are frequently utilized in ships, aircraft, and weapon systems. However, little reported literature can be found addressing the degradation modeling for a system of this type. This paper proposes a general degradation modeling framework for hybrid deteriorating systems by employing an additive Wiener process model that consists of a linear degradation part and a nonlinear part. Furthermore, we derive the analytical solution of the remaining useful life distribution approximately for the presented model. For a specific system in service, the posterior estimates of the stochastic parameters in the model are updated recursively by using the condition monitoring observations based on a Bayesian framework with the consideration that the stochastic parameters in the linear and nonlinear deteriorating parts are correlated. Thereafter, the posterior distribution of stochastic parameters is used to update in real-time the distribution of the remaining useful life where the uncertainties in the estimated stochastic parameters are incorporated. Finally, a numerical example and a practical case study are provided to verify the effectiveness of the proposed method. Compared with two existing methods in literature, our proposed degradation modeling method increases the one-step prediction accuracy slightly in terms of mean squared error, but gains significant improvements in the estimated remaining useful life.

Degradation Data Analysis Using Wiener Processes With Measurement Errors

Degradation signals that reflect a system's health state are important for diagnostics and health management of complex systems. However, degradation signals are often compounded and contaminated by measurement errors, making data analysis a difficult task. Motivated by the wear problem of magnetic heads used in hard disk drives (HDDs), this paper investigates Wiener processes with measurement errors. We explore the traditional Wiener process with positive drifts compounded with i.i.d. Gaussian noises, and improve its estimation efficiency compared with the existing inference procedure. Furthermore, to capture the possible heterogeneity in a population, we develop a mixed effects model with measurement errors. Statistical inferences of this model are discussed. The mixed effects model subsumes several existing Wiener processes as its limiting cases, and thus it is useful for suggesting an appropriate Wiener process model for a specific dataset. The developed methodologies are then applied to the wear problem of magnetic heads of HDDs, and a light intensity degradation problem of light-emitting diodes.

Residual life estimation based on bivariate Wiener degradation process with measurement errors

An adaptive method of residual life estimation for deteriorated products with two performance characteristics (PCs) was proposed, which was sharply different from existing work that only utilized one-dimensional degradation data. Once new degradation information was available, the residual life of the product being monitored could be estimated in an adaptive manner. Here, it was assumed that the degradation of each PC over time was governed by a Wiener degradation process and the dependency between them was characterized by the Frank copula function. A bivariate Wiener process model with measurement errors was used to model the degradation measurements. A two-stage method and the Markov chain Monte Carlo (MCMC) method were combined to estimate the unknown parameters in sequence. Results from a numerical example about fatigue cracks show that the proposed method is valid as the relative error is small.

Optimal design of accelerated degradation tests based on Wiener process models

Optimal accelerated degradation test (ADT) plans are developed assuming that the constant-stress loading method is employed and the degradation characteristic follows a Wiener process. Unlike the previous works on planning ADTs based on stochastic process models, this article determines the test stress levels and the proportion of test units allocated to each stress level such that the asymptotic variance of the maximum-likelihood estimator of the qth quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are also developed for checking the validity of the relationship between the model parameters and the stress variable. Finally, using an example, sensitivity analysis procedures are presented for evaluating the robustness of optimal and compromise plans against the uncertainty in the pre-estimated parameter value, and the importance of optimally determining test stress levels and the proportion of units allocated to each stress level are illustrated.

Semiparametric inference on a class of Wiener processes

Abstract. This article studies the estimation of a nonhomogeneous Wiener process model for degradation data. A pseudo-likelihood method is proposed to estimate the unknown parameters. An attractive algorithm is established to compute the estimator under this pseudo-likelihood formulation. We establish the asymptotic properties of the estimator, including consistency, convergence rate and asymptotic distribution. Random effects can be incorporated into the model to represent the heterogeneity of degradation paths by letting the mean function be random. The Wiener process model is extended naturally to a normal inverse Gaussian process model and similar pseudo-likelihood inference is developed. A score test is used to test the presence of the random effects. Simulation studies are conducted to validate the method and we apply our method to a real data set in the area of health structure monitoring.

Joint analysis of current status and marker data: an extension of a bivariate threshold model.

This paper considers joint analysis of current status and marker data using a threshold model based on first hitting times. A failure time is defined as the time at which a subject's latent health status process first decreases to zero. We extend the bivariate Wiener process model in Whitmore et al. (1998) to the case when only current status data are available. We develop maximum likelihood estimation procedures and provide simulation studies. We apply our methods to a motivating example involving liver tumors in mice.

Self-Tuning Pid Neural Network Controller to Control Nonlinear Ph Neutralization in Waste Water Treatment

The conventional PID control (linear) is popular control scheme that is used in almost pH control at waste water treatment process. pH model with respect to titration liquid flow rate has been known to be intrinsically difficult and nonlinear, especially when the process is conducted to non-linear range pH reference (pH 4-8), the settling time reach a long time. Therefore in this paper the nonlinear controller with self-tuning PID scheme is performed handle pH by training a neural-network based on backpropagation error signals. The pH process model is combination between CSTR (Continuous Stirred Tank Reactor) linear dynamic with H3PO4, HF, HCl, H2S flow rate and nonlinear static electro neutrality (wiener process model). The simulation result has a good and suitable performance under several tests (set-point and load change). The simulation result show the pH control system can follow the change of pH set-point and load with the average steady state error 0.00034. The settling time achieved at 50 second faster than conventional PID controller scheme.

Randomness and variability of the neuronal activity described by the Ornstein–Uhlenbeck model

Normalized entropy as a measure of randomness is explored. It is employed to characterize those properties of neuronal firing that cannot be described by the first two statistical moments. We analyze randomness of firing of the Ornstein–Uhlenbeck (OU) neuronal model with respect either to the variability of interspike intervals (coefficient of variation) or the model parameters. A new form of the Siegert's equation for first-passage time of the OU process is given. The parametric space of the model is divided into two parts (sub-and supra-threshold) depending upon the neuron activity in the absence of noise. In the supra-threshold regime there are many similarities of the model with the Wiener process model. The sub-threshold behavior differs qualitatively both from the Wiener model and from the supra-threshold regime. For very low input the firing regularity increases (due to increase of noise) cannot be observed by employing the entropy, while it is clearly observable by employing the coefficient of variation. Finally, we introduce and quantify the converse effect of firing regularity decrease by employing the normalized entropy.

BENCHMARKING FOR PRODUCTIVITY IMPROVEMENT: A HEALTH‐CARE APPLICATION*

A methodology is developed and applied to compare the performance of publicly funded agencies providing treatment for alcohol abuse in Maine. The methodology estimates a Wiener process that determines the duration of completed treatments, while allowing for agency differences in the effectiveness of treatment, costs of treatment, standards for completion of treatment, patient attrition, and the characteristics of patient populations. Notably, the Wiener process model separately identifies agency fixed effects that describe differences in the effectiveness of treatment (“treatment effects”), and effects that describe differences in the unobservable characteristics of patients (“population effects”). The estimated model enables hypothetical comparisons of how different agencies would treat the same populations. The policy experiment of transferring the treatment practices of more cost‐effective agencies suggests that Maine could have significantly reduced treatment costs without compromising health outcomes by identifying and transferring best practices.

On Convergence of a P-Algorithm Based on a Statistical Model of Continuously Differentiable Functions

This paper is a study of the one-dimensional global optimization problem for continuously differentiable functions. We propose a variant of the so-called P-algorithm, originally proposed for a Wiener process model of an unknown objective function. The original algorithm has proven to be quite effective for global search, though it is not efficient for the local component of the optimization search if the objective function is smooth near the global minimizer. In this paper we construct a P-algorithm for a stochastic model of continuously differentiable functions, namely the once-integrated Wiener process. This process is continuously differentiable, but nowhere does it have a second derivative. We prove convergence properties of the algorithm.

The Stochastically Subordinated Poisson Normal Process for Modelling Financial Assets.

The method of stochastic subordination, or random time indexing, has been recently applied to Wiener process price processes to model financial returns. Previous emphasis in stochastic subordination models has involved explicitly identifying the subordinating process with an observable quantity such as number of trades. In contrast, the approach taken here does not depend on the specific identification of the subordinated time variable, but rather assumes a class of time models and estimates parameters from data. In addition, a simple Markov process is proposed for the characteristic parameter of the subordinating distribution to explain the significant autocorrelation of the squared returns. It is shown, in particular, that the proposed model, while containing only a few more parameters than the commonly used Wiener process models, fits selected financial time series particularly well, characterising the autocorrelation structure and heavy tails, as well as preserving the desirable self-similarity structure, and the existence of risk-neutral measures necessary for objective derivative valuation. Also, it will be shown that the model proposed fits financial times series data better than the popular generalised autoregressive conditional heteroscedasticity (GARCH) models. Additionally, this paper will develop a skew model by replacing the normal variates with Levy stable variates.

The stochastically subordinated Poisson normal process for modelling financial assets

The method of stochastic subordination, or random time indexing, has been recently applied to Wiener process price processes to model financial returns. Previous emphasis in stochastic subordination models has involved explicitly identifying the subordinating process with an observable quantity such as number of trades. In contrast, the approach taken here does not depend on the specific identification of the subordinated time variable, but rather assumes a class of time models and estimates parameters from data. In addition, a simple Markov process is proposed for the characteristic parameter of the subordinating distribution to explain the significant autocorrelation of the squared returns. It is shown, in particular, that the proposed model, while containing only a few more parameters than the commonly used Wiener process models, fits selected financial time series particularly well, characterising the autocorrelation structure and heavy tails, as well as preserving the desirable self-similarity structure, and the existence of risk-neutral measures necessary for objective derivative valuation. Also, it will be shown that the model proposed fits financial times series data better than the popular generalised autoregressive conditional heteroscedasticity (GARCH) models. Additionally, this paper will develop a skew model by replacing the normal variates with Levy stable variates.

Identification and Adaptive Control of Wiener Type Nonlinear Processes

This study is concerned with a problem of the on-line identification and adaptive control of Wiener type nonlinear processes. It is assumed that the linear dynamic part of the process can be approximated by a pulse transfer function model of a known order and stable inverse followed by a known pure time-delay. The static nonlinearity is assumed to be monotonicaI function of its argument and it is approximated by a piecewise-polynomial function model where the number of intervals, polynomial orders and the breakpoints should be chosen a priori. The output signal from the linear part is not available for measurement. The combined model is nonlinear in its parameters and it is characterized by a larger number of parameters than the original Wiener model.Deterministic globally stable recursive parameter identification algorithms and the model reference adaptive control systems are developed for the Wiener process model. The asymptotic properties in the presence of the input and output measurement noises and the influence of the error due to the approximation of a static nonlinearity by a piecewise-polynomial function of a finite order on the convergence properties of the proposed identification schemes are analysed. Finally, the on-line identification and adaptive control of the pH-process in a continuous flow reactor were performed. The simulation and the experimental test results are presented.

Renormalised canonical perturbation theory for stochastic propagators

A canonical transformation which removes the coherent oscillatory motion of a particle in a stochastic potential (the renormalised oscillation-centre transformation) is constructed by a new classical perturbation method using Lie operators and Green function techniques. A frequency and wavevector dependent particle-wave collision operator is calculated explicitly for stationary, homogeneous electrostatic turbulence in the short wavelength limit. The width of the resonance is proportional to the one-third power of the quasilinear diffusion coefficient, in agreement with Dupree's result (1966). However the k dependence is quite different from that expected from a simple Wiener process model. At large k spatial diffusion dominates over velocity diffusion in sharp contrast with previous theories.