Non-stationary Probability Density Function Research Articles

Non-stationary probabilistic analysis of non-linear ship roll motion due to modulated periodic and random excitations

This paper is about the study on the non-stationary probability density function (PDF) solution of the non-linear ship rolling system under modulated periodic and random excitations. The quadratic and cubic damping terms are introduced to describe the non-linear damping moment and the power terms in state variables are adopted to describe the righting moment in the ship rolling system, respectively. The relevant Fokker–Planck–Kolmogorov (FPK) equation of the ship rolling system is constructed and the non-stationary probabilistic solutions are obtained by the presented solution procedure. In the solution process, the time variable t is introduced to the traditional exponential-polynomial-closure (EPC) method to obtain the non-stationary PDF solutions of the ship rolling system. The obtained non-stationary PDF solutions of the non-linear ship rolling systems demonstrate that the presented solution procedure can give high solution accuracy even in the PDF tails when they are compared with Monte Carlo simulation (MCS). Moreover, owing to the strong system nonlinearity and the interaction of the periodic excitation and random excitation, the achieved non-stationary PDF solutions of the system responses can deviate significantly from the Gaussian PDF. It is also found that the PDFs of responses are not even symmetrical about the means of responses due to the interaction of the periodic excitation and the random excitation even if the system only contains odd nonlinear terms. In addition, the computational efficiency is increased by over five hundred times by the presented solution procedure compared to MCS.

Transient probability density of nonlinear oscillator under parametric harmonic and external modulated stochastic excitations

In this study, the exponential-polynomial-closure (EPC) method is applied to determine and study the non-stationary probability density function (PDF) of the nonlinear system simultaneously stimulated by parametric harmonic and modulated stochastic excitations. The non-stationary probabilistic responses are represented as a time-related exponential polynomial function with unknown variables. Integrating the weighted residual error can result in a group of first-order differential equations. Those equations can be solved by numerical techniques to obtain the undefined variables. Four examples are given to show that the non-stationary behaviors of the presented oscillators can be addressed by the presented EPC procedure effectively, especially in the probabilistic tails which are of particular importance in reliability analysis. Moreover, the outcomes show that the response means equal to zero and the response PDFs are symmetrical with respect to the zero means. In particular, the presented EPC procedure can heavily decrease the running effort with no sacrifice to the solution correctness when compared with MCS.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

Finite volume computation for the non-stationary probability density function of an impulsively controlled 1-D diffusion process

Finite volume computation for the non-stationary probability density function of an impulsively controlled 1-D diffusion process

Nonstationary response of a nonlinear economic cycle model under random disturbance

Stochastic mechanics is widely applied in the field of economics and econometrics. One important application is to study economic fluctuations in economic cycle theory. However, nonstationary response of nonlinear economic cycles with random disturbance is less investigated. This paper is devoted to studying the nonstationary response of a nonlinear economic cycle model. First, a nonlinear economic cycle model is developed according to the Goodwin model and the Puu nonlinear economic cycle model. Gaussian white noise is regarded as a spontaneous function in the developed nonlinear economic cycle model. Then, a path integration method is adopted based on Gauss–Legendre scheme and short-time Gaussian approximation for obtaining the nonstationary probability density function (PDF) of the nonlinear economic cycle model. The obtained results are verified by the simulation result. Finally, the evolution of the nonstationary PDF is studied in detail in the numerical analysis. The effects of system parameters are compared and discussed. The results show that the addition of the quadratic term of income velocity leads to the nonzero means of the PDFs of income and income velocity. Due to the presence of the quadratic and cubic terms, these PDFs have non-Gaussian distributions.

Transient responses of stochastic systems under stationary excitations

Stochastic systems to stationary random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. It has been a challenge to obtain exact transient response properties. A novel approximate technique for determining non-stationary probability density function (PDF) of randomly excited nonlinear oscillators is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. Based on the results from statistical linearization, residual error of the FPK equation associated with proposed approximation solution can be treated by weighted residual method. As a result, nonlinear ordinary differential equations are produced. Numerical method is adopted to solve these equations and approximate PDF solutions are then obtained. In order to verify the efficiency of the proposed procedure, four examples of stochastic vibrating systems with additional excitations or/and parametric excitations are considered. It is shown that the results obtained by the proposed procedure agree well with those from Monte Carlo simulation.

Nonstationary solutions of nonlinear dynamical systems excited by Gaussian white noise

Nonlinear dynamical systems to random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. A novel approximate technique for determining nonstationary probability density function (PDF) of randomly excited nonlinear Oscillator is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. By applying statistical linearization and weighted residual method, residual error of the FPK equation associated with approximated solution is reduced to a series of nonlinear first-order ordinary differential equations, which can be solved by the numerical method. Finally, a class of nonlinear vibrating systems with additive excitations or/and parametric excitations are considered. The obtained PDF has tail regions of logarithmic form, which are important for reliability and failure analysis, and agrees very well with the simulated ones. In particular, the computational time spent by the proposed procedure is a very small fraction of the one taken by the MCS method. This technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods.

Probabilistic Analysis of Random Structural Intensity for Structural Members under Stochastic Loadings

The concept of structural intensity (SI) is extended to the random domain by introducing a physical quantity denominated random structural intensity (RSI). This quantity is formulated for mechanical systems whose dynamical responses are stochastic due to random excitations. In order to fully characterize the stochastic behavior of a system under random loadings, it is imperative to obtain the probability density function (PDF) of RSI. Based on the elastic theory and the definition of SI, RSI is expressed as functions of system responses. In general, the PDF of system responses is governed by Fokker–Planck–Kolmogorov (FPK) equation under the assumption that random dynamic loadings are idealized as white noise excitations. Therefore, the PDF of RSI is derived with the joint PDF of system responses. In the present study, four demonstrating cases of beams and plates under separately concentrated and uniform random loadings are studied to investigate the properties of RSI. Stationary and non-stationary PDFs of RSI at arbitrary section of beam and plate are obtained. Numerical results show that the PDF of RSI is transient at early stage of stationary loading and then converges to the exact stationary ones as time increases. With the obtained PDFs of RSI, energy transmission path over the beam and plate can be determined, which is guided from the locations with lower probabilities of RSI to the ones with higher probabilities of RSI. Furthermore, virtual energy flow sinks on the plate and beam can be found, which are identified by the locations with the maximum probabilities of RSI.

Regime shifts under forcing of non-stationary attractors: Conceptual model and case studies in hydrologic systems

We present here a conceptual model and analysis of complex systems using hypothetical cases of regime shifts resulting from temporal non-stationarity in attractor strengths, and then present selected published cases to illustrate such regime shifts in hydrologic systems (shallow aquatic ecosystems; water table shifts; soil salinization). Complex systems are dynamic and can exist in two or more stable states (or regimes). Temporal variations in state variables occur in response to fluctuations in external forcing, which are modulated by interactions among internal processes. Combined effects of external forcing and non-stationary strengths of alternative attractors can lead to shifts from original to alternate regimes. In systems with bi-stable states, when the strengths of two competing attractors are constant in time, or are non-stationary but change in a linear fashion, regime shifts are found to be temporally stationary and only controlled by the characteristics of the external forcing. However, when attractor strengths change in time non-linearly or vary stochastically, regime shifts in complex systems are characterized by non-stationary probability density functions (pdfs). We briefly discuss implications and challenges to prediction and management of hydrologic complex systems.

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

Assessing uncertainties in the building of ensemble RCMs over Spain based on dry spell lengths probability density functions

Spain is one of the European countries with most environmental problems related to water scarcity and droughts. Additionally, several studies suggest trends of increasing temperature and decreasing rainfall, mainly for the Iberian Peninsula, due to climate variability and change. While Regional Climate Models (RCM) are a valuable tool for understanding climate processes, the causes and plausible impacts on variables and meteorological extremes present a wide range of associated uncertainties. The multi-model ensemble approach allows the quantification and reduction of uncertainties in the predictions. The combination of models (RCM in this case), generally increases the reliability of the predictions, although there are different weighting methodologies. In this paper, a strategy is presented for the building of non-stationary PDF (probability density functions) ensembles with the aim of evaluating the spatial pattern of future risk of drought for an area. At the same time, the uncertainty associated with the metric used in the construction of the PDF ensembles is assessed. A comparative study of methodologies based on the application of the Reliability Ensemble Averaging (REA), assessing its factors using two performance measures, on the one hand the Perkins Score Methods, on the other hand the Kolmogorov-Smirnov test, is proposed. The evaluation of the sensitivity of the methodologies used in the construction of ensembles, as proposed in this paper, although without completely eliminating uncertainty, allows a better understanding of the sources and magnitude of the uncertainties involved. Despite the differences between the spatial distribution results from each metric (which can be in the order of 40 % in some areas), both approaches concluded about a plausible significant and widespread increase throughout continental Spain of the mean value of annual maximum dry spell lengths (AMDSL) between the years 1990 and 2050. Finally, the more parsimonious approach in the building of ensembles PDF, based on AMDSL in peninsular Spain, is identified.

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker–Planck–Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.

Nonparametric density estimation of streaming data using orthogonal series

Streaming data represent a serious challenge because implicit in the nature of streaming data, data are not exchangeable and are not storable. This means data must be processed on the fly. Density estimation is an essential tool used to make sense of data collected by large scale systems. In this paper, we present a recursive method for constructing and updating an estimate of the nonstationary probability density function. Our approach is shown to work well with simulated data as well as with real data.

Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

Approximate solution of the Fokker–Planck–Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.

Non-linear stochastic response distributions by local statistical linearization

A novel computational approach based on the statistical equivalent linearization and Gaussian superposition is introduced. The methodology allows to extend the concept of statistical equivalent linearization to proceed from estimates of the second moment properties of the stochastic response to estimates for the non-Gaussian probability distribution. This paper introduces the concept of local equivalent linearization designed to compute a close approximation of the non-stationary probability density function p( x ,t) of a non-linear system exited by random excitation. Non-Gaussian distributions are represented as a sum of normal distributions. The suggested approach employs the well-developed equivalent linearization procedure or Gaussian closure to compute the non-Gaussian distribution of the response of a non-linear system. Since equivalent linearization is applicable for higher dimensions and finite element models, the suggested approach lends itself to providing numerical solutions for higher-dimensional cases.

Stationary and non-stationary probability density function for non-linear oscillators

A method for the evaluation of the stationary and non-stationary probability density function of non-linear oscillators subjected to random input is presented. The method requires the approximation of the probability density function of the response in terms of C-type Gram-Charlier series expansion. By applying the weighted residual method, the Fokker-Planck equation is reduced to a system of non-linear first order ordinary differential equations, where the unknowns are the coefficients of the series expansion. Furthermore, the relationships between the A-type and C-type Gram-Charlier series coefficient are derived.

On averages of radar rain echoes in nonstationary conditions

The authors examine a problem, recently opened up by Jameson and Kostinski (1996), of statistics for radar rain echoes in a nonstationary case such as the observation of rain by a moving antenna beam. The present authors study this problem under the framework of the standard fluctuation theory of Marshall and Hitschfeld (1953). They obtain the exact nonstationary probability density function (pdf) for an average of n mutually independent samples of the received power in the square-law detection, and an approximate nonstationary pdf in the logarithmic detection. They find that the concept of bias does not appropriately describe the effect of antenna beam movement during the sampling. They introduce a new quantity to assess the effect of antenna beam movement on rain observation by radars.

MITOTIC INDEX, INFLUX AND MEAN TRANSIT TIME IN THE HAMSTER CHEEK POUCH EPITHELIUM, A PARTIALLY SYNCHRONIZED CELL SYSTEM*†

Mitotic activity was followed in the epithelium of the hamster cheek pouch for about 12 hr in two experiments under different conditions of noise and light intensity. A mathematical model based on a non-stationary probability density function for the transit time through mitosis was developed, making an analysis of this partially synchronized cell system possible. The most important result of the two experiments is the indication of a non-stationary mean transit time for cells in mitosis. In the first experiment (70 dB, 103 lux), which produced high mitotic indices (maximum 2.0%), the influx changed from 0. to 0.9%/hr (mean of 7-hourly determinations: 0.4%/hr), and the mean transit time from 1.9 to 5.5 hr. In the second experiment (70 dB, 265 lux), which had a lower level of MI (maximum 1.3%), the influx changed from 0 to 1.8%/hr (mean of 13-hourly determinations: 0.4%/hr), and the mean transit time from 0.3 to 3.3 hr. It was difficult to say precisely whether the variation in influx or in mean transit time was the main factor in the increase and decrease of MI. The data could not be simulated with a stationary mean transit time. It is suggested that changes in MI due to different external circumstances are mainly the result of variation in the mean transit time.