Univariate Stochastic Processes Research Articles

Estimation of evolutionary power spectra of univariate stochastic processes by energy-based reckoning

In this paper a novel procedure is developed for evolutionary power spectra (EPS) estimation of univariate nonstationary stochastic processes. Specifically, the EPS is determined by estimating the statistical moments of the energy of a lightly damped linear filter excited by the nonstationary stochastic process. In this context, a smoothing procedure is incorporated by using the Savitzky-Golay (S-G) moving average filter to obtain reliable EPS based even on a limited number of available records. Further, a refinement of the approach is implemented relying on a polynomial model of temporal variation of the energy function of the filter output. Several numerical applications, involving both simulated nonstationary processes with known spectra and historical accelerograms, are used to assess the reliability and accuracy of the proposed procedure. Comparisons with the EPS estimated by a wavelets-based procedure are also provided.

Extreme value analysis of high-dimensional Gaussian vector processes

The analysis of the extreme value pertaining to first-passage failure is a key consideration for engineering systems undergoing random vibration. Most of the literature focus on a univariate stochastic process, however for system reliability analysis, the extreme value of multiple dependent processes is required. Due to the problem complexity, analytical methods are scarce, and they are limited to low-dimensional problems (dimension refers to the number of component processes). This paper presents an analytical method for extreme analysis of multivariate stationary Gaussian processes. The method can efficiently solve high-dimensional problems, thus it has diverse applications such as analysing dynamic systems with many degrees-of-freedom, or evaluating space-time extremes of random fields by discretizing the spatial domain. A maximum process is defined, representing the maxima of the components at each time instant. The exact upcrossing rate of the maximum process is derived, and the Poisson approximation is invoked to estimate the extreme value distribution. The method is fast as it involves multiple one-dimensional integrals, which can be computed using Gaussian quadrature. Moreover, an accurate approximation is proposed that obviates the need for numerical integration. The proposed method is implemented on two examples, specifically a dynamic system and spreading ocean waves. Monte Carlo simulations, which are computationally demanding, are performed to verify the accuracy and correctness of the proposed method.

Estimating Density Dependence, Environmental Variance, and Long-Term Selection on a Stage-Structured Life History.

AbstractA method for analyzing long-term demographic data on density-dependent stage-structured populations in a stochastic environment is derived to facilitate comparison of populations and species with different life histories. We assume that a weighted sum of stage abundances, N, exerts density dependence on stage-specific vital rates of survival and reproduction and that N has a small or moderate coefficient of variation. The dynamics of N are approximated as a univariate stochastic process governed by three key parameters: the density-independent growth rate, the net density dependence, and environmental variance in the life history. We show how to estimate the relative weighs of stages in N and the key parameters. Life history evolution represents a stochastic maximization of a simple function of the key parameters. The long-term selection gradient on the life history can be expressed as a vector of sensitivities of this function with respect to density-independent, density-dependent, and stochastic components of the vital rates. To illustrate the method, we analyze 38 years of demographic data on a great tit population, estimating the key parameters, which accurately predict the observed mean, coefficient of variation, and fluctuation rate of N; we also evaluate the long-term selection gradient on the life history.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

A multivariate statistical representation of railway track irregularities using ARMA models

Railway track irregularities occur naturally from service operations and are characterised by deviations of the track relative to its nominal geometry. Since the dynamic behaviour of a rail vehicle depends on the geometry of the track, studying irregularities is essential for safety, comfort, and maintenance. Conventionally, an irregularity is synthesised from a univariate stochastic process, which is a suitable way to represent excitations occurring in one dimension. However, most three-dimensional analyses of rail vehicles require a multivariate description of the irregularities. Therefore, synthesising multiple variables using univariate processes neglects possible relationships among them. This work addresses track irregularities from a multivariate series perspective to model the spatial autocorrelation of each irregularity and their spatial correlation with other types of irregularities. Several multivariate models with different complexity levels are estimated from the irregularities of a 3.6-km long straight track segment. A comparison among models shows that they should be selected based on not only statistical information criteria, but also their ability to reproduce relevant physical characteristics. Finally, a case study shows that the vehicle response to irregularities that are synthesised using the proposed multivariate process matches the response to measured data better than irregularities synthesised using independent univariate processes.

Scenario-Based Asset Allocation With Fat Tails And Non-Linear Correlation

This paper highlights the shortfalls of Modern Portfolio Theory (MPT). Amongst other flaws, MPT assumes that returns are normally distributed, that correlations are linear and risks are symmetrical. We propose a dynamic and flexible scenario-based approach to portfolio selection that incorporates an investor's economic forecast. Extreme Value Theory (EVT) is used to capture the skewness and kurtosis inherent in asset class returns and account for the volatility clustering and extreme co-movements across asset classes. The estimation consists of using an asymmetric GJR-GARCH model to extract filtered residuals for each asset class return. Subsequently, a marginal cumulative distribution function (CDF) of each asset class is constructed by using a Gaussian-kernel estimation for the interior, together with a generalised Pareto distribution (GPD) for the upper and lower tails. The distribution of exceedance method is applied to find residuals in the tails. A Student's t copula is then fitted to the data to induce correlation between the simulated residuals of each asset class. A Monte Carlo technique is applied to simulate standardised residuals, which represent a univariate stochastic process when viewed in isolation but maintain the correlation induced by the copula. The results are mean-CVaR optimised portfolios, which are derived based on an investor's forward-looking expectation.

Cross-codifference for bidimensional VAR(1) time series with infinite variance

In this paper, we consider the problem of a measure that allows us to describe the spatial and temporal dependence structure of multivariate time series with innovations having infinite variance. By using recent results obtained in the problem of temporal dependence structure of univariate stochastic processes, where the auto-codifference was used, we extend its idea and propose a cross-codifference measure for a general bidimensional vector autoregressive time series of order 1 (bidimensional VAR(1)). Next, we derive analytical results for VAR(1) model with Gaussian and stable sub-Gaussian innovations, that are characterized by finite and infinite variance, respectively. We emphasize that obtained expressions perfectly agree with the empirical counterparts. Moreover, we show that for the considered time series the cross-codifference simplifies to the well-established cross-covariance in the case when the innovations of time series are given by Gaussian white noise. The last part of the work is devoted to the statistical estimation of VAR(1) time series parameters based on the empirical cross-codifference. Again, we demonstrate via Monte Carlo simulations that the proposed methodology works correctly.

Nonparametric estimation of a multiple order conditional within-subject covariance function for a continuous times univariate stochastic process

We introduce a nonparametric estimation of a multiple order conditional within-subject correlation of a continuous times stochastic process X={X(t),t∈[0,T]} defined on a probability space (Ω,A,P). We prove the asymptotic normality of the conditional within-subject covariance estimators.

Digital simulation of 3D turbulence wind field of Sutong Bridge based on measured wind spectra

Time domain analysis is an essential implement to study the buffeting behavior of long-span bridges for it can consider the non-linear effect which is significant in long-span bridges. The prerequisite of time domain analysis is the accurate description of 3D turbulence winds. In this paper, some hypotheses for simplifying the 3D turbulence simulation of long-span cable-stayed bridges are conducted, considering the structural characteristics. The turbulence wind which is a 3D multivariate stochastic vector process is converted into four independent 1D univariate stochastic processes. Based on recorded wind data from structural health monitoring system (SHMS) of the Sutong Bridge, China, the measured spectra expressions are then presented using the nonlinear least-squares fitting method. Turbulence winds at the Sutong Bridge site are simulated based on the spectral representation method and the Fast Fourier transform (FFT) technique, and the relevant results derived from target spectra including measured spectra and recommended spectra are compared. The reliability and accuracy of the presented turbulence simulation method are validated through comparisons between simulated and target spectra (measured and recommended spectra). The obtained turbulence simulations can not only serve further analysis of the buffeting behavior of the Sutong Bridge, but references for structural anti-wind design in adjacent regions.

Multivariate time series models for studies on stochastic generators in power systems

This paper presents an applied statistics method for the synthesis of multivariate time series of stochastic generation for planning purposes in power systems. It is shown that a suited model should represent satisfactorily the individual univariate stochastic processes and also reproduce the stochastic dependence structure between them. In order to fulfill all requirements, the method proposes the transformation of a set of recorded time series to the multivariate normal domain, the identification of a vector autoregressive model, the synthetization of a multivariate time series with desired length, and subsequently the back-transformation into the initial domain. The method can capture density functions, chronological persistence, diurnal periodicities and dependence structures of and between an arbitrary number of distributed system infeeds. Application can be found in the expansion of short data records into statistically significant synthetic time series for power system studies with distributed and uncertain resources. A simple practical example is given for illustration.

Assessing Time‐Reversibility Under Minimal Assumptions

Abstract. This article considers a simple procedure for assessing whether a weakly dependent univariate stochastic process is time‐reversible. Our approach is based on a simple index of the deviation from zero of the median of the one‐dimensional marginal law of differenced data. An attractive feature of the method is that it requires no moment assumptions. Instead of relying on Gaussian asymptotic approximations, we consider using subsampling and resampling methods to construct confidence intervals for the time‐reversibility parameter, and show that such inference procedures are asymptotically valid under a mild mixing condition. The small‐sample properties of the proposed procedures are examined by means of Monte Carlo experiments and an application to real‐world data is also presented.

Study of Dependence for Some Stochastic Processes

This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem.

Testing nonlinear stochastic models on phytoplankton biomass time series

Time series of algae mass in German water reservoir lakes being collected for several years in irregular sampling, half weekly on average, are tested in their stationary distribution against several univariate stochastic processes with linear and nonlinear dynamics, with pure additive and with state dependent noise. In each case the parameters of the corresponding Fokker–Planck equations in the stationary state are estimated by maximisation of the log-likelihood. For statistical evaluation the Kolmogorov–Smirnov-test is applied with additional graphical tools, being especially suited for short data sets typical for ecological systems. To obtain the optimally adapted stochastic process, an additional parameter has to be estimated describing the autocorrelation. We use two methods: first, we use a novel method based on the estimation of conditional probabilities directly from the data. The resulting stochastic model with state dependent noise is simulated. Second, we use the Lomb normalised Fourier spectrum for unevenly sampled data, e.g. no measurements during winter time to minimise the overall deviation between the empirical spectrum and spectra obtained from stochastic simulations, the sampling of which follows in detail the one of the empirical data. The parameter value of the first method is used as a starting point for the adaptation of the cumulative spectrum. In this way a stochastic process is obtained which describes the stationary distribution as well as the spectrum of the empirical data both indistinguishable at high statistical confidence.

Approximate Solutions of Green's Type for Univariate Stochastic Processes

SUMMARY A study is made of approximations of Green's type (often called W.K.B. approximations) to the solution of equations governing univariate stochastic processes. After considering some specific processes, the general homogeneous process in continuous time is discussed. A numerical comparison is made between the approximate and exact probabilities for the multiplicative birth and death process. Diffusion processes are briefly examined. The method is applied to the equations both for probabilities and for the moment generating function, and the duality between the approximate solutions is indicated. Finally, approximate formulae for the cumulants are obtained and the limitations of the method are pointed out.