Response of linear dynamic systems to non-Erlang renewal impulses: Stochastic equations approach

Linear dynamical systems under random trains of impulses driven by a class of non-Erlang renewal processes are considered. The class considered is the one where the renewal events are selected from an Erlang renewal process. The original train of impulses is recast, with the aid of an auxiliary stochastic variable, in terms of two independent Poisson processes. Thus, by augmenting the state vector of the dynamic system with the auxiliary stochastic variables, the original non-Markov problem is converted to a Markov one.

The differential equations for the response statistical moments can then be derived from the generalised Ito's differential rule.

Numerical results obtained for a few different models and various sets of parameters, show that the present approach allows to account for a variety of inter arrival time's probability distributions. Transient mean value and variance of the response of a linear oscillator have been obtained from the equations for moments.     

State-space based analysis and forecasting of macroscopic road safety trends in Greece

In this paper, macroscopic road safety trends in Greece are analyzed using state-space models and data for 52 years (1960–2011). Seemingly unrelated time series equations (SUTSE) models are developed first, followed by richer latent risk time-series (LRT) models. As reliable estimates of vehicle-kilometers are not available for Greece, the number of vehicles in circulation is used as a proxy to the exposure. Alternative considered models are presented and discussed, including diagnostics for the assessment of their model quality and recommendations for further enrichment of this model. Important interventions were incorporated in the models developed (1986 financial crisis, 1991 old-car exchange scheme, 1996 new road fatality definition) and found statistically significant. Furthermore, the forecasting results using data up to 2008 were compared with final actual data (2009–2011) indicating that the models perform properly, even in unusual situations, like the current strong financial crisis in Greece. Forecasting results up to 2020 are also presented and compared with the forecasts of a model that explicitly considers the currently on-going recession. Modeling the recession, and assuming that it will end by 2013, results in more reasonable estimates of risk and vehicle-kilometers for the 2020 horizon. This research demonstrates the benefits of using advanced state-space modeling techniques for modeling macroscopic road safety trends, such as allowing the explicit modeling of interventions. The challenges associated with the application of such state-of-the-art models for macroscopic phenomena, such as traffic fatalities in a region or country, are also highlighted. Furthermore, it is demonstrated that it is possible to apply such complex models using the relatively short time-series that are available in macroscopic road safety analysis.     

Applied Regression Analysis and Experimental Design

In industrial hygiene, a worker’s exposure to chemical, physical, and biological agents is increasingly being modeled using deterministic physical models that study exposures near and farther away from a contaminant source. However, predicting exposure in the workplace is challenging and simply regressing on a physical model may prove ineffective due to biases and extraneous variability. A further complication is that data from the workplace are usually misaligned. This means that not all timepoints measure concentrations near and far from the source. We recognize these challenges and outline a flexible Bayesian hierarchical framework to synthesize the physical model with the field data. We reckon that the physical model, by itself, is inadequate for enhanced inferential and predictive performance and deploy (multivariate) Gaussian processes to capture uncertainties and associations. We propose rich covariance structures for multiple outcomes using latent stochastic processes. This article has supplementary material available online.     

Modelling and estimating the reliability of stochastic dynamical systems with Markovian switching

In this paper, a general framework for the modelling of physical phenomena with stochastic dynamical systems switched by jump Markov processes is given. A methodology of the associated estimation procedures is provided. A particular attention is paid to the estimation of the underlying jump process, which is not observable.As an application, a stochastic model is proposed for the fatigue crack growth problem. The estimation of the model parameters is made on a real crack growth data set. We are thus able to simulate some crack growth paths which are used for reliability analysis through Monte Carlo techniques.     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

Singularity analysis, balance equations and soliton solution of the nonlocal complex Ginzburg-Landau equation

The modified complex Ginzburg-Landau equation (mCGLE) which includes a delayed response term in the integral form is analysed. In particular, a singularity analysis of mCGLE is presented. It is shown that this equation fails to pass the Painlevé test when the non-conservative terms are nonzero. Nevertheless, exact solutions to this equation do exist. Stationary solutions can be treated using the segment balance method which is an extension of conservation laws to non-conservative systems. This method is used to derive an exact soliton solution of mCGLE.     

Parameter estimation from observations of first-passage times of the Ornstein–Uhlenbeck process and the Feller process

Renewal point processes show up in many different fields of science and engineering. In some cases the renewal points become the only observable parts of an anticipated hidden random variation of some physical quantity. The hypothesis might be that a hidden random process originating from zero or some other low value only becomes visible at the time of first crossing of some given value level, and that the process is restarted from scratch immediately after the level crossing. It might then be of interest to reveal the defining properties of this hidden process from a sample of observed first-passage times. In this paper the hidden process is first anticipated as a non-stationary Ornstein–Uhlenbeck (OU) process with unknown parameters that have to be estimated only by use of the information contained in a sample of first-passage times. The estimation method is a direct application of the Fortet integral equation of the OU process. A non-stationary Feller process is considered subsequently. As the OU process, the Feller process has a known transition probability distribution that allows the formulation of the integral equation. The described integral equation estimation method also provides a subjective graphical test of the applicability of the OU process or the Feller process when applied to a reasonably large sample of observed first-passage data.

These non-stationary processes have several applications in biomedical research, for example as idealized models of the neuron membrane potential. When the potential reaches a certain threshold the neuron fires, whereupon the potential drops to a fixed initial value, from where it continuously builds up again until the next firing. Also in civil engineering there are hidden random phenomena such as internal cracking or corrosion that after some random time break through to the material surface and become observable. However, the OU process has as a model of physical phenomena the defect of not being bounded to the negative side. This defect is not present for the Feller process, which therefore may provide a useful modeling alternative to the OU process.     

Response spectral density of linear systems with external and parametric non-Gaussian, delta-correlated excitation

An exact, closed form, analytical expressions for a response spectral density of a certain type of systems, subjected to non-Gaussian, stationary, delta-correlated noise are derived. A new, extended mean square stability conditions are derived for such systems.     

Ray tracing and scalar diffraction calculations of wavefronts,caustics and complex amplitudes in optical systems

The procedures for precise calculation of wavefronts, caustics and complex amplitudes in optical systems are developed. Numerical methods are compared with analytical formulae for caustics. The conditions for the validity of the integration on wavefronts for obtaining the complex amplitudes (and hence intensities defining the PSFs) within scalar diffraction theory are discussed in detail. To illustrate the precision of the results obtained with the developed techniques, an experiment showing quite unexpected results is studied and explained in detail.     

Identification of stiffness,dissipation and input parameters of multi degree of freedom civil systems under unmeasured base excitations

A time domain dynamic identification technique based on a statistical moment approach has been formulated for civil systems under base random excitations in the linear state. This technique is based on the use of classically damped models characterized by a mass proportional damping. By applying the Itô stochastic calculus, special algebraic equations that depend on the statistical moments of the response can be obtained. These equations can be used for the dynamic identification of the mechanical parameters that define the structural model, in the case of unmeasured input as well, and the identification of the input itself. Furthermore, the above equations demonstrate the possibility of identifying the dissipation characteristics independently from the input characteristics. This paper discusses how the equations for solving the identification problem can be obtained and gives an application.