Nonlinear systems driven by polynomials of filtered Poisson processes

The perturbation method is applied to determine approximately the mean, variance, skewness and kurtosis of the transient and stationary response of nonlinear systems driven by polynomials of filtered Poisson processes. The analysis is based on the classical perturbation method, the Itô differentiation formula, and properties of the response of linear systems subjected to polynomials of filtered Poisson processes. Two examples are presented to demostrate the efficiency and accuracy of this approximate analysis.     

Higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses

The higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses are evaluated by means of knowledge of the first order statistics and without any further integration. This is made possible by a coordinate transformation which replaces the original system by a quasi-linear one with parametric Poisson delta-correlated input; and, for these systems, a simple relationship between first order and higher order statistics is found in which the transition matrix of the dynamical new system, incremented by the correction terms necessary to apply the Itô calculus, appears.     

First passage time of filtered Poisson process with exponential shape function

Solving some integro-differential equation we find the Laplace transform of the first passage time for filtered Poisson process generated by pulses with uniform or exponential distributions. Also, the martingale technique is applied for approximations of expectations and distributions of the first passage times. The approximations accuracy is verified with the help of Monte-Carlo simulations.     

Higher order statistics of the response of MDOF linear systems under polynomials of filtered normal white noises

This paper exploits the work presented in the companion paper in order to evaluate the higher order statistics of the response of linear systems excited by polynomials of filtered normal processes. In fact, by means of a variable transformation, the original system is replaced by a linear one excited by external and linearly parametric white noise excitations. The transition matrix of the new enlarged system is obtained simply once the transition matrices of the original system and of the filter are evaluated. The method is then applied in order to evaluate the higher order statistics of the approximate response of nonlinear systems to which the pseudo-force method is applied.     

Control Charts with Variable Dimension for Linear Combination of Poisson Variables

This article analyzes the simultaneous control of several correlated Poisson variables by using the Variable Dimension Linear Combination of Poisson Variables (VDLCP) control chart, which is a variable dimension version of the LCP chart. This control chart uses as test statistic, the linear combination of correlated Poisson variables in an adaptive way, i.e. it monitors either p₁ or p variables (p₁ < p) depending on the last statistic value. To analyze the performance of this chart, we have developed software that finds the best parameters, optimizing the out‐of‐control average run length (ARL) for a shift that the practitioner wishes to detect as quickly as possible, restricted to a fixed value for in‐control ARL. Markov chains and genetic algorithms were used in developing this software. The results show performance improvement compared to the LCP chart. Copyright © 2015 John Wiley & Sons, Ltd.     

Monitoring of zero-inflated Poisson processes with EWMA and DEWMA control charts

The zero-inflated Poisson (ZIP) distribution is an extension of the ordinary Poisson distribution and is used to model count data with an excessive number of zeros. In ZIP models, it is assumed that random shocks occur with probability p, and upon the occurrence of random shock, the number of nonconformities in a product follows the Poisson distribution with parameter λ. In this article, we study in more detail the exponentially weighted moving average control chart based on the ZIP distribution (regarded as ZIP-EWMA) and we also propose a double EWMA chart with an upper time-varying control limit to monitor ZIP processes (regarded as ZIP-DEWMA chart). The two charts are studied to detect upward shifts not only in each parameter individually but also in both parameters simultaneously. The steady-state performance and the performance with estimated parameters are also investigated. The performance of the two charts has been evaluated in terms of the average and standard deviation of the run length, and compared with Shewhart-type and CUSUM schemes for ZIP distribution, it is shown that the proposed chart is very effective especially in detecting shifts in p when λ remains in control (IC) and in both parameters simultaneously. Finally, one real example is given to display the application of the ZIP charts on practitioners.     

Level‐cut inhomogeneous filtered Poisson field for two‐phase microstructures

Level‐cut homogeneous filtered Poisson fields developed in (J. Appl. Phys. 2003; 94 (6):3762–3770) to model two‐phase microstructures are defined, and their properties are briefly reviewed. Filtered Poisson fields are sums of randomly scaled and oriented kernels that are centered at the points of homogeneous Poisson fields. The cuts of these fields above specified thresholds are called level‐cut homogeneous filtered Poisson fields. It is shown that an arbitrary inhomogeneous Poisson field becomes homogeneous if observed in new coordinates, and that the mapping relating inhomogeneous and homogeneous Poisson fields can be constructed in a simple manner. This mapping and the model in (J. Appl. Phys. 2003; 94 (6): 3762–3770) provide an efficient algorithm for generating arbitrary inhomogeneous two‐phase microstructures. Developments in (Int. J. Numer. Meth. Engng 2008; DOI: 10.1002/nme.2340 ), using arguments essentially identical to those in (J. Appl. Phys. 2003; 94 (6):3762–3770) to define and generate inhomogeneous Poisson fields, overlook the natural extension of results in (J. Appl. Phys. 2003; 94 (6): 3762–3770) to these fields provided by the mapping constructed in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

Zernike-gauss polynomials and optical aberrations of systems with gaussian pupils

In the first two Notes of this series,(l,2) we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.(3-5) Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the pointspread functions of uniform pupils.(6) Propagation of Gaussian laser beams also involves such pupils.     

Estimating the Change Point of a Poisson Rate Parameter with a Linear Trend Disturbance

Knowing when a process changed would simplify the search and identification of the special cause. In this paper, we compare the maximum likelihood estimator (MLE) of the process change point designed for linear trends to the MLE of the process change point designed for step changes when a linear trend disturbance is present. We conclude that the MLE of the process change point designed for linear trends outperforms the MLE designed for step changes when a linear trend disturbance is present. We also present an approach based on the likelihood function for estimating a confidence set for the process change point. We study the performance of this estimator when it is used with a cumulative sum (CUSUM) control chart and make direct performance comparisons with the estimated confidence sets obtained from the MLE for step changes. The results show that better confidence can be obtained using the MLE for linear trends when a linear trend disturbance is present. Copyright © 2005 John Wiley & Sons, Ltd.     

Zernike annular polynomials and optical aberrations of systems with annular pupils

Zernike annular polynomials that represent orthogonal andbalanced aberrations suitable for systems with annular pupilsare described. Their numbering scheme is the same asfor Zernike circle polynomials. Expressions for standard deviationof primary and balanced primary aberrations are given.     

Zernike circle polynomials and optical aberrations of systems with circular pupils

Zernike circle polynomials, their numbering scheme, and relationship to balanced optical aberrations of systems with circular pupils are discussed.     

Stochastic dynamics of MDOF structural systems under non-normal filtered inputs

The stochastic dynamics of structural systems driven by a filtered train of pulses, randomly distributed in time, is analysed by the space moments method. Usually the analyses, regarding MDOF structural systems, require a dramatic computation effort because the number of moment equations increases with power law with respect to the order of the moments. In this paper, by means of simple algebraic manipulations, the analysis of MDOF systems is reduced to the analysis of a set of Langevin equations with complex coefficients. This way, which is shown to considerably reduce the computational effort related to the stochastic analysis, seems to be the only reasonable one to perform the analysis of MDOF systems.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.     

A univariate Chebyshev polynomials method for structural systems with interval uncertainty

This paper presents an effective univariate Chebyshev polynomials method (UCM) for interval bounds estimation of uncertain structures with unknown-but-bounded parameters. The interpolation points required by the conventional collocation methods to generate the surrogate model are the tensor product of each one-dimensional (1D) interpolating point. Therefore, the computational cost is expensive for uncertain structures containing more interval parameters. To deal with this issue, the univariate decomposition is derived through the higher-order Taylor expansion. The structural system is decomposed into a sum of several univariate subsystems, where each subsystem only involves one uncertain parameter and replaces the other parameters with their midpoint value. Then the Chebyshev polynomials are utilized to fit the subsystems, in which the coefficients of these subsystems are confirmed only using the linear combination of 1D interpolation points. Next, a surrogate model of the actual structural system composed of explicit univariate Chebyshev functions is established. Finally, the extremum of each univariate function that is obtained by the scanning method is substituted into the surrogate model to determine the interval ranges of the uncertain structures. Numerical analysis is conducted to validate the accuracy and effectiveness of the proposed method.     

Estimating and simulating Poisson processes having trends or multiple periodicities

We develop and evaluate procedures for estimating and simulating nonhomogeneous Poisson processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. The experimental performance evaluation for this estimation procedure involves applying the procedure to 100 independent replications of nine selected point processes that possess up to four trigonometric rate components together with a polynomial rate component whose degree ranges from zero to three. On each replication of each process, the fitting procedure is applied to estimate the parameters of the process; and then the corresponding estimates of the rate and mean-value functions are computed over the observation interval. Evaluation of the fitting procedure is based on plotted tolerance bands for the rate and mean-value functions together with summary statistics for the maximum and average absolute estimation errors in these functions computed over the observation interval. The experimental results provide substantial evidence of the numerical stability and usefulness of the fitting procedure in simulation applications.     

The Wong-Zakai theorem for dynamical systems with parametric Poisson white noise excitation

The theorem of Wong and Zakai is applied to obtain a mathematical model for systems under random impulse excitation, in case that the excitation process tends to be delta-correlated. By means of the Wong-Zakai transformation, a class of reducible non-linear stochastic integro-differential equations is identified. Exact stationary probability density functions for reducible stochastic integro-differential equations are calculated.     

Testing for Poisson arrivals in INAR(1) processes

In the framework of integer-valued autoregressive processes of order 1 [INAR(1)], two new tests for the null hypothesis of Poisson-distributed innovations are developed. The tests focus on time reversibility, as this feature is shown to be satisfied exclusively by Poisson INAR(1) processes. The necessary asymptotic variances are explicitly calculated using the joint cumulants of these processes. The finite-sample behavior of the test statistics and the power of the tests are investigated in a simulation study. The results show that the newly developed tests perform better than existing ones in certain situations.     

An alternative expression for stochastic dynamical systems with parametric Poisson white noise

Di Paola and Falsone's formula is widely used in expressing a correction term to the usual Ito integral in stochastic dynamical systems with parametric Poisson white noise. An alternative expression is presented here. Comparing with Di Paola and Falsone's original expression, the alternative one is applicable under more general conditions, and shows significantly improved performance in numerical implementation. The alternative expression turns out to be a special case of the Marcus integrals.