Showing papers on "Gaussian process published in 1990"

Cube Root Asymptotics

Abstract: We establish a new functional central limit theorem for empirical processes indexed by classes of functions. In a neighborhood of a fixed parameter point, an $n^{-1/3}$ rescaling of the parameter is compensated for by an $n^{2/3}$ rescaling of the empirical measure, resulting in a limiting Gaussian process. By means of a modified continuous mapping theorem for the location of the maximizing value, we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure. These statistics include the short, Rousseeuw's least median of squares estimator, Manski's maximum score estimator, and the maximum likelihood estimator for a monotone density. The limit theory depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.

Large Deviation Techniques in Decision, Simulation, and Estimation

Abstract: Cramir's Theorem and Extensions Sanov's Theorem and the Contraction Principle Gaussian Processes and Wentzell-Freidlin Theory Large Deviations for Markov Processes Applications to Detection Theory Asymptotic Expansions Quick Simulation Applications to Parameter Estimation Applications to Information Theory Appendices Solutions to Exercises References Index.

Simulation of random fields via local average subdivision

Abstract: A fast and accurate method of generating realizations of a homogeneous Gaussian scalar random process in one, two, or three dimensions is presented. The resulting discrete process represents local averages of a homogeneous random function defined by its mean and covariance function, the averaging being performed over incremental domains formed by different levels of discretization of the field. The approach is motivated first by the need to represent engineering properties as local averages (since many properties are not well defined at a point and show significant scale effects), and second to be able to condition the realization easily to incorporate known data or change resolution within sub‐regions. The ability to condition the realization or increase the resolution in certain regions is an important contribution to finite element modeling of random phenomena. The Ornstein‐Uhlenbeck and fractional Gaussian noise processes are used as illustrations.

Stein's method for diffusion approximations

Abstract: Stein's method of obtaining distributional approximations is developed in the context of functional approximation by the Wiener process and other Gaussian processes. An appropriate analogue of the one-dimensional Stein equation is derived, and the necessary properties of its solutions are established. The method is applied to the partial sums of stationary sequences and of dissociated arrays, to a process version of the Wald-Wolfowitz theorem and to the empirical distribution function.

Cumulant-based order determination of non-Gaussian ARMA models

Abstract: The objective is the order determination of non-Gaussian and nonminimum phase autoregressive moving average (ARMA) models, using higher-order cumulant statistics. The two methods developed assume knowledge of upper bounds on the ARMA orders. The first method performs a linear dependence search among the columns of a higher-order statistics matrix by means of the Gram-Schmidt orthogonalization procedure. In the second method, the order of the AR part is found as the rank of the matrix formed by the higher-order statistics sequence. For numerically robust rank determination the singular value decomposition approach is adopted. The argument principle and samples of the polyspectral phase are used to obtain the relative degree of the ARMA model, from which the order of the MA part can be determined. Statistical analysis is included for determining the correct MA order with high probability, when estimates of third-order cumulants are only available. Simulations are used to verify the performance of the methods and compare autocorrelation with cumulant-based order determination approaches. >

Modeling and estimation of discrete-time Gaussian reciprocal processes

Abstract: Discrete-time Gaussian reciprocal processes are characterized in terms of a second-order two-point boundary-value nearest-neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics. This second-order model is the analog for reciprocal processes of the standard first-order state-space models for Markov processes. The model is used to obtain a solution to the smoothing problem for reciprocal processes. The resulting smoother obeys second-order equations whose structure is similar to that of the Kalman filter for Gauss-Markov processes. It is shown that the smoothing error is itself a reciprocal process. >

Asymptotically optimal estimation of MA and ARMA parameters of non-Gaussian processes from high-order moments

Abstract: A description is given of an asymptotically-minimum-variance algorithm for estimating the MA (moving-average) and ARMA (autoregressive moving-average) parameters of non-Gaussian processes from sample high-order moments. The algorithm uses the statistical properties (covariances and cross covariances) of the sample moments explicitly. A simpler alternative algorithm that requires only linear operations is also presented. The latter algorithm is asymptotically-minimum-variance in the class of weighted least-squares algorithms. >

Testing for threshold autoregression

Abstract: We consider the problem of determining whether a threshold autoregressive model fits a stationary time series significantly better than an autoregressive model does. A test statistic $\lambda$ which is equivalent to the (conditional) likelihood ratio test statistic when the noise is normally distributed is proposed. Essentially, $\lambda$ is the normalized reduction in sum of squares due to the piecewise linearity of the autoregressive function. It is shown that, under certain regularity conditions, the asymptotic null distribution of $\lambda$ is given by a functional of a central Gaussian process, i.e., with zero mean function. Contiguous alternative hypotheses are then considered. The asymptotic distribution of $\lambda$ under the contiguous alternative is shown to be given by the same functional of a noncentral Gaussian process. These results are then illustrated with a special case of the test, in which case the asymptotic distribution of $\lambda$ is related to a Brownian bridge.

On Likelihood Ratio Tests for Threshold Autoregression

Abstract: SUMMARY This paper addresses the null distribution of the likelihood ratio statistic for threshold autoregression with normally distributed noise. The problem is non-standard because the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We reduce the problem to the first-passage probability associated with a Gaussian process which, in some special cases, turns out to be a Brownian bridge. It is also shown that, in some specific cases, the asymptotic null distribution of the test statistic depends only on the 'degrees of freedom' and not on the exact null joint distribution of the time series.

Image identification and restoration based on the expectation-maximization algorithm

Abstract: In this paper, the problem of identifying the image and blur parameters and restoring a noisy blurred image is addressed. Specifying the blurring process by its point spread function (PSF), the blur identification problem is formulated as the maximum likelihood estimation (MLE) of the PSF. Modeling the original image and the additive noise as zeromean Gaussian processes, the MLE of their covariance matrices is also computed. An iterative approach, called the EM (expectation-maximization) algorithm, is used to find the maximum likelihood estimates ofthe relevant unknown parameters. In applying the EM algorithm, the original image is chosen to be part of the complete data; its estimate is computed in the E-step of the EM iterations and represents the restored image. Two algorithms for identification/restoration, based on two different choices of complete data, are derived and compared. Simultaneous blur identification and restoration is performed by the first algorithm, while the identification of the blur results from a separate minimization in the second algorithm. Experiments with simulated and photographically blurred images are shown.

On the capacity of a twisted-wire pair: Gaussian model

Abstract: The performance of a twisted-pair channel is assumed to be dominated by near-end crosstalk (NEXT) from other pairs in the same cable. Both intrabuilding local and central office loop channels can be modeled as NEXT-dominated channels. The capacity of this type of channel is found, using a Gaussian model. It is shown that the capacity is independent of the transmitted power spectral density. The results also indicate that present systems operate far below theoretical capacity. The capacity of a twisted-pair channel with both NEXT and white Gaussian noise present is also addressed. >

On estimating noncausal nonminimum phase ARMA models of non-Gaussian processes

Abstract: The authors address the problem of estimating the parameters of non-Gaussian ARMA (autoregressive moving-average) processes using only the cumulants of the noisy observation. The measurement noise is allowed to be colored Gaussian or independent and identically non-Gaussian distributed. The ARMA model is not restricted to be causal or minimum phase and may even contain all-pass factors. The unique parameter estimates of both the MA and AR parts are obtained by linear equations. The structure of the proposed algorithm facilitates asymptotic performance evaluation of the parameter estimators and model order selection using cumulant statistics. The method is computationally simple and can be viewed as the least-squares solution to a quadratic model fitting of a sampled cumulant sequence. Identifiability issues are addressed. Simulations are presented to illustrate the proposed algorithm. >

Structural Load Modeling and Combination for Performance and Safety Evaluation

Abstract: 1. Introduction. Benefit versus risk of engineering facilities. Uncertainties in demand and capacity of engineering facilities. Treatment of uncertainty in design of engineering facilities. Objectives and emphasis. Organization of subject materials. 2. Basic Random Variable and Random Process Models. Commonly used load occurrence models. Bernoulli sequence. Poisson process. Renewal process. Polya process. Multi-variate point process. Commonly used load intensity models. Uni- and multi-variate normal distribution. Lognormal distribution. Gamma and exponential distribution. Extreme value distribution. Continuous Gaussian process. Point process with deterministic shape response function. Generation of random load intensity and random load process on digital computer and Monte-Carlo method. Generation of random variables. Generation of random processes. Convergence of Monte-Carlo method. 3. Modeling of Time Varying Load and Load Effect. Fluctuation of load and load effect. Loadings with macro-scale time variability only. Loadings with both macro- and micro-time variability. Pulse process. Poisson pulse process. Generalization. Other pulse process. Intermittent continuous process. Examples of load and load effect as pulse and intermittent processes. Appendix 3-A: Input-Output relationship of linear systems. Under dynamic random excitation. 4. Combination of Loads and Load Effects. Linear combination. Load coincidence (L.C.) method. Method of point crossing. Method of upcrossing rate. Other methods. Nonlinear combination. Outcrossing rate analysis. Resistance uncertainty. Load coincidence method. First and second order methods. Point crossing method. 5. Modeling and Effect of Load Dependencies. Within-load dependencies. Dependence between intensity and duration. Occurrence dependence (clustering). Intensity dependence. Between-load dependencies. Occurrence clustering among loads. Intensity dependence between loads. General case. Duration of coincidence of dependent loadings. Appendices: Monte-Carlo simulation and combination of dependent pulse processes. Sum of two independent Gauss-Markov processes. Derivation of function h 12 (3) (t,t'). Integration of Eq. 5.50. 6. Load Combination Rules. Risk consistency of current rules. Load reduction factor method (LRF). SRSS rule. Companion action factor method (CAF). Turkstra's rule (TR). Accuracy of load combination rules. Appendices: Derivation of joint distribution function of lifetime maximum value, R, and arbitrary-point-in-time value, S, of a pulse or intermittent process. Derivation of nonexceedance probability according to Turkstra's rule. Index.

Slepian Models and Regression Approximations in Crossing and Extreme Value Theory

Abstract: In crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of the Slepian model and the regression method and shows how they can be used to obtain good numerical approximations to various crossing variables. It gives a detailed account of the regression method for Gaussian processes with auxiliary variables chosen in a recursive way. It also presents a package of computer programs for the numerical calculations, and gives numerical examples on excursion lengths as well as wavelength and amplitude distributions. Further examples deal with an engineering 'jump-and-bump' problem, and excursions for a chi-2-process.

Linear modeling of multidimensional non-Gaussian processes using cumulants

Abstract: Extending the notion of second-order correlations, we define thecumulants of stationary non-Gaussian random fields, and demonstrate their potential for modeling and reconstruction of multidimensional signals and systems. Cumulants and their Fourier transforms calledpolyspectra preserve complete amplitude and phase information of a multidimensional linear process, even when it is corrupted by additive colored Gaussian noise of unknown covariance function. Relying on this property, phase reconstruction algorithms are developed using polyspectra, which can be computed via a 2-D FFT-based algorithm. Additionally, consistent ARMA parameter estimators are derived for identification of linear space-invariant multidimensional models which are driven by unobservable, i.i.d., non-Gaussian random fields. Contrary to autocorrelation based multidimensional modeling approaches, when cumulants are employed, the ARMA model is allowed to be non-minimum phase, asymmetric non-causal or non-separable.

Methods for Time‐Dependent Reliability and Sensitivity Analysis

Abstract: A general and efficient method for reliability and sensitivity analysis is presented. The methods are for the analysis of components and systems in design situations where uncertainties are represented by a vector of random variables and a stationary Gaussian vector process. A formulation as a first-passage problem for a vector process outcrossing a safe set is first applied for a fixed value of the random variable vector. This gives a conditional failure probability, and a fast integration technique based on a first- or second-order reliability method is then applied to compute the unconditional failure probability. Extensive use of analytical gradient information is made in the iteration algorithms and in the calculation of sensitivity factors. A nested first-order reliability method is proposed together with a method based on an integrated optimization for the two first-order reliability analyses. For a preliminary analysis, a “crude” first-order reliability method is described. The computation time is only a few times the computation time for a first-order reliability analysis of a similar time-independent problem. Software developed from existing special software for first-order reliability analysis or existing general optimization software can be applied.

Extreme Values and High Boundary Crossings of Locally Stationary Gaussian Processes

Abstract: We consider the large values of a locally stationary Gaussian process which satisfies Berman's condition on the long range dependence. We present some limit results on the exceedances of the process above a certain general smooth high boundary. This allows deriving the limiting distribution of the maximum up to time T, for example, in the case of a standardized process with a constant boundary or in the case of a nonstandardized process with a smooth trend

Frequency domain Cramer-Rao bound for Gaussian processes

Abstract: An expression is derived for the asymptotic frequency-domain Cramer-Rao bound (CRB) on the parameters of Gaussian processes with information in their means. The general result extends Whittle's (1953) formula which is applicable to Gaussian processes with information in their covariance matrices or spectra. It is useful in many fields such as system identification, radar, and sonar. Examples of its application to system deconvolution and time-delay estimation in colored noise are given. >

Random response of a single‐degree‐of‐freedom vibro‐impact system with clearance

Abstract: The closed form solutions of the stationary random response of a single-degree-of-freedom vibro-impact system with clearance are formulated in this paper. The Hertz contact law from elasticity is used to model the contact phenomena between the mass and constraint during vibration. The excitation is assumed to be a stationary white Gaussian process with zero mean. Through solving the time-independent Fokker-Planck equation, the stationary responses are obtained analytically. The effects of contact stiffness and clearance on the response are discussed probabilistically. It is found that, when the clearance is about twice the square root of the mean square response of the corresponding linear system, the contact phenomena are almost negligible.

A stochastic model for the distribution of HIV latency time based on T4 counts

Abstract: SUMMARY A mathematical model is proposed for the description of the T4 level, as a function of the time since initial infection, in the blood of an HIv-infected individual. Two random variables are defined for each infected individual, the time T from the moment of infection to the moment when the infection is first discovered, and the level R of T4 cells in the blood at the time of the discovery. Here T is the length of the latency period, and is not typically observable. On the basis of a stochastic model employing a stationary Gaussian process, the joint density of these two random variables is derived, and hence the conditional distribution of T, given R. It is shown that if T has an exponential marginal distribution, then the conditional distribution of T is a censored normal distribution. The theory is applied to data collected from intravenous drug users in New York City.

On shape asymmetry of Gaussian molecules

Abstract: A method of calculating average moments, to an arbitrary order m, of the principal components of the gyration tensor of a Gaussian molecule imbedded in any k‐dimensional space is presented. These average moments are then used in generalized shape parameters Am of degree m≤k, which measure the asymmetry of the shapes of Gaussian molecules. Simple formulas for A2, A3, and the average moments up to third order are given. Explicit expressions for A2 and A3 for linear and circular chains, regular stars with infinitely long arms, double rings of a large number of beads, and combs with many side chains are obtained, and the shape characteristics of these molecules are discussed. It is found that Gaussian molecules are, on the average, prolate rather than oblate even in an infinite dimensional space, with the exception of regular stars with densely radiated long arms which exhibit perfect symmetry. The problem of analytic characterizations of shape asymmetry of Gaussian molecules or non‐self‐avoiding random walks...

Accuracy of peak deconvolution algorithms within chromatographic integrators

Abstract: The soundness of present-day algorithms to deconvolve overlapping skewed peaks was investigated. From simulated studies based on the exponentially modified Gaussian model (EMG), chromatographic peak area inaccuracies for unresolved peaks are presented for the two deconvolution methods, the tangent skim and the perpendicular drop method. These inherent inaccuracies, in many cases exceeding 50%, are much greater than those calculated from ideal Gaussian profiles. Multiple linear regression (MLR) was used to build models that predict the relative error for either peak deconvolution method. MLR also provided a means for determining influential independent variables, defining the required chromatographic relationships needed for prediction. Once forecasted errors for both methods are calculated, selection of either peak deconvolution method can be made by minimum errors. These selection boundaries are contrasted to method selection criteria of present data systems algorithms.

Stationary points of the constant modulus algorithm for real Gaussian signals

Abstract: An analysis of the stationary points for four versions of the constant modulus algorithm (CMA) when the received signal is a zero-mean, real Gaussian process is presented. With this type of signal, it is well known that CMA generally cannot equalize a communication channel; the analysis of the CMA performance function demonstrates how this occurs. It is shown that in each case the stationary points are determined by a condition on the variance of the equalizer output, and they depend only on the modulus factor r and the received signal correlation matrix R. As a result, there can be an infinity of weight vector solutions, each corresponding to a stationary point of the CMA performance function. >

Nonuniform image motion estimation from noisy data

Abstract: Image motion estimation is viewed as a problem in nonlinear demodulation. The motion D(x) modulates the intensity of the previous frame s(x) to result in the intensity of the present frame s(x-D(x)). An iterative algorithm based on the generalized maximum likelihood criterion is developed and implemented to demodulate D(x). An advantage of this scheme is the incorporation of the covariance function matrix, which is specifically useful in the stationary case to determine the integration region alpha /sub x/. Simulation demonstrated the ability of the algorithm to deal with cases such as Markov-2 motion. >