Showing papers on "Gaussian process published in 1993"

Checking the Cox model with cumulative sums of martingale-based residuals

Abstract: SUMMARY This paper presents a new class of graphical and numerical methods for checking the adequacy of the Cox regression model The procedures are derived from cumulative sums of martingale-based residuals over follow-up time and/or covariate values The distributions of these stochastic processes under the assumed model can be approximated by zero-mean Gaussian processes Each observed process can then be compared, both visually and analytically, with a number of simulated realizations from the approximate null distribution These comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are Special attention is given to checking the functional form of a covariate, the form of the link function, and the validity of the proportional hazards assumption An omnibus test, consistent against any model misspecification, is also studied The proposed techniques are illustrated with two real data sets

Proper complex random processes with applications to information theory

Abstract: The covariance of complex random variables and processes, when defined consistently with the corresponding notion for real random variables, is shown to be determined by the usual complex covariance together with a quantity called the pseudo-covariance. A characterization of uncorrelatedness and wide-sense stationarity in terms of covariance and pseudo-covariance is given. Complex random variables and processes with a vanishing pseudo-covariance are called proper. It is shown that properness is preserved under affine transformations and that the complex-multivariate Gaussian density assumes a natural form only for proper random variables. The maximum-entropy theorem is generalized to the complex-multivariate case. The differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum if and only if the random vector is proper, Gaussian, and zero-mean. The notion of circular stationarity is introduced. For the class of proper complex random processes, a discrete Fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. An application of the theory is presented. >

Gaussian multiaccess channels with ISI: capacity region and multiuser water-filling

Abstract: The capacity region of a two-user Gaussian multiaccess channel with intersymbol interference (ISI) in which the inputs pass through respective linear systems and are superimposed before being corrupted by an additive Gaussian noise process is discussed. A geometrical method for obtaining the optimal input power spectral densities and the capacity region is presented. This method can be viewed as a nontrivial generalization of the single-user water-filling argument. It is shown that, as in the traditional memoryless multiaccess channel, frequency-division multiaccess (FDMA) with optimally selected frequency bands for each user achieves the total capacity of the multiuser Gaussian multiaccess channel with ISI. However, the capacity region of the two-user channel with memory is, in general, not a pentagon unless the channel transfer functions for both users are identical. >

Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction

Abstract: This article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain T, given values of yo and all of its first derivatives at a set of design sites (points) in T. The intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in T represents a particular configuration of the input parameters. It is assumed that the first derivatives are already available (e.g., from a sensitivity analysis) or can be produced by the code that implements the model. A Bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary Gaussian stochastic process is used. The calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example. The issue of experimental design is also discussed, in particular the criterion of maximizing the reduction in entropy...

Vector quantization for the efficient computation of continuous density likelihoods

Abstract: In speech recognition systems based on continuous observation density hidden Markov models, the computation of the state likelihoods is an intensive task. The author presents an efficient method for the computation of the likelihoods defined by weighted sums (mixtures) of Gaussians. This method uses vector quantization of the input feature vector to identify a subset of Gaussian neighbors. It is shown that, under certain conditions, instead of computing the likelihoods of all the Gaussians, one needs to compute the likelihoods of only the Gaussian neighbours. Significant (up to a factor of nine) likelihood computation reductions have been obtained on various data bases, with only a small loss of recognition accuracy. >

The behavior of LMS and NLMS algorithms in the presence of spherically invariant processes

Abstract: The behavior of least-mean-square (LMS) and normalized least-mean-square (NLMS) algorithms with spherically invariant random processes (SIRPs) as excitations is shown. Many random processes fall into this category, and SIRPs closely resemble speech signals. The most pertinent properties of these random processes are summarized. The LMS algorithm is introduced, and the first- and second-order moments of the weight-error vector between the Wiener solution and the estimated solution are shown. The behavior of the NLMS algorithm is obtained, and the first- and second-order moments of the weight-error vector are calculated. The results are verified by comparison with known results when a white noise process and a colored Gaussian process are used as input sequences. Some simulation results for a K/sub 0/-process are then shown. >

On the performance of box-counting estimators of fractal dimension

Abstract: SUMMARY Box-counting estimators are popular for estimating fractal dimension. However, very little is known of their stochastic properties, despite increasing statistical interest in their application. We show that, if the irregular curve to which the estimators are applied is modelled by a Gaussian process, concise formulae may be developed for asymptotic bias and variance of box-counting estimators. These formulae point to critical differences between a naive form of the box-counting estimator, based directly on the capacity definition of fractal dimension, and a regression-inspired version of that estimator.

Goodness of Fit Problem and Scanning Innovation Martingales

Abstract: This paper is mainly devoted to the following statistical problem: in the case of random variables of any finite dimension and both simple or parametric hypotheses, how to construct convenient "empirical" processes which could provide the basis for goodness of fit tests-more or less in the same way as the uniform empirical process does in the case of simple hypothesis and scalar random variables. The solution of this problem is connected here with the theory of multiparameter martingales and the theory of function-parametric processes. Namely, for the limiting Gaussian processes some kind of filtration is introduced and so-called scanning innovation processes are constructed-the adapted standard Wiener processes in one-to-one correspondence with initial Gaussian processes. This is done for the function-parametric versions of the processes.

Models for Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes and the Inverse Gaussian Distribution

Abstract: Variable-stress accelerated life testing trials are experiments in which each of the units in a random sample of units of a product is run under increasingly severe conditions to get information quickly on its life distribution. We consider a fatigue failure model in which accumulated decay is governed by a continuous Gaussian process W(y) whose distribution changes at certain stress change points to < t l < < …

Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum

Abstract: We present a comprehensive normal-mode decomposition analysis for the recently introduced [ J. Opt. Soc. Am. A10, 95 ( 1993)] class of twisted Gaussian Schell-model fields in partially coherent beam optics. The formal analogies to quantum mechanics in two dimensions are exploited. We also make effective use of a dynamical SU(2) symmetry of these fields to achieve the mode decomposition and to determine the spectrum. The twist phase is nonseparable in nature, rendering it nontrivially two dimensional. The consequences of this, resulting in the need to use Laguerre–Gaussian functions rather than products of Hermite–Gaussians, are carefully analyzed. An important identity involving these sets of special functions is established and is used in deriving the spectrum.

Numerical estimation of adsorption energy distributions from adsorption isotherm data with the expectation-maximization method

Abstract: The expectation-maximization (EM) method of parameter estimation is used to calculate adsorption energy distributions of molecular probes from their adsorption isotherms. EM does not require prior knowledge of the distribution function, or the isotherm, requires no smoothing of the isotherm data, and converges with high stability toward the maximum-likelihood estimate. The method is therefore robust and accurate at high iteration numbers. The EM algorithm is tested with simulated energy distributions corresponding to unimodel Gaussian, bimodal Gaussian, Poisson distributions, and the distributions resulting from Misra isotherms. Theoretical isotherms are generated from these distributions using the Langmuir model, and then chromatographic band profiles are computed using the ideal model of chromatography. Noise is then introduced in the theoretical band profiles comparable to those observed experimentally. The isotherm is then calculated using the elution-by-characteristic points method. The energy distribution given by the EM method is compared to the original one. The results are contrasted to those obtained with the House and Jaycock algorithm HILDA and shown to be superior in terms of both robustness, accuracy, and information theory. 20 refs., 6 figs., 4 tabs.

Nonuniversal correlations for random matrix ensembles

Abstract: The eigenvalue correlations of the generalized Gaussian and Laguerre random matrix ensembles are calculated exactly. The fluctuations are shown to be nonuniversal in certain intervals of the spectrum. A physical example from quantum transport where such nonuniversal effects occur is discussed.

System identification using a linear combination of cumulant slices

Abstract: A linear approach to identifying the parameters of a moving-average (MA) model from the statistics of the output is presented. First, it is shown that, under some constraints, the impulse response of the system can be expressed as a linear combination of cumulant slices. Then, this result is used to obtain a well-conditioned linear method for estimating the MA parameters of a nonGaussian process. The linear combination of slices used to compute the MA parameters can be constructed from different sets of cumulants of different orders, provided a general framework in which all the statistics can be combined. It is not necessary to use second-order statistics (autocorrelation slice), and therefore the proposed algorithm still provides consistent estimates in the presence of colored Gaussian noise. Another advantage of the method is that while most linear methods give totally erroneous estimates if the order is overestimated, the proposed approach does not require a previous estimation of the filter order. The simulation results confirm the good numerical conditioning of the algorithm and its improvement in performance in comparison to existing methods. >

Comparisons of theory and experiment in light scattering from a randomly rough surface

Abstract: For a conducting surface with one-dimensional roughness, we compare experimental and theoretical results for the four unique elements of the Stokes-scattering matrix that provide a complete description of the diffusely scattered light. The rough surface has been fabricated with new techniques and is strictly one dimensional, and scattered intensities at infrared wavelengths show clear backscattering enhancement that arises from multiple scattering within surface corrugations. To obtain theoretical results for the Stokes matrix elements, we numerically apply an impedance boundary-condition method, appropriate for the roughness and the high conductivity of the experimental surface, to a statistical ensemble of rough surfaces. The experimental surface has been found to have nearly Gaussian first-order height statistics, and experimental measurements of the matrix elements are compared with theoretical results for a surface consistent with a Gaussian process. These comparisons suggest that there is more multiple scattering in the experimental data than is accounted for by the theoretical calculations. We attribute this observation to the properties of the second derivative of the experimental surface, which are found to be inconsistent with those of a Gaussian process. In further calculations that take account of the unusual properties of the experimental surface, excellent agreement of theoretical and experimental results is obtained.

Robust estimation of structured covariance matrices

Abstract: In the context of the narrowband array processing problem, robust methods for accurately estimating the spatial correlation matrix using a priori information about the matrix structure are developed. By minimizing the worse case asymptotic variance, robust, structured, maximum-likelihood-type estimates of the spatial correlation matrix in the presence of noises with probability density functions in the in -contamination and Kolmogorov classes are obtained. These estimates are robust against variations in the noise's amplitude distribution. The Kolmogorov class is demonstrated to be the natural class to use for array processing applications, and a technique is developed to determine exactly the size of this class. Performance of bearing estimation algorithms improves substantially when the robust estimates are used, especially when nonGaussian noise is present. A parametric structured estimate of the spatial correlation matrix that allows direct estimation of the arrival angles is also demonstrated. >

A generalization of the entropy power inequality with applications

Abstract: The authors prove the following generalization of the entropy power inequality: h(ax)>or=h(Ax) where h(.) denotes (joint-) differential-entropy x=x/sub 1/...x/sub n/, is a random vector with independent components, x=x...x/sub n/, is a Gaussian vector with independent components such that h(x/sub i/)=h(x/sub i/), i=1...n, and A is any matrix. This generalization of the entropy-power inequality is applied to show that a non-Gaussian vector with independent components becomes "closer" to Gaussianity after a linear transformation, where the distance to Gaussianity is measured by the information divergence. Another application is a lower bound, greater than zero, for the mutual-information between nonoverlapping spectral components of a non-Gaussian white process. They also describe a dual generalization of the Fisher information inequality. >

Outcrossings from Convex Polyhedrons for Nonstationary Gaussian Processes

Abstract: This technical note concerns the estimation of the outcrossing rate of nonstationary Gaussian vector processes from a convex polyhedral limit-state surface enclosing the origin. The theoretically most attractive approach for this type of problem in time-dependent structural reliability analysis is to employ the concept of first-passage probability in stochastic-process theory. Because the use of the generalized Rice formula poses some difficulty, the outcrossing problem is formulated in another way; i.e., using the original Rice formula directly. This situation allows the multidimensional integral to be reduced to a one-dimensional integral. This then allows the formulation (which applies also to smooth nonstationary processes) to be extended to time-dependent domain boundaries. An example is given to illustrate the method.

Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics

Abstract: Extending the work of part I of this series [ J. Opt. Soc. Am. A10, 2008– 2016 ( 1993)], we analyze the structure of the eigenvalue spectrum as well as the propagation characteristics of the twisted Gaussian Schell-model beams. The manner in which the twist phase affects the spectrum, and hence the positivity property of the cross-spectral density, is brought out. Propagation characteristics of these beams are simply deduced from the elementary properties of their modes. It is shown that the twist phase lifts the degeneracy in the eigenvalue spectrum on the one hand and acts as incoherence in disguise on the other. An abstract Hilbert-space operator corresponding to the cross-spectral density of the twisted Gaussian Schell-modelbeam is explicitly constructed, bringing out the useful similarity between these cross-spectral densities and quantum-mechanical thermal-state-density operators of isotropic two-dimensional oscillators, with a term proportional to the angular momentum added to the Hamiltonian.

The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions

Abstract: Recent experimental and computational observations demonstrate the occurrence of large‐scale intermittency for diffusing passive scalars, as manifested by broader than Gaussian probability distribution functions. Here, a family of explicit exactly solvable examples is developed which demonstrates these effects of large‐scale intermittency at any positive time through simple formulas for the higher flatness factors without any phenomenological approximations. The exact solutions involve advection–diffusion with velocity fields involving a uniform shear flow perturbed by a random fluctuating uniform shear flow. Through an exact quantum mechanical analogy, the higher‐order statistics for the scalar in these models are solved exactly by formulas for the quantum‐harmonic oscillator. These explicit formulas also demonstrate that the large time asymptotic limiting probability distribution function for the normalized scalar can be either broader than Gaussian or Gaussian depending on the relative strength of the mean flow and the fluctuating velocity field.

Skewed, exponential pressure distributions from Gaussian velocities

Abstract: A simple analytical argument is given to show that the distribution function of the pressure and that of its gradient have exponential tails when the velocity is Gaussian A calculation of moments implies a negative skewness for the pressure Explicit analytical results are given for the case of the velocity being restricted to a shell in wave number Numerical pressure distributions are presented for Gaussian velocities with realistic spectra For real turbulent flows, one expects that the pressure distribution should retain exponential tails while the pressure gradients should develop stretched‐exponential distributions In the context of the theory, available numerical and laboratory data are examined for the pressure, along with data for the wall shear stress in a boundary layer

Simulation of Spatially Incoherent Random Ground Motions

Abstract: Two techniques for the artificial generation of spatially incoherent Gaussian seismic ground motions are proposed and validated. The simulated motions are homogeneous and stationary, and may be one-, two-, or three-dimensional in space. They satisfy a prescribed, or target, local power spectrum and a target coherency function. Nonstationarity characteristics are introduced by superimposing a time-dependent envelope function to produce a uniformly modulated nonstationary process. The first technique is asymptotic and approximates the coherency function by a Fourier series: it is general and suits any form of spectrum and coherency. The second technique is approximate in the sense that it satisfies the autospectrum everywhere but satisfies the cross spectrum, or the coherency, between successive stations only. The latter technique is computationally very efficient, and may be accurate enough for discretely supported systems such as single-span structures and multispan, simply supported bridges. The techniques proposed may be useful in response analysis of structures, or structural components, for spatially incoherent random processes in the fields of earthquake, ocean, and wind engineering.

A note on small ball probability of a Gaussian process with stationary increments

Abstract: Let {X(t), 0≤t≤1} be a Gaussian process with mean zero and stationary increments. Let σ2(h) =EX 2(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.

On the asymptotic statistical behavior of empirical cepstral coefficients

Abstract: The asymptotic covariance matrix of the empirical cepstrum is analyzed. It is shown that for Gaussian processes, ceptral coefficients derived from smoothed periodograms are asymptotically uncorrelated and their variances multiplied by the sample size tend to unity. For an autoregressive process and its autoregressive cepstrum estimate, somewhat weaker results hold. >