Showing papers on "Gaussian process published in 1969"

A general likelihood-ratio formula for random signals in Gaussian noise

Abstract: It is shown that the likelihood ratio for the detection of a random, not necessarily Gaussian, signal in additive white Gaussian noise has the same form as that for a known signal in white Gaussian noise. The role of the known signal is played by the casual least-squares estimate of the signal from the observations. However, the "correlation" integral has to be interpreted in a special sense as an Ito stochastic integral. It will be shown that the formula includes all known explicit formulas for signals in white Gaussian noise. However, and more important, the formula suggests an "estimator-correlator" philosophy for engineering approximation of the optimum receiver. Some extensions of the above result are also discussed, e.g., additive finite-variance, not necessarily Gaussian, noise plus a white Gaussian noise component. Purely colored Gaussian noise can be treated if whitening filters can be specified. The analog implementation of Ito integrals is briefly discussed. The proofs of the formulas are based on the concept of an innovation process, which has been useful in certain related problems of linear and nonlinear least-squares estimation, and on the concept of covariance factorization.

Asymptotic properties of the maximum in a stationary Gaussian process.

Abstract: and their behavior as t -> 00. In §2, the form of the asymptotic distribution of Z(t) is given for all a. This generalizes the result of [6] wherein a = 1, and the result of Cramer [4] and Volkonski and Rozanov [9], wherein a = 2. In §3, the almost sure asymptotic behaviour of Z(t) is investigated for all a. For the case a = 2, the result extends that of Shur [8]. The results of this work depend heavily upon some of the results involving upcrossings given in [7]. They are valid for smallest as well as for largest values, provided appropriate but obvious modifications are made.

The minimum of a stationary Markov process superimposed on a U-shaped trend

Abstract: 1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by

Calculation of Bayes' Recognition Error for Two Multivariate Gaussian Distributions

Abstract: An algorithm is presented for calculating recognition error when applying pattern vectors to an optimum Bayes' classifier. The pattern vectors are assumed to come from two classes whose populations have Gaussian statistics with unequal covariance matrices and arbitrary a priori probabilities. The quadratic discriminant function associated with a Bayes' classifier is used as a one-dimensional random variable from which the probability of error is calculated, once the distribution of the discriminant function is obtained.

Harmonic analysis of local times and sample functions of Gaussian processes

Abstract: v(A) = u(x-'(A)) on the linear Borel sets A. If v is absolutely continuous with respect to the usual linear Borel measure, then its derivative is called the local time of x. In this paper we deduce some properties of the variation of x from smoothness properties of the local time when u is fixed and x is a sample function of a Gaussian process. The main results are Theorems 5.1 and 5.2 which imply: If X(t), 0 C It - s 1, for some I, 0< f < 2, then the sample functions (a) are of unbounded y-variation for y < 2/1; (b) nowhere satisfy a Holder condition of order 2/(2m + E + 1) if : < 2/(2m + E + 1), where m is a nonnegative integer and 0 < E < 1; (c) are nowhere differentiable if : < 1; (d) return infinitely often to every neighborhood of almost all the points they visit. The result (c) was stated without proof in [5] for a stationary process whose covariance function r is not twice differentiable at the origin. A related result was proved by Yeh [7] under the additional assumption that X(.) is continuous. The paper of Marcus [4] contains results on Holder conditions for processes with stationary increments satisfying the condition that E(X(t) - X(0))2 is concave in some neighborhood of 0, or related conditions. Using a previous result on analytic local times [2], we prove in ?6 that Levy's Brownian motion over Hilbert space has sample functions unbounded over certain compact ellipsoids (cf. [1]). 1. Square integrable local times. Put

An algorithm for the decomposition of a distribution into Gaussian components.

Abstract: An algorithm for the decomposition of a mixture of Gaussian components is given, partially in Algol-60 language, and experiences with its use on a digital computer are described. The algorithm contains three steps: (1) finding the mean of each component by the aid of a Fourier transform of the given density function and the method of decreasing the standard deviations (Medgyessi [1961]); (2) determining the standard deviations and frequencies of the components using the continued fraction approximation of the error function; and (3) testing results by the Kolmogorov-Smirnov method. Illustrative examples are given.

The correlation function of Gaussian noise passed through nonlinear devices

Abstract: This paper is concerned with the output autocorrelation function R^{y} of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes in R^{y} when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governing R^{y} is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devices y = x^{m} with m either an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculating R^{y} .

Comparison of linearly and quadratically modified spectral estimates of Gaussian signals

Abstract: Although the use of quadratically modified periodograms as spectral estimators for the Gaussian case is well established in the literature, the linearly modified form has grown in popular use even when the data involved can be reasonably presumed to be Gaussian. The mean and variance of the two forms of estimators are calculated as well as the equivalent duration of the data window. It is shown that variance is always greater or the equivalent duration of the sample data length is always less for the linear case when the results are normalized with respect to the effective bandwidth. In addition, a generalized form of the Hanning window is considered.

Negative binomial processes

Abstract: This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk , and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

Error bounds for the white Gaussian and other very noisy memoryless channels with generalized decision regions

Abstract: For a class of generalized decision strategies, which afford the possibility of erasure or variable-size list decoding, asymptotically tight upper and lower error bounds are obtained for orthogonal signals in additive white Gaussian noise channels. Under the hypothesis that a unique signal set is asymptotically optimal for the entire class of strategies, these bounds are shown to hold for the optimal set in both the white Gaussian channel and the class of input-discrete very noisy memoryless channels.

On the deflection of a quadratic-linear test statistic

Abstract: The deflection of a bounded quadratic-linear test statistic is considered for the following binary detection problem. Hypothesis H_{1} --received waveform is a sample function from a random process with known covariance and mean functions, but unknown probability distributions, versus H_{2} --received waveform is a sample function from a Gaussian process (noise) having known covariance and mean functions. Sample functions are assumed to belong to a real and separable Hilbert space. The test statistic is assumed to be the sum of a bounded quadratic operation and a bounded linear operation on the data. Necessary and sufficient conditions for the deflection to be bounded over all non-null bounded quadratic-linear operations are given, and additional results are obtained under the assumption that the deflection is bounded. Several relations are shown to exist between the deflection problem and the optimum discrimination problem when both processes are Gaussian. In particular, it is shown that nonsingular discrimination occurs if and only if a generalized deflection is bounded, and that in some cases the problem of realizing the log-likelihood ratio is equivalent to the problem of attaining the least upper bound for the deflection.

A bound on the probability that a Gaussian process exceeds a given function (Corresp.)

Abstract: An upper bound is calculated for the probability that a mean-zero Gaussian random process remains above a specified signal throughout a prescribed interval of time.

On the bound of first excursion probability

Abstract: Method has been developed to improve the lower bound of the first excursion probability that can apply to the problem with either constant or time-dependent barriers. The method requires knowledge of the joint density function of the random process at two arbitrary instants.

On the extraction of pattern features from continuous measurements

Abstract: A suboptimum method of extracting features, by linear operations, from continuous data belonging to M pattern classes is presented. The set of features selected minimizes bounds on the probability of error obtained from the Bhattacharyya distance and the Hajek divergence. The random processes associated with the pattern classes are assumed to be Gaussian with different means and covariance functions. For M=2, in the two special cases in which, respectively, the means and the covariance functions are the same, both the above distance measures yield the same answer. The results obtained represent an extension of the existing results for two pattern classes with the same means and different covariance functions.

On the continuity of stationary Gaussian processes

Abstract: Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = { X ( t ), − ∞ t ∞ } with mean zero and with the covariance function p ( t ) = EX ( t + s ) X ( s ). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of e-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

Characteristics of Gaussian random processes by representations in terms of independent random variables

Abstract: Certain isospectral classes of random processes and certain linear representations of these processes are considered. It is shown that the Gaussian member of each class is the only one having the desirable property of being representable as a linear combination of statistically independent components. For example, among all strictly stationary, band-limited, white-noise processes, only the Gaussian process has mutually independent Nyquist samples. Characterizations among classes wider than isospectral classes are also obtained. For example, let x(t) have a rational spectral density with Karhunen-Loeve (K-L) coefficients \{a_{i}(T_{k})\} for the interval [- T_{k}, I_{k}] . It is argued that if the coefficients \{a_{i}(T_{k})\} are mutually independent for each of a sequence of intervals \{T_{k}\} with T_{k} \rightarrow \infty , then x(t) is Gaussian. This conclusion makes use of a more general characterization of Gaussian processes that is obtained using a characterization of the Gaussian distribution among infinitely divisible distributions. It also uses a conjecture about the behavior of the K-L representation of a known function m(t), t \in (- \infty, \infty) as T_{k} \rightarrow \infty . Finally, certain non-Gaussian processes defined as sums of a random number of random pulses are considered. Necessary and sufficient conditions for the independence of linear functionals of this process are obtained.

Optimal adaptive estimation: Structure and parameter adaptation-Part I: Linear models and continuous data

Abstract: A Bayesian approach to optimal adaptive estimation with continuous data is presented. Both structure and parameter adaptation are considered and specific recursive adaptation algorithms are derived for gaussian process models and linear dynamics. Specifically, for the class of adaptive estimation problems with linear dynamic models and gaussian excitations, a form of the "partition" theorem will be given that is applicable both for structure and parameter adaptation. The "partition" or "decomposition" theorem effects the partition of the essentially nonlinear estimation problem into two parts, a linear non-adaptive part consisting of ordinary Kalman estimators and a nonlinear part that incorporates the adaptive or learning nature of the adaptive estimator. In addition, a simple performance measure is introduced for the on-line performance evaluation of the adaptive estimator. The on-line performance measure utilizes quantities available from the adaptive estimator and hence a minimum of additional computational effort is required for evaluation.

Digital simulation of multidimensional Gauss-Markov random processes

Abstract: A general technique is presented to simulate on a digital computer a multidimensional stationary or nonstationary Gauss-Markov random process of specified autocorrelation function. The choice of a suitable time step is also discussed. The results are of interest in the simulation studies of systems which are forced by random inputs, e.g, control and communication systems.

Phase Free Estimation of Coherence

Abstract: A phase free estimate of the coherence of a bivariate Gaussian process is presented. The technique is based on the usual independent, complex normal approximation to the distribution of the finite Fourier transform of a multivariate stationary time series, and the complex Wishart approximation to the distribution of spectrum estimates. If the spectral densities and coherence can be assumed to be constant over a wider frequency band than the phase can be assumed to be constant, the concept of inner and outer spectral windows would seem appropriate. Maximum likelihood estimates of the coherence are obtained using phase free marginal distributions at the inner window level. The results of simulations are presented showing the likelihood for various inner windows.

Earthquake Response Statistics of Nonlinear Systems

Abstract: A stationary, Gaussian random process, having a prescribed nonuniform power spectral density function determined by a special simulation procedure which realizes the local site geology is used to analyze the response statistics of nonlinear yield type structures subjected to earthquake excitations. The structures investigated are single-mode oscillators covering long and short periods, high and low viscous dampings, and having history-dependent and -independent interanal energy dissipation mechanisms as represented respectively by stiffness degrading and ordinary elasto-plastic models. Emphasis of the analysis is placed on the probability distribution of the single highest displacement (SHD) response of structures and the accumulation of damages caused by consecutive earthquake loadings.