Showing papers on "Gaussian process published in 1973"
TL;DR: It is shown that the procedure described by Hannan (1969) for the estimation of the parameters of one-dimensional autoregressive moving average processes is equivalent to a three-stage realization of one step of the NewtonRaphson procedure for the numerical maximization of the likelihood function, using the gradient and the approximate Hessian.
Abstract: SUMMARY Closed form representations of the gradients and an approximation to the Hessian are given for an asymptotic approximation to the log likelihood function of a multidimensional autoregressive moving average Gaussian process. Their use for the numerical maximization of the likelihood function is discussed. It is shown that the procedure described by Hannan (1969) for the estimation of the parameters of one-dimensional autoregressive moving average processes is equivalent to a three-stage realization of one step of the NewtonRaphson procedure for the numerical maximization of the likelihood function, using the gradient and the approximate Hessian. This makes it straightforward to extend the procedure to the multidimensional case. The use of the block Toeplitz type characteristic of the approximate Hessian is pointed out.
1,112 citations
TL;DR: In this paper, the authors consider density estimates of the usual type generated by a weight function and obtain limit theorems for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Abstract: We consider density estimates of the usual type generated by a weight function Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved
703 citations
Journal Article•
TL;DR: The paper is devoted to an investigation of the structure and properties of n-copulas and their connection with random variables.
Abstract: If G is an n-dimensional joint distribution function with 1-dimensional margins F1,...,Fn, then there exists a function C (called an \"w-copula\") from the unit »-cube to the unit interval such that G(xx, ..., xn) = C(F1(x1), ..., Fn(xn)) for all real n-tuples (xlt ..., xn). The paper is devoted to an investigation of the structure and properties of n-copulas and their connection with random variables.
642 citations
TL;DR: In this article, a survey on sample function properties of Gaussian processes with the main emphasis on boundedness and continuity, including Holder conditions locally and globally, is presented, along with a discussion of sample functions.
Abstract: This is a survey on sample function properties of Gaussian processes with the main emphasis on boundedness and continuity, including Holder conditions locally and globally. Many other sample function properties are briefly treated.
350 citations
213 citations
TL;DR: In this paper, it is shown that the main calculation in our previous work involves a property of Gaussian processes which is of independent interest, namely local nondeterminism, which is a property that is related to the one in this paper.
Abstract: 1. This work grew from a study of the conditions under which a Gaussian stochastic process has a \"smooth\" local time for almost all sample functions [l]-[4]. It is shown here that the main calculation in our previous work involves a property of Gaussian processes which is of independent interest—local nondeterminism. Let X(t\\ — oo < t < oo, be a Gaussian process with mean 0, and J an open interval on the t-axis. Suppose that
121 citations
TL;DR: In this paper, it was shown that the simulated random processes are asymptotically Gaussian processes as the number of terms, N, of sine or cosine functions approaches infinity.
82 citations
TL;DR: A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process, which hinges on expanding the functions of interest in a way that allows computation by means of recursive relations.
Abstract: A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf). The analysis hinges on expanding the functions of interest in a way that allows computation by means of recursive relations. Specifically, all the expansions are expressed in terms of sums of products of Gaussian hypergeometric functions and Laguerre polynomials. Computer results obtained on a CDC 6600 are presented. If a and b are the amplitudes of the two sine waves, normalized to the rms noise level, the expansions presented are useful up to values of a,b of about 17 dB, in double precision on the CDC 6600. A different approximation is also given for the case of very high SNR.
54 citations
TL;DR: In this article, the authors studied continuous mean continuous Gaussian random fields X(p) with an N-dimensional parameter and having a correlation function p(p, q) for which 1 p p, q is asymptotic to a regularly varying (at zero) function of the distance dis with exponent 0 < a < 2.
Abstract: In this paper we study continuous mean. zero Gaussian random fields X(p) with an N-dimensional parameter and having a correlation function p(p, q) for which 1 p(p, q) is asymptotic to a regularly varying (at zero) function of the distance dis (p, q) with exponent 0 < a < 2. For such random fields, we obtain the asymptotic tail distribution of the maximum of X(p) and an asymptotic almost sure property for X(p) as IPI N. Both results generalize ones previously given by the authors for N = 1.
52 citations
TL;DR: Using reproducing-kernel Hilbert space (RKHS) techniques, new results are obtained for an explicit formula for the minimum-variance unbiased estimate of the arrival time of a step function in white Gaussian noise and a new interpretation of the Bhattacharyya bounds on the variance of an unbiased estimates of a function of regression coefficients.
Abstract: Using reproducing-kernel Hilbert space (RKHS) techniques, we obtain new results for three different parameter estimation problems. The new results are 1) an explicit formula for the minimum-variance unbiased estimate of the arrival time of a step function in white Gaussian noise, 2) a new interpretation of the Bhattacharyya bounds on the variance of an unbiased estimate of a function of regression coefficients, and 3) a concise formula for the Cramer-Rao bound on the variance of an unbiased estimate of a parameter determining the covariance of a zero-mean Gaussian process.
51 citations
01 Jan 1973
TL;DR: In this article, a class of Gaussian processes with a multidimensional time parameter was studied and a simple computation using the conditional mean of one Gaussian variable given another determines the limiting mean.
Abstract: Publisher Summary This chapter discusses two-dimensional random fields. This chapter focuses on the obtainment of a few results for a class of Gaussian processes with a multidimensional time parameter. The results are obtained by an appropriate modification of the techniques employed in the case of a one-dimensional time parameter and are indicated in some detail in the two-dimensional case. The chapter also explores the mean and covariance properties of a Gaussian conditional distribution, and presents the equation of a covariance matrix leading to the desired covariance function. A simple computation using the conditional mean of one Gaussian variable given another determines the limiting mean.
TL;DR: In this paper, a variety of analytical approximations applicable to stationary random processes is extended to non-stationary random processes, with the aid of numerical examples, and the merits of each approximation are examined by comparing with the results of simulation.
TL;DR: In this article, the vibration of a two-degree-of-freedom elastic system due to wind loading is investigated by a Monte Carlo technique, and the response analysis is performed in time domain by numerically simulating the resulting wind forces.
Abstract: The vibration of a two-degree-of-freedom elastic system due to wind loading is investigated by a Monte Carlo technique. The response analysis is performed in time domain by numerically simulating the resulting wind forces. The fluctuating wind velocity field is idealized as a stationary Gaussian random process with mean zero. For wind loading and response analysis, both across-wind and along-wind directions are considered. The results are used to study the effect of mechanical and aerodynamic parameters of the systems and to compare the current formulation with the approximate treatment commonly used.
01 Jan 1973
TL;DR: The aim of this paper is to discuss the multiplicity of the sum of two independent Gaussian processes where x 1 ( t ) is a Wiener process and is a simple Markov process.
Abstract: The aim of this paper is to discuss the multiplicity of the sum of two independent Gaussian processes where x 1 ( t ) is a Wiener process and is a simple Markov process.
TL;DR: In this article, Markov Processes on Lattices are considered and it is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and given this form Gaussian processes of this kind are easily constructed.
Abstract: Stationary processes which are defined on the points of a square lattice and are Markovian in various senses are considered. It is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and that given this form Gaussian processes of this kind are easily constructed. Certain non-Gaussian processes satisfying the various Markovian properties are also constructed and the difference from nearestneighbour systems emphasized. It is conjectured, but not proved, that the assumption of linearity of regression also implies Gaussianity. MARKOV PROCESSES ON LATTICES; SPECTRAL THEORY
TL;DR: A number of interesting zero-one laws on path properties of Gaussian processes are derived by using a zero- one law for Gaussian process which extends a result of G. Kallianpur and N. C. Jain.
Abstract: : A number of interesting zero-one laws on path properties of Gaussian processes are derived by using a zero-one law for Gaussian processes which extends a result of G. Kallianpur and N. C. Jain. (Author)
TL;DR: This paper deals with the problem of recovering a band-limited process after it has been distorted by an instantaneous non-linearity and subsequently band- limited, and an iterative procedure for the recovery of the input is presented.
Abstract: This paper deals with the problem of recovering a band-limited process after it has been distorted by an instantaneous non-linearity and subsequently band-limited. Several uniqueness theorems for the input-output relationships are derived. In contrast with the deterministic case, no requirement that the nonlinearity be monotonic is made here. An iterative procedure for the recovery of the input is presented. Applications to two-level quantizers are considered, and a new result on the determination of a band-limited Gaussian process from its zero crossings is obtained.
TL;DR: In this article, the covariance function of a stationary Gaussian process with continuous sample functions is shown to satisfy the following conditions: (a) r(t)= 1 |í|a/7(í) + o(|t|a /Y(í)) as 1^0, where 0 < a < 2 and H varies slowly at zero, and (b) rU) = 0(l/log') as t −>°c.
Abstract: In this article the following results are established. Theorem A. Let \\X(t): 0 < t < coj be a stationary Gaussian process with continuous sample functions and E(X(t)i = 0. Suppose that the covariance function r(t) satisfies the following conditions. (a) r(t)= 1 |í|a/7(í) + o(|t|a/Y(í)) as 1^0, where 0 < a < 2 and H varies slowly at zero, and (b) rU) = 0(l/log') as t —>°c. Then for any nondecreasing positive function 4>(t) defined on La, °o) with d>(oo) = oo, P(X(t) > 4>U) i.o. for some sequence t —> ooj = 0 or 1 according r°° — 1 2 / as the integral I(4>) = J g(4>(t)) (') exp(— 4> (t)/2)dt is finite or infinite, where g(x) = l/cr~ (l/x) is a regularly varying function with exponent i/a. and 3?2(7) = l\\t\\aH(t\\ Theorem C. Let \\X : n > l| be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some y > 0, r(n) = 0(l/zi ) as n —► °°. Let ¡ lj be a nondecreasing sequence of positive numbers with lim 4>(n) = <*>; suppose that S(l/<î5(z2))exp(-ç*2(n)/2) = oc. Then lim L 'l / Z E[L]= 1 \\ 4>(k)\\.
TL;DR: In this paper, the notion of non-anticipative representation of one Gaussian process with respect to the other is defined, and the main theorem establishes the existence of such a representation under very general conditions.
Abstract: Given two equivalent Gaussian processes the notion of a non-anticipative representation of one of the processes with respect to the other is defined. The main theorem establishes the existence of such a representation under very general conditions. The result is applied to derive such representations explicitly in two important cases where one of the processes is (i) a Wiener process, and (ii) a $N$-ple Gaussian Markov process. Radon-Nikodym derivatives are also discussed.
TL;DR: The following path properties of real separable Gaussian processes with parameter set an arbitrary interval are established as discussed by the authors : at every fixed point the paths of a Gaussian process are continuous, or differentiable, with probability zero or one.
TL;DR: In this paper, an approximation to the first-order probability density function of the amplitude response of a linear system to random pulse excitation was obtained, by using a saddle point technique.
01 Jan 1973
TL;DR: In this article, a class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space undergoing random changes of direction at random times while maintaining constant speed, and an invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved.
Abstract: A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space (d > 1) undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if d > 2, and not being almost surely nonnegative if d = 1. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rose*n, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.
TL;DR: A principle of invariance is used to derive a detector with performance that is invariant (or insensitive) to system gain, or equivalently channel attenuation, and the invariance feature of the detector makes it a constant false alarm rate (CFAR) receiver.
Abstract: The detection of information-bearing Gaussian processes immersed in additive white Gaussian noise (WGN) is an important problem that arises in many signal processing applications. When the level of the WGN is unknown, classical approaches to the problem fail. In this paper a principle of invariance is used to derive a detector with performance that is invariant (or insensitive) to system gain, or equivalently channel attenuation. The detector structure can be realized and detection thresholds set without prior knowledge of the WGN level. When the observation interval is large the detector has the structure of a spectral estimator-correlator, the output of which is compared to an adaptive threshold. The invariance feature of the detector makes it a constant false alarm rate (CFAR) receiver; an ad hoc structure for suboptimal CFAR Gauss-Gauss detection is discussed as well.
TL;DR: In this article, the maximum likelihood estimates for the levels of the mean value function and the covariance function of a Gaussian random process are investigated and conditions for asymptotic stability of the estimates and physical interpretations are presented.
Abstract: Maximum-likelihood estimates for the levels of the mean value function and the covariance function of a Gaussian random process are investigated. The stability of these estimates is examined as the actual covariance function of the process deviates from the form assumed in the estimators. It is found that the time-bandwidth product for stationary processes represents an upper bound on the number of estimator terms that can be safely used when estimating with uncertainty about the process covariance function. This result is consistent with other interpretations of the time-bandwidth product and tempers the conclusion that, in principle, an infinite number of estimator terms can be used to obtain a perfect estimate of the covariance level. In practice, the estimate of the level can never be perfect, and the accuracy of the estimate depends on the observation interval. Finally, conditions are established to ensure asymptotic stability of the estimates and physical interpretations are presented.