Showing papers on "Gaussian process published in 1972"
TL;DR: In this paper, it was shown that many important reliability measures related to stationary random motion require the knowledge of two spectral parameters which depend on the first few moments of the reduced spectral density function.
Abstract: It is shown that many important reliability measures related to stationary random motion require the knowledge of two spectral parameters which depend on the first few moments of the reduced spectral density function The first is a characteristic frequency, the second a unitless measure of the variability in the frequency content, ie, of the bandwidth or the dispersion of the spectral density about its central frequency For general stationary random processes, the spectral parameters are simply related to the mean square values of the process, its envelope, and their respective time derivatives For Gaussian processes, other statistical properties, eg, average barrier crossing rates, clump sizes, and maximum response characteristics, are importantly related to these parameters A derivation is given for the spectral moments of the stationary response of damped linear multidegree-of-freedom systems for which classical modal analysis is possible
296 citations
TL;DR: It is shown that the leading term in the variance computations is identical with the Cramer-Rao lower bound calculated using the Fisher information matrix.
Abstract: We are interested in estimating the moments of the spectral density of a comp[ex Gaussian signal process \{ q^{(1)} (t) \} when the signal process is immersed in independent additive complex Gaussian noise \{q^{(2)} (t) \} . Using vector samples Q = \{ q(t_1),\cdots ,q(t_m)\} , where q(t) = q^{(1)}(t) + q^{(2)}(t) , estimators for determining the spectral moments or parameters of the signal-process power spectrum may be constructed. These estimators depend upon estimates of the covariance function R_1 (h) of the signal process at only one value of h
eq 0 . In particular, if m = 2 , these estimators are maximum-likelihood solutions. (The explicit solution of the likelihood equations for m > 2 is still an unsolved problem.) using these solutions, asymptotic (with sample size) formulas for the means and variances of the spectral mean frequency and spectral width are derived. It is shown that the leading term in the variance computations is identical with the Cramer-Rao lower bound calculated using the Fisher information matrix. Also considered is the case Where the data set consists of N samples Of continuous data, each of finite duration. In this case asymptotic (with N ) formulas are also derived for the means and variances of the spectral mean frequency and spectral width.
267 citations
IBM1
TL;DR: It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities and it is proved that the conditional expectation is linear with exactly the same form as the Gaussian case.
Abstract: A random variable is said to have elliptical distribution if its probability density is a function of a quadratic form. This class includes the Gaussian and many other useful densities in statistics. It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities. This property is not changed under linear transformation of the random variables. It is also proved in this paper that the conditional expectation is linear with exactly the same form as the Gaussian case. Many estimation results of the Gaussian case can be readily extended. Problems of computing optimal estimation, filtering, stochastic control, and team decisions in various linear systems become tractable for this class of random processes.
140 citations
01 Jan 1972
TL;DR: In this article, the authors presented a strengthening and generalization to higher dimensions of the real variable lemma presented in [4] and obtained a criterion for the continuity of sample functions of Gaussian processes with a multidimensional time parameter.
Abstract: In this paper we shall present a strengthening and generalization to higher dimensions of the real variable lemma presented in [4]. As a consequence we shall obtain a criterion for the continuity of sample functions of Gaussian processes with a multidimensional time parameter. Remarkably enough, the difficulty ofthe arguments here is almost independent of dimensions, indeed the proofs in this paper are considerably simpler and yield stronger results than those in [4]. As in [4] our point of departure is a real variable lemma giving an a priori modulus of continuity for functions satisfying certain integral inequalities. As in [4], the basic ingredients are two functions p(u), defined in [-1, 1] and T(u), defined in (o, + oo). However here, in addition to the conditions
115 citations
TL;DR: In this paper, the main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's scheme.
Abstract: The main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's scheme [20], and consequent error estimates for weak and strong invariance principles. In particular, letting k→∞ we embed the sample df in the Gaussian process with 2-dimensional time to which it has long been known to converge weakly. We discuss previous sample df embeddings, which have yielded related results; while some of our estimates are slight improvements, the emphasis here will be on the naturality of the embedding per se (although it will be indicated why it is probably far from the final word on the subject.
99 citations
TL;DR: The structure of a stationary Gaussian process near a local maximum with a prescribed height has been explored in several papers by the present author, see [5] and as discussed by the authors, which include results for moderateu as well as foru→±∞.
Abstract: The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]–[7], which include results for moderateu as well as foru→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ R n}, with mean zero and the covariance functionr(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type. In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as $$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$ WhereA(t) andb(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with heightu fort=0; see [4] for terminology. In Section 2 we examine the process ξu(t) asu→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial int 1…,t n with random coefficients. This result is quite analogous with the one-dimensional case. In Section 3 we study the locations of the local minima of ξu(t) asu → ∞. In the non-isotropic caser(t) may have a local minimum at some pointt 0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu neart 0, and that τu-t 0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case. (Less)
89 citations
TL;DR: In this article, the authors considered separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho( t, s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or < 2.
Abstract: : The authors consider two problems for separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho(t,s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or <2. In showing the existence of such (stationary) processes for 0 = or < alpha < 2, the authors relate the magnitude of the tails of the spectral distributionsto the behavior of the covariance function at the origin. For 0 < alpha = or < 2, the authors obtain the asymptotic distribution of the maximum of X(t). This second result is used to obtain a result for X(t) as t approaches infinity similar to the 'so called' law of the iterated logarithm. (Author)
87 citations
TL;DR: In this paper, the authors consider a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t)-X(0)|2.
Abstract: Let X(t), t≥0, be a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t) - X(0)|2. Here dH(λ), for some bounded monotone H. We summarize the main results.
73 citations
TL;DR: It is shown that by working with the proper metric, any Gaussian process can be used as the innovations process and if the problem of deciding between two Gaussian processes is nonsingular, then there exists a causal and causally invertible transformation between them.
Abstract: The concept of a white Gaussian noise (WGN) innovations process has been used in a number of detection and estimation problems. However, there is fundamentally no special reason why WGN should be preferred over any other process, say, for example, an nth-order stationary autoregressive process. In this paper, we show that by working with the proper metric, any Gaussian process can be used as the innovations process. The proper metric is that of the associated reproducing kernel Hilbert space. This is not unexpected, but what is unexpected is that in this metric some basic concepts, like that of a causal operator and the distinction between a causal and a Volterra operator, have to be carefully reexamined and defined more precisely and more generally. It is shown that if the problem of deciding between two Gaussian processes is nonsingular, then there exists a causal (properly defined) and causally invertible transformation between them. Thus either process can be regarded as a generalized innovations process. As an application, it is shown that the likelihood ratio (LR) for two arbitrary Gaussian processes can, when it exists, be written in the same form as the LR for a known signal in colored Gaussian noise. This generalizes a similar result obtained earlier for white noise. The methods of Gohberg and Krein, as specialized to reproducing kernel spaces, are heavily used.
60 citations
TL;DR: In this article, the authors define the quasi-Markov property and give a complete characterization of a stationary Gaussian quasi-markov process on the whole real line, and show that it is a Markovian singular process.
TL;DR: In this article, moment approximations to the density function of the wavelength were given, i.e., the time between a randomly chosen local maximum with height u and the following minimum in a stationary Gaussian process with a given covariance function.
Abstract: We give moment approximations to the density function of the wavelength, i. e., the time between "a randomly chosen" local maximum with height u and the following minimum in a stationary Gaussian process with a given covariance function. For certain processes we give similar approximations to the distribution of the amplitude, i. e., the vertical distance between the maximum and the minimum. Numerical examples and diagrams illustrate the results.
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TL;DR: In this paper, the steady state distribution and mean squared error of a delta modulator with an ideal integrator can be computed exactly when the input signal to the modulator is a stationary Gaussian process with a rational power spectral density.
Abstract: We show how the steady-state distribution and the mean squared error of a delta modulator with an ideal integrator can be computed exactly when the input signal to the modulator is a stationary Gaussian process with a rational power spectral density. Curves are presented for the mean squared error as a function of the quantizer step size and the sampling interval for several different input spectra. The mathematical development makes use of the Markov properties of the system and involves series ex-expansions in n-dimensional Hermite functions. The key integral equation is generalized to treat the case of a realizable filter in the feedback path, but an analytic method of solving this equation has not been found.
TL;DR: In this article, the authors studied the correspondence between Gaussian processes with paths in linear function spaces and Gaussian measures on function spaces, and showed that if a Gaussian process has paths in these spaces, then it induces a Gaussian measure on them, and conversely, every Gaussian measure on these spaces is induced by a GPs with paths.
Abstract: The subject of this paper is the study of the correspondence between Gaussian processes with paths in linear function spaces and Gaussian measures on function spaces. For the function spaces $C(I), C^n\lbrack a, b\rbrack, AC\lbrack a, b\rbrack$ and $L_2(T, \mathscr{A},
u)$ it is shown that if a Gaussian process has paths in these spaces then it induces a Gaussian measure on them, and, conversely, that every Gaussian measure on these spaces is induced by a Gaussian process with paths in these spaces.
TL;DR: In this article, a 1-1 correspondence is established between Gaussian measures on Lp, 1 ≤ p < ∞, and Gaussian processes with paths in Lp.
TL;DR: Results of the comparison indicate that the method based on triangular factorization of the covariance matrix generally requires less memory space and computer time than the other two methods.
Abstract: Three methods for generating outcomes on multivariate normal random vectors with a specified variance-covariance matrix are presented. A comparison is made to determine which method requires the least computer execution time and memory space when utilizing the IBM 360/67. All methods use as a basis a standard Gaussian random number generator. Results of the comparison indicate that the method based on triangular factorization of the covariance matrix generally requires less memory space and computer time than the other two methods.
TL;DR: In this article, a bound on the accuracy of causally estimating a Gaussian process from nonlinear observations is derived, where both additive Gaussian noise and Poisson observations are included.
TL;DR: In this article, it was shown that the variance of the number of zeros of a stationary Gaussian process in a bounded interval is finite if the covariance function is integrable around the origin.
Abstract: For a real, stationary Gaussian process $X(t)$, it is well known that the mean number of zeros of $X(t)$ in a bounded interval is finite exactly when the covariance function $r(t)$ is twice differentiable. Cramer and Leadbetter have shown that the variance of the number of zeros of $X(t)$ in a bounded interval is finite if $(r"(t) - r"(0))/t$ is integrable around the origin. We show that this condition is also necessary. Applying this result, we then answer the question raised by several authors regarding the connection, if any, between the existence of the variance and the existence of continuously differentiable sample paths. We exhibit counterexamples in both directions.
01 Jan 1972
TL;DR: Theorem 1.1.1 as mentioned in this paper states that real, continuous Gaussian processes have stationary increments and the law of the iterated logarithm is a consequence of the stationary increments.
Abstract: We will be concerned with real, continuous Gaussian processes. In (A) of Theorem 1.1, a result on the growth rate of the supremum as t oo is given. The processes covered by Theorem 1.1 all have stationary increments. The law of the iterated logarithm, given as (C) below, is a consequence of (A). Theorem 1.1 will be stated and discussed in this section. The proof of this theorem and supporting propositions are given in Section 2. An analogous result for small times is given in Section 3. That the method of proof can also be successfully employed in dealing with certain Gaussian processes not possessing stationary increments is illustrated by Theorem 4.1. The results of Section 1 were announced in [5]. Let (Ye, t _ 0) be a real, separable Gaussian process with Yo = 0, E[Yj] _ 0, and set
TL;DR: In this article, an extension of Fernique's lemma to larger classes of Gaussian processes is presented, where fewer restrictions are imposed on the covariance functions, and the upper bounds are obtained for continuous Gaussian process sequences.
Abstract: Upper bounds are obtained for $|X(t)|/Q(t)$ as $t \rightarrow \infty$, where $X(t)$ is a continuous Gaussian process with $EX^2(t) \leqq Q^2(t), Q(t)$ non-decreasing. Our results are extensions of some work of Pickands (1967), Nisio (1967) and Orey (1971) to larger classes of Gaussian processes, i.e. fewer restrictions are imposed on the covariance functions. The results follow from Fernique's lemma (1964) and a recent lemma on the maximum of Gaussian sequences due to Landau, Shepp, Fernique and the author (see Marcus, Shepp (1971) for further references to this lemma).
TL;DR: In this article, the exact distribution of the sample correlation coefficient for a bivariate, equal variance, complex Gaussian distribution was derived by extending a method of Mehta and Gurland to the complex case.
Abstract: Let $u_n$ denote the sample correlation coefficient for $n$ observations from a bivariate, equal variance, complex Gaussian distribution. In this note we derive the exact distribution of $u_n$ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of $E|u_n|^k$ as $n \rightarrow \infty$ is determined via the method of steepest descent. Applicability of the results to the analysis of certain estimators of spectral parameters of stationary time series is discussed.
01 Jan 1972
01 Jan 1972
TL;DR: In this paper, the problem of estimating the unobservable process 0, by means of observations on the values of 4' = (40, 4I, * * *, (,)).
Abstract: where AI (t) and A2 (t) are Gaussian and, in general, mutually dependent random vectors; while the vectors ai(t, co) and Ai(t, co) and the matrices b(t, co) and B(t, co) are, for each t, JF, = u{co: ,0, * * * } measurable. The system (1.2) is to be solved for the initial conditions (00, '0), which are assumed to be independent of the processes A1(t) and A2(t), t = 0, 1, * . In the sequel, O will be treated as a vector with unobservable components, and c, as a vector with observable components. The statistical problems we wish to consider involve the construction of optimal (in the mean square sense) estimates of the unobservable process 0, in terms of observations on the process Xt. One can distinguish the following three basic problems of estimation, which will be called the problems of filtering, interpolation, and extrapolation. Filtering. By filtering is understood the problem of estimating the unobservable vector 0, by means of observations on the values of 4' = (40, 4I, * * *, (,). We put Ha(t) = P{Q I aYotI} (where for vectors x = (x1, ... ,Xk), y = (YI, * , Yk) the inequality x _ y is taken to mean that xi _ yi for all i = 1, , k), let m(t) = M(tI|JYW4), and
TL;DR: In this article, the joint detection and estimation of Gauss-Markov processes in white Gaussian noise is considered, and explicit Bayes' optimal recursive estimation and detection rules are derived, and the resulting optimal receiver is amenable to a causal estimator-correlator-type interpretation.
Abstract: This paper considers the joint detection and estimation of Gauss-Markov processes in white Gaussian noise, where the operations of detection and estimation are strongly coupled. Explicit Bayes' optimal recursive estimation and detection rules are derived, and it is shown that the resulting optimal receiver is amenable to a causal estimator-correlator-type interpretation.
TL;DR: In this article, it was shown that any zero-mean Gaussian measure on a separable Frechet space can be obtained as the σ extension of the canonical Gaussian cylinder measure of separable Hilbert spaces, which is applied to obtain Karhunen-Loeve-type expansions for Gaussian processes.
TL;DR: In this paper, the weak convergence of weighted empirical cumulatives based on the ranks of independent, not necessarily identically distributed, observations to a continuous Gaussian process is proved.
Abstract: The weak convergence of weighted empirical cumulatives based on the ranks of independent, not necessarily identically distributed, observations to a continuous Gaussian process is proved. The results contain a shorter proof of a central limit theorem by Dupac and Hajek (1969) Ann. Math. Statist. Analogous results are proved for signed rank processes.
20 Dec 1972
TL;DR: In this paper, the authors presented an approach for analyzing certain time series processes, such as the autoregressive, the Markov and the stationary Gaussian process, under certain conditions, which could generate a time series for air pollution data.
Abstract: : For the purposes of evaluating air quality, it is important to know the probability that maximum pollutant concentrations will exceed state standards stated for various averaging times. Extreme value theory can be used if it is reasonable to assume that observations on air pollutant concentrations are independent. Since it is well known that air pollution data is highly correlated, it is reasonable to look upon this data as a time series in which the successive observations are correlated. In the report, the authors present a new approach for analyzing certain time series processes. They show that, under certain conditions, several stochastic processes which could generate a time series for air pollutant data are associated. The processes considered are the autoregressive, the Markov and a stationary Gaussian process with a specified autocorrelation function. It is shown that the extreme value distribution provides a lower bound on the distribution function of the maxima of averages of observations generated by an associated stochastic process. (Author)
20 Oct 1972
TL;DR: It is found that the required signal-to- noise ratios are significantly greater in some cases than those predicted from a simple deflection criterion of system performance.
Abstract: : The required signla-to-noise ratios for short-term detection of a narrowband Gaussian signal process in Gaussian noise are computed for false alarm probabilities 10 sup (-n), n = 1(1)8; detection probabilities 0.5, 0.7, 0. 9, 0.99; and observation-time analysis-bandwidth products in the range of 0 to 100. No assumptions are made about Gaussian statistics of the decision variable, or about long observation intervals. It is found that the required signal-to- noise ratios are significantly greater in some cases than those predicted from a simple deflection criterion of system performance.