Showing papers on "Gaussian process published in 1974"
TL;DR: The high-frequency component of the random solution of a model problem is statistically orthogonal to the Gaussian component as discussed by the authors, which is a consequence of the existence of an equilibrium range.
Abstract: The high-frequency component of the random solution of a model problem is shown to be statistically orthogonal to the Gaussian component. This is shown to be a consequence of the existence of an equilibrium range. It is concluded that random flow fields can be viewed as being approximately Gaussian only in a very special sense and, in particular, that Wiener–Hermite expansions can provide a useful description only of large-scale hydrodynamical phenomena.
100 citations
Book•
01 Jan 1974
TL;DR: The major results in this survey are Gaussian processes with equal covariance functions including nonrandom signals in gaussian noise and rate of expected weight of evidence.
Abstract: List of symbols.- Information in events and weight of evidence.- Entropy.- Singularity between two probability measures.- Expected mutual information.- Expected weight of evidence.- Divergence.- Expected mutual information, expected weight of evidence, and divergence for random processes.- Relationships between certain random processes and the singularity between probability measures.- Other expressions for expected mutual information.- Expressing expected weight of evidence for gaussian generalized processes in terms of integral operators.- Comparison between I(RT:ST), WT(HN/HS+N), and JT(N, S+N) for gaussian signals and noise.- Expected mutual information rate.- Rate of expected weight of evidence.- Gaussian processes with equal covariance functions including nonrandom signals in gaussian noise.- Summary of the major results in this survey for gaussian processes, including gaussian signals and noise.- Conclusions and areas for additional research.
74 citations
TL;DR: The algorithm calculates the exact cumulative distribution of the two-sided Kolmogorov-Smirnov statistic for samples with few observations for data sampling and discrete system simulation.
Abstract: The algorithm calculates the exact cumulative distribution of the two-sided Kolmogorov-Smirnov statistic for samples with few observations. The general problem for which the formula is needed is to assess the probability that a particular sample comes from a proposed distribution. The problem arises specifically in data sampling and in discrete system simulation. Typically, some finite number of observations are available, and some underlying distribution is being considered as characterizing the source of the observations.
69 citations
TL;DR: It is proved that the optimal controller for a discrete-time linear dynamic system with quadratic performance criterion is a linear function of the state variables when the additive random vector is a nonlinear function ofThe state and/or control variables.
62 citations
TL;DR: A proof of the strong consistency of the maximum likelihood estimate of the parameters of Gaussian random processes possessing linear autoregressive moving average or state space representations is outlined.
Abstract: We outline a proof of the strong consistency of the maximum likelihood estimate of the parameters of Gaussian random processes possessing linear autoregressive moving average or state space representations.
61 citations
TL;DR: The limit set of a sequence (2 log n)−1/2X n (t), n≧3) of Gaussian processes in C [0, 1] was shown in this paper.
Abstract: In this paper, we find the limit set of a sequence (2 log n)−1/2X n (t), n≧3) of Gaussian processes in C [0,1], where the processes X n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \), where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.
56 citations
TL;DR: In this paper, weak convergence of the empirical process (in the topology on $D^p\lbrack 0, 1 \rbrack) to an appropriate Gaussian process is established under a simple condition on the mixing constants.
Abstract: For a stationary $\phi$-mixing sequence of stochastic $p(\geqq 1)$-vectors, weak convergence of the empirical process (in the $J_1$-topology on $D^p\lbrack 0, 1 \rbrack)$ to an appropriate Gaussian process is established under a simple condition on the mixing constants $\{\phi_n\}$. Weak convergence for random number of stochastic vectors is also studied. Tail probability inequalities for Kolmogorov Smirnov statistics are provided.
47 citations
TL;DR: In this article, three types of estimator of the second spectral moment of a stationary Gaussian process are considered: integral estimator, crossing estimator and nonzero level estimator.
Abstract: Three types of estimator of the second spectral moment of a stationary Gaussian process are considered. The integral estimator is based on the integral of the squared derivative of the process, while crossing estimators make use of the number of upcrossings of zero or nonzero levels. It is shown that the zero-crossing estimator can often compete with the integral estimator in efficiency and that it can be considerably improved by the additional use of nonzero levels.
42 citations
TL;DR: In this paper, it was shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent.
Abstract: Wichura (1969) studied an invariance principle for partial sums of a multi-dimensional array of independent random variables. It is shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent. Weak convergence of generalized $U$-statistics for random sample sizes is also studied. The case of (generalized) von Mises' functional is treated briefly.
41 citations
TL;DR: The theory and procedures for the determination of an optimum pth-order nonlinear time-invariant, model of a given system is developed in this article, where the model determined is optimum in the sense that, with the input being from a Gaussian process, the mean-square difference between its output and that of the given system was the smallest possible.
Abstract: The theory and procedures for the determination of an optimum pth-order nonlinear time-invariant, model of a given system is developed in this article The model determined is optimum in the sense that, with the input being from a Gaussian process, the mean-square difference between its output and that of the given system is the smallest possible An analysis of the mean-square error also is presented In addition, a Parseval theorem for nonlinear systems is derived from which a measure of pth-order non-linearity for a given system is obtained
36 citations
TL;DR: These findings treat, in the presence of a Gaussian vector, the classical problem of “random flights” dating back to Rayleigh, and some calculations for the 2- and 3-dimensional problem are presented, and an application to coherent phase-shift-keying communications systems is discussed.
Abstract: This paper investigates the statistical properties of the sum, S, of an n-dimensional Gaussian random vector, N, plus the sum of M vectors, X 1 , …, X M , having random amplitudes and independent arbitrary orientations in n-dimensional space. We derive expressions for the probability density function (p.d.f.) and distribution function (d.f.) of S and of its length, |S|, as series expansions involving only the moments of |X i |, i = 1, …, M. In addition, we find the p.d.f. and d.f. of the projection of S onto 1-dimensional space. Our results are generalizations of the n = 2-dimensional problem of finding the statistical properties of a sum of constant-amplitude sinusoids having independent uniformly distributed phase angles plus Gaussian noise. The latter problem has been treated by Rice1 and Esposito and Wilson,2 but our results can also deal with sinusoids having random amplitudes. When n = 3, our findings treat, in the presence of a Gaussian vector, the classical problem of “random flights” dating back to Rayleigh. Some calculations for the 2- and 3-dimensional problem are presented, and an application to coherent phase-shift-keying communications systems is discussed.
TL;DR: The LQD is shown to be robust for small as well as large sample sizes, and both the large and small sample performance of a limiter-quadratic detector are evaluated.
Abstract: In this correspondence we evaluate both the large and small sample performance of a limiter-quadratic detector (LQD) for the detection of a Gaussian signal in non-Gaussian noise. The LQD is shown to be robust for small as well as large sample sizes.
TL;DR: In this paper, the influence of wave-current interactions on peak distribution of fluid force on an element of a cylinder in a random, Gaussian, zero mean, stationary deep-water gravity wave field was examined.
Abstract: When waves encounter current, characteristics of waves, wave field kinematics, and fluid force undergo changes. Evaluation of fatigue damage of marine structures in random fluid field requires knowledge of peak distribution of fluid force. This paper examines the influence of wave-current interactions on peak distribution of fluid force on an element of a cylinder in a random, Gaussian, zero mean, stationary deep-water gravity wave field under the influence of a steady current. The influence of Gaussian assumption of fluid force on its peak distribution is also studied. Results are obtained and presented graphically for a 40 mph wind for current speeds U = O (no current), U = 3 fps (concurring current), and U = –3 fps (adverse current). When current is present, comparisons are made between the cases when wave-current interactions are considered and ignored. It is shown that interactions have pronounced effects on the peak distribution of fluid force especially in adverse current. Furthermore, Gaussian assumption of fluid force is decisively unsatisfactory.
TL;DR: In this paper, it was shown that the number of high level crossings by p dependent stationary Gaussian processes has asymptotically independent Poisson distributions if the observation interval and the height of the level increase in a coordinated way.
Abstract: It is shown that the numbers of high level crossings by p dependent stationary Gaussian processes have asymptotically independent Poisson distributions if the observation interval and the height of the level increase in a coordinated way. The processes may be highly correlated, even linearly dependent.
TL;DR: Contained herin is a derivation of two figures of merit for evaluating the performance of a product correlator and an associated discriminator curve and the accuracy and probability of false match are found to take very simple forms.
Abstract: Contained herin is a derivation of two figures of merit for evaluating the performance of a product correlator and an associated discriminator curve. The general formulas are valid for both area and one-dimensional scenes. An explicit evaluation of the formulas is presented when the correlation functions are also Gaussian. In this case, the accuracy and probability of false match are found to take very simple forms.
TL;DR: In this paper, Strassen's version of the law of the iterated logarithm is extended to the two-parameter Gaussian process with covariance function R((s 1,t 1 ),(s 2,t 2 )) = min (s 1,s 2 ) min (t 1,t 2 ).
TL;DR: In this paper, the probability of error asymptotically for testing one Gaussian stochastic process against another when the mean vectors are zero and the choice between two given covariance matrices is studied.
Abstract: We study the probability of error asymptotically for testing one Gaussian stochastic process against another when the mean vectors are zero and we have the choice between two given covariance matrices. It is shown that under certain conditions the probabilities of error form asymptotically a geometric progression with a ratio that is derived. The approach employs Laplace's method of approximating integrals and does not appeal to Fourier analysis; in this sense it can be said to be elementary.
TL;DR: In this article, the peak statistics of rigid-blade flapping responses to atmospheric turbulence at high advance ratios were studied and it was shown that high-level peaks which cause significant damage to system components are most likely to occur within narrow ranges of the azimuth angle.
Abstract: Studies are made of the peak statistics of rigid-blade flapping responses to atmospheric turbulence at high advance ratios. The rotor model is characterized by a finite dimensional linear system with periodically varying parameters and with feedback controls. System inputs represent a modulated nonstationary Gaussian process. The response description includes: 1) the ratio between the total number of peaks and of zero-level up-crossings per period, 2) the ranges of lower level thresholds within which the response process could deviate from being a narrow-band process, and 3) the accuracy of the approximate formulae of peak distribution conditional on the occurrence of a peak. The conditional probability density of peak magnitude indicates that high-level peaks which cause significant damage to system components are most likely to occur within narrow ranges of the azimuth angle. The Rayleigh density law provides a satisfactory approximation to this conditional prpbability density only within such narrow azimuth ranges. A similar approximation by a widely used narrow-band formula is not satisfactory. Numerical analysis further substantiates an earlier finding that the flapping response is a quasi-coherent narrow-band process with small phase angle variance.
TL;DR: An approximately optimum signal design method that yields the minimum probability of error is proposed for on-off signaling over linear, time-varying stochastic channels.
Abstract: The problems of optimum performance and signal design for on-off signaling over linear, time-varying stochastic channels are considered. The channel is modeled as a nonstationary linear time-varying Gaussian filter with additive white Gaussian noise at its output. A new expression for the probability of error is presented and is minimized under the constraint of fixed received energy. An approximately optimum signal design method that yields the minimum probability of error is proposed.
TL;DR: In this article, the influence of wave-current interactions on various wave statistics is examined analytically, including peak magnitude, number of zero crossings and maxima, velocities of zeros, specular points, and points of inflexion of sea surface.
Abstract: The influence of wave-current interactions on various wave statistics is examined analytically. The statistics considered are those of peak magnitude, numbers of zero crossings and maxima, velocities of zeros, specular points, and points of inflexion of sea surface. The sea surface is considered both as a function of space and time. The random wave field is assumed to be unidirectional, zero mean, Gaussian, stationary and homogeneous, and in deep water. The current is considered to be steady in time, nonuniform in space, uniformly distributed in depth, and in or opposite to the direction of the waves. Numerical results are obtained and presented graphically. Results indicate that the interactions phenomena have a relatively mild influence on the maxima of sea surface but the rate of zero crossings and maxima of sea surface, treated as function of space, and the velocity of zeros, specular points, and points of inflexion, are affected by the presence of current to an appreciable extent and can be utilized effectively for current measurements by remote sensing devices.
TL;DR: In this paper, the authors derived Bayesian estimators for a system of stochastic linear differnce equations where the random disturbances U is a timely non-independent GAUssian process.
Abstract: For a system of stochastic linear differnce equations where the random disturbances U is a timely non-independent GAUssian process we derive BAYESian estimators for A, B and the unknown parameters the convariance function of U t and conditions for the consistance this estimators. Further we derive BAYEsian predictors of X(T + 1) by the observation of and the recursive form the BAYEsian estimators for time independent random disturbances U t . For the derivation of the BAYEsian estimators and predictors conjugate prior distributions are used.
TL;DR: The method is described for evaluating the ratio of total nonzeros created between Gauss-Jordan elimination (GJE) and Gaussian elimination (GE) for large random sparse matrices and an approximate constant fill-in ratio of two has been verified.
Abstract: The method is described for evaluating the ratio of total nonzeros created between Gauss-Jordan elimination (GJE) and Gaussian elimination (GE) for large random sparse matrices. It has been found that, within the lower and upper bounds of nonzero densities for the matrices, an approximate constant fill-in ratio of two has been verified. It was also found that, within those bounds, the fill-in ratio is independent of the nonzero densities and the matrices' order.
TL;DR: The performance of random reference correlators is examined in terms of sample rate and signal-to-noise ratio and a method to improve the performance is suggested that requires three-level quantization.
Abstract: The performance of random reference correlators is examined in terms of sample rate and signal-to-noise ratio. A method to improve the performance is suggested that requires three-level quantization. Comparisons are made using weakly correlated jointly Gaussian processes as inputs to the correlator and independent samples in each channel.
TL;DR: A condition on the eigenvalues of two Gaussian processes is given that is sufficient for the boundedness of the difference of their \varepsilon -entropies and it is proved that the difference between the two is bounded.
Abstract: A condition on the eigenvalues of two Gaussian processes is given that is sufficient for the boundedness of the difference of their \varepsilon -entropies. It is also proved that the difference between the \varepsilon -entropies of two strongly equivalent Gaussian processes is bounded. The relation between the \varepsilon -entropies of equivalent (or strongly equivalent) Gaussian processes is used for obtaining the asymptotic behavior (as \varepsilon goes to zero) of the \varepsilon -entropy of certain processes.
TL;DR: In this article, the general multiplicity distribution for models with only short-range correlations was approximated by a Gaussian times a power series in 1/n/n, which corresponds to an expansion of the general distribution around its single maximum.
Abstract: The general multiplicity distribution for models with only short-range correlations can be approximated by a Gaussian times a power series in 1/〈n〉, which corresponds to an expansion of the general distribution around its single maximum. This approximate distribution is compared with experimental prong cross-sections for inelastic p-p scattering and the necessity of an additional (diffractive) mechanism is investigated.
TL;DR: In this article, the Levy parameters of the inverse gaussian distribution are obtained and the asymptotic behavior at infinity of Bessel processes and the inverse Gaussian distribution corresponds to no random passage time.
Abstract: The Levy parameters of the inverse gaussian distribution are obtained. Indices for inverse gaussian processes and Bessel processes are computed and used to compare small time sample path properties of the two classes of stochastic processes. The asymptotic behavoir at infinity of inverse gaussian and Bessel processes is discussed. It is shown that the inverse gaussian distribution corresponds to no random passage time.
TL;DR: It is shown that an observer recording only the presence or absence of a Gaussian process in a window can reconstruct its correlation function.
01 Jan 1974
TL;DR: The probability density function for the amplitude of a Gaussian-shaped pulse of unknown arrival time is derived together with its mean and variance, when the time argument of the pulse is randomly distributed according to aGaussian probability distribution or a uniform distribution.
Abstract: The probability density function for the amplitude of a Gaussian-shaped pulse of unknown arrival time is derived together with its mean and variance, when the time argument of the pulse is randomly distributed according to a Gaussian probability distribution or a uniform distribution. Then using the amplitude density as a prior distribution, the marginal density function for the pulse plus stationary Gaussian noise and its attendant expression for the probability of detection are derived. Applications to the calculation of detection probabilities in the presence of pulse peak location errors are cited.