Showing papers on "Gaussian published in 1975"

Approximate non-Gaussian filtering with linear state and observation relations

Abstract: Two approaches to the non-Gaussian filtering problem are presented. The proposed filters retain the computationally attractive recursive structure of the Kalman filter and they approximate well the exact minimum variance filter in cases where either 1) the state noise is Gaussian or its variance small in comparison to the observation noise variance, or 2) the observation noise is Gaussian and the system is one step observable. In both cases, the state estimate is formed as a linear prediction corrected by a nonlinear function of past and present observations. Some simulation results are presented.

Critical properties of many-component systems

Abstract: Critical properties are discussed for systems with order parameters given by n vectors S/sub a/, each with m components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces n to an explicit parameter in an averaged partition function for the m-component model. This leads to a simple discussion of properties of the system as a function of n. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when n is a negative even integer and that n = 0 corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary m-component model which is random when n = 0 and constrained when n ..-->.. infinity. A direct derivation of the large-n limit is given and leads to a variety of exactly solvable models. Expressions for the order n/sup -1/ correction are obtained in terms of correlation functions. This expansion is validmore » at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena. (auth)« less

On the estimation and testing of spatial interaction in Gaussian lattice processes

Abstract: SUMMARY Maximum likelihood estimation for rectangular lattice autonormal schemes is discussed. Explicit results for the first-order isotropic scheme are given and compared with those obtained using a simple coding technique. The efficiency of the coding technique in testing for randomness on various types of lattice is assessed.

Application of Optimal Control Theory to Civil Engineering Structures

Abstract: Modern control theory has been successfully applied to control the motions of aerospace vehicles. An exploratory study is made herein to investigate the feasibility of applying such a theory to control the vibration of civil engineering structures under random loadings. It is assumed that random excitations to structures, such as wind loads and earthquakes, can be modeled by passing either a stationary Gaussian white noise or a nonstationary Gaussian shot noise through a filter. The performance index to be minimized consists of the covariances of both the structural responses and the control forces. Under these conditions, the optimal control law is a linear feedback control. The optimal control forces are obtained by solving a matrix Riccati equation. Applications of the optimal control to a multi-degree-of-freedom structure, under stationary wind loads and nonstationary earthquakes, are demonstrated. It is shown that significant reduction in covariances of the structural responses can be achieved by the use of an active control system.

Numerical Comparison of Five Mean Frequency Estimators

Abstract: Five techniques of estimating power spectrum mean frequency are examined. Performance is given in terms of estimate bias, accuracy, and noise immunity. Techniques examined are: 1) fast Fourier transform, 2) covariance argument approximation, 3) vector phase change, 4) scalar phase change, and 5) time derivative form of covariance. Estimator evaluation is made from numerical results obtained with a computer-simulated signal having a Gaussian spectral density which serves as the population with known parameters in the statistical analysis, and 2) real data from a pulsed Doppler radar. Both data sets consist of uniformly time-spaced digital samples of a complex signal. Absolute and relative performance of each estimator are noted, and numerical results are compared with theoretical calculations made by other investigators. Insofar as the pulsed Doppler meteorological return is represented by the signal type examined (narrow, symmetrical spectral densities), the covariance technique of mean frequency...

Non-Gaussian fluctuations in electromagnetic radiation scattered by random phase screen. I. Theory

Abstract: The statistical, spatial and temporal coherence properties of electromagnetic radiation scattered into the far field by a deep random phase screen are investigated. It is shown that significant departures from Gaussian behaviour can occur even when the phase correlation length is much smaller than the dimensions of the scattering region-a situation in which the central limit theorem might be expected to apply. Formulae are derived relating these departures to elementary properties of the scattering structure, which may therefore be determined by measurement of the fluctuations in the scattered radiation. Application of the results to scattering from very rough surfaces is discussed.

Critical indices for Dyson's asymptotically-hierarchical models

Abstract: It is known that the investigation of the critical point for models of the type of Dyson's hierarchical models is reduced to the solution of some non-linear integral equation. In our previous publication the Gaussian solution was investigated. Here we construct non-Gaussian solutions of the equation and find the expressions for critical indices connected with them. Our procedure permits us to construct meaningful e-expansions.

The Behavior of Robust Estimators on Dependent Data

Abstract: : The report studies the effect of serial dependence on the efficiency of various robust estimators of the location parameter. In order to show that the asymptotic distribution of these estimators is a normal distribution a slightly stronger mixing condition than Rosenblatt's strong mixing is introduced and it is shown that the empiric c.d.f. formed from such a process approaches a Gaussian process. In particular, first order autoregressive processes with Gaussian, Cauchy and double-exponential marginal distributions are shown to obey the conditions. The behavior of robust estimators on Gaussian processes is studied in greater detail. One general result states that for any Gaussian process with serial correlation (rho sub k) > 0 and summation (rho sub k) < infinity, the efficiency of any linear combination of the order statistics relative to the sample mean is greater than its efficiency in the case of independent observations. The same result holds for the Hodges-Lehmann estimator. These results are applied to two models of contamination and show that the estimators which have been developed to be robust against outliers are robust against dependence. (Author)

Ionospheric scintillation by a random phase screen: Spectral approach

Abstract: The theory developed by Briggs and Parkin, given in terms of an anisotropic gaussian correlation function, is extended to a spectral description specified as a continuous function of spatial wavenumber with an intrinsic outer scale as would be expected from a turbulent medium. Two spectral forms were selected for comparison: (1) a power-law variation in wavenumber with a constant three-dimensional index equal to 4, and (2) Gaussian spectral variation. The results are applied to the F-region ionosphere with an outer-scale wavenumber of 2 per km (approximately equal to the Fresnel wavenumber) for the power-law variation, and 0.2 per km for the Gaussian spectral variation. The power-law form with a small outer-scale wavenumber is consistent with recent F-region in-situ measurements, whereas the gaussian form is mathematically convenient and, hence, mostly used in the previous developments before the recent in-situ measurements. Some comparison with microwave scintillation in equatorial areas is made.

Strong scintillations in astrophysics. I - The Markov approximation, its validity and application to angular broadening

Abstract: The Markov approximation to the propagation of waves in an extended, irregular medium is discussed in an astrophysical context. A new derivation is presented which is simple and which shows that the assumption of Gaussian statistics used by previous authors is irrelevant. We discuss the relevance of the approximation and show that it may apply in many situations of interest, including interstellar scintillations of pulsar signals. The approximation does not require the assumption of weak scattering or Gaussian correlation functions. The Markov equation for the angular spectrum is particularly simple, and solutions are discussed for typical turbulence spectra. It is found that the equation for the angular spectrum is very nearly that used by previous authors, and the present discussion shows that these results are much more general than previously thought. A possible observational test for distinguishing between Gaussian and power-law interstellar density spectra is discussed.

A theory of spontaneous fluctuations in macroscopic systems

Abstract: A theory is developed for the dynamics of spontaneous fluctuations in systems which, on the average, obey nonlinear transport equations. The theory is a generalization of the Ornstein–Uhlenbeck theory of near equilibrium fluctuations (Langevin‐type theories) and yields a stochastic process which is nonstationary with a Gaussian conditional probability. The three assumptions on which the theory is based are predominately kinetic in nature and in order to apply the theory it is necessary to formulate rate equations in terms of elementary events. Examples of this are given for nonlinear chemical reactions and the complete nonlinear Boltzmann equation.

Ab initio calculation of spin-orbit coupling constants for gaussian lobe and gaussian-type wave functions

Abstract: The spin-orbit coupling constants of a series of diatomic compounds containing first row atoms have been calculated using ab initio molecular wave functions with gaussian lobe and gaussian-type basis sets. In most cases, fair agreement with the experimental values has been achieved. It is pointed out that multicentre terms have almost no effect on the spin-orbit coupling constant. Furthermore, suggestions for a one-electron approximation using effective nuclear charges are presented.

Quantization Error in Predictive Coders

Abstract: Predictive coders have been suggested for use as analog data compression devices. Exact expressions for reconstructed signal error have been rare in the literature. In fact most results reported in the literature are based on the assumption of Gaussian statistics for prediction error. Predictive coding of first-order Gaussian Markov sequences are considered in this paper. A numerical iteration technique is used to solve for the prediction error statistics expressed as an infinite series in terms of Hermite polynomials. Several interesting properties of predictive coding are thereby demonstrated. First, prediction error is in fact close to Gaussian, even for the binary quantizer. Sencond, quantizer levels may be optimized at each iteration according to the calculated density. Finally, the existence of correlation between successive quantizer outputs is shown. Using the series solutions described above, performance in terms of meansquare reconstruction error versus bit rate can be shown to parallel the theoretical rate distortion function for the first-order Markov process by about 0.6 bits/sample at low bit rates.

Gaussian basis sets suitable for accurate valence‐shell calculations using the model potential method

Abstract: Pople’s extended Gaussian‐type basis, usually termed 4‐31G, was adapted for use with the model potential method. Molecular calculations were performed successfully for N2, H2O, CH4, NH3, HCN, PH3, H2S, and ClF. Substantial savings were achieved in computing cost.

Many‐body perturbation theory applied to H2

Abstract: Diagrammatic many‐body perturbation theory is applied to the H2 molecule using a discrete basis set composed of Gaussian orbitals. Three different zero‐order potentials are tried. Corrections through third order are calculated, and higher orders are estimated. The energy obtained is accurate to about 1 kcal/mole. The technique of partial summation of certain classes of diagrams by denominator shifts is investigated. Dipole polarizabilities and transition moments are also calculated, with an accuracy of 5% or better. The use of the geometric approximation to estimate high‐order corrections is discussed.

Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. II. Application to dynamic scattering in liquid crystal

Abstract: For pt.I see ibid., vol.8, 369 (1975). The authors have studied experimentally the statistical, spatial coherence and temporal coherence properties of non-Gaussian fluctuations in light scattered by a thin layer of liquid crystal (MBBA) in its dynamic scattering state. The results are consistent with the deep phase-screen theory developed in the previous paper. With 20 V applied to the sample, experimental values of the parameters of the model are: mean square phase deviation phi 2=45.6+or-8.0 rad2, phase correlation length xi =2.63+or-0.24 mu m and phase coherence time 2.2+or-0.2s. Indications are found that, while phase fluctuations in the emergent wave-front are probably dominant, amplitude fluctuations are not entirely negligible. It is argued, however, that the effects of amplitude fluctuations on the values of the above parameters are probably quite small. Taking a broader view, the results confirm that, in many scattering experiments, detailed information concerning the scattering process can be obtained from measurements in the non-Gaussian regime.

A generalized theory of multiplicative stochastic processes using cumulant techniques

Abstract: The rules for the construction of the nth order cumulant for time−dependent, stochastic, matrices or operators which do not commute with themselves at unequal times are derived. The results are identical with van Kampen’s rules. In the Gaussian case, Kubo’s concept of a generalized Gaussian process is criticized. Under certain conditions Kubo’s idea becomes asymptotically valid, while the same conditions justify use of the author’s earlier delta function theory. A generalized density matrix equation is presented and its behavior during the approach to equilibrium is discussed. A finite correlation time, τc, does not necessarily invalidate a monotonic approach to equilibrium.

On higher-order spectra of turbulence

Abstract: Measurements of higher-order spectra of turbulent velocity fluctuations in the atmospheric boundary layer over the open ocean and land produce the interesting result that, in the wavenumber range designated originally by Kolmogorov as an inertial subrange, the functional dependence of the spectra on wavenumber is practically independent of the order of the spectrum. These results confirm the observation of Dutton & Deaven that their extension by a dimensional similarity argument of the original Kolmogorov theory to higher-order spectra was not valid. In the present work, we derive an alternative generalization of the Kolmogorov ideas for spectra of arbitrary order. The results of this generalization describe the dependence upon wavenumber of the available data quite well. We also present theoretical calculations based on a Gaussian model for the fluctuating velocity field which furnish quantitative predictions for spectra of arbitrary order that are also in good agreement with the measurements, both in functional form and in absolute value.Comparison of results based on the Gaussian model with laboratory measurements obtained in a free shear layer shows that the Gaussian theory predicts accurately all the available normalized higher-order spectra for all frequencies. When the corresponding measured higher-order moments are close to those expected for a Gaussian process, the Gaussian theory also correctly predicts the absolute magnitudes of the higher-order spectra.

Optimal allocation in sequential tests comparing the means of two Gaussian populations

Abstract: SUMMARY The invariant sequential probability ratio test used in testing for a difference between the means of two Gaussian populations is set up. The error probabilities for this test are effectively constant over a rich class of data-dependent allocation rules. The additional risk, average sample number plus (y - 1) times the expected number of observations to the inferior population, for y > 1, is introduced and the optimal allocation rule is found for the continuous-time analogue to this problem. Analytical results show this rule to be asymptotically optimal in discrete time, and simulations indicate its near optimal per- formance for the finite case. The problem of two-population hypothesis testing with data-dependent allocation of observations has been treated by several authors. Also, the applications of this decision model, especially to clinical testing, have been well documented. Recent results show that when the test is sequential and the termination rule is of the sequential probability ratio test type, the probability of correct hypothesis selection is constant, ignoring overshoot, for a rich class of data-dependent allocation rules; see Flehinger, Louis, Robbins & Singer (1972) and an as yet unpublished paper of mine. This constancy permits one to search the class for a rule which performs well with respect to some additional cost structure, one usually based on the number of observations taken on the superior and inferior populations. Flehinger & Louis (1972) and Robbins & Siegmund (1974) give simulations showing that a substantial reduction in the expected number of observations on the inferior popu- lation is possible using data-dependent allocation rules, as opposed to equal assignment, for the case of comparing two Gaussian populations with known variances. The simulation results of Flehinger & Louis (1971) show the same reduction for the exponential distri- bution. In the present paper the risk function formed from the average sample number plus (y - 1) x the expected number of observations allocated to the inferior population, for y > 1, is introduced into the Gaussian testing model. Here y is the relative cost of taking an observation from the inferior as opposed to the superior population, and varying y allows one to balance the two components of risk. In ? 2 first the Gaussian allocation and testing problem is set up and previous results are summarized. Using the above risk function, in Appendix A the optimal allocation and its risk are obtained for the continuous-time idealization, that of comparing the drifts of two Brownian motions. Back in ? 2 this optimal rule is related to the discrete testing situation.

Quality of Gaussian basis sets: Direct optimization of orbital exponents by the method of conjugate gradients

Abstract: Expressions are given for calculating the energy gradient vector in the exponent space of Gaussian basis sets and a technique to optimize orbital exponents using the method of conjugate gradients is described. The method is tested on the (9s5p) Gaussian basis space and optimum exponents are determined for the carbon atom. The analysis of the results shows that the calculated one‐electron properties converge more slowly to their optimum values than the total energy converges to its optimum value. In addition, basis sets approximating the optimum total energy very well can still be markedly improved for the prediction of one‐electron properties. For smaller basis sets, this improvement does not warrant the necessary expense.

Worst sources and robust codes for difference distortion measures

Abstract: It has long been known that for a mean-square error distortion measure the Gaussian distribution requires the largest rate of all sources of a given variance. It has also been stated that a code designed for the Gaussian source and yielding distortion d when used with a Gaussian source will yield distortion \leq d when used with any independent-letter source of the same variance. In this paper, we extend these results in two directions: a) instead of assuming that the source has a fixed variance, we fix an arbitrary moment; b) instead of mean-square error distortion measures, we consider nearly arbitrary continuous difference distortion measures. For each moment constraint, we show that there is a given distribution that has the largest rate for (nearly) any difference distortion measure and that a code designed for this source yielding distortion d yields distortion \leq d for any ergodic source satisfying the same moment constraint. Furthermore, digital encoding of the output of this encoder may yield a lower rate when this encoder is used with a source for which it was not designed. We also extend these results to the case of a random process or random field of known correlation function under a difference distortion measure.