Showing papers on "Gaussian published in 1979"
TL;DR: In this article, a data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation.
Abstract: SUMMARY In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify the usefulness of this formula for small samples. A data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation. The sensitivity of the procedure is investigated using several probability models which violate the Gaussian assumption.
1,633 citations
01 Jan 1979
TL;DR: In this paper, a stationary Gaussian sequence Xn, n = −1,0,1,... and a real function H(x) was given, where ANs were appropriate norming constants.
Abstract: SummaryLet a stationary Gaussian sequence Xn, n=... −1,0,1, ... and a real function H(x) be given. We define the sequences
$$Y_n^N = \frac{1}{{A_N }} \cdot \sum\limits_{j = \left( {n - 1} \right)N}^{nN - 1} {H\left( {X_j } \right)}$$
,n=... −1,0,1...; N=1,2, ... where ANare appropriate norming constants. We are interested in the limit behaviour as N→∞. The case when the correlation function r(n)=EX0Xn tends slowly to 0 is investigated. In this situation the norming constants A>
N tend to infinity more rapidly than the usual norming sequence A>
N=√N. Also the limit may be a non-Gaussian process. The results are generalized to the case when the parameter-set is multi-dimensional.
560 citations
01 Apr 1979
TL;DR: In this article, a theory of edge detection is presented, in which intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales.
Abstract: A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientation-dependent. Thus, intensity changes at a given scale are best detected by finding the zero values of delta 2G(x,y)*I(x,y) for image I, where G(x,y) is a two-dimensional Gaussian distribution and delta 2 is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zero-crossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination boundaries, and these all have the property that they are spatially. Because of this, the zero-crossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround delta 2G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
405 citations
TL;DR: An ab initio Hartree-Fock gradient program is described, characterized by efficiency of the gradient evaluation, and capability of handling higher angular momentum basis functions.
Abstract: An ab initio Hartree-Fock gradient program is described. It is characterized by (1) efficiency of the gradient evaluation, and (2) capability of handling higher angular momentum (d andf) basis functions. The latter are constructed from shifted Cartesian Gaussian p-type primitives. A satisfactory solution is presented for the problems connected with the neglect of small integrals in a gradient program. Methods for increasing the efficiency of the SCF procedure are discussed.
266 citations
TL;DR: A class of generalized M-estimates is proposed which has attractive mean-squared-error robustness properties towards both IO and AO type deviations from the Gaussian model.
Abstract: Outliers in time series can adversely affect both the least squares estimates and ordinary M-estimates of autoregressive parameters. Attention is focused here on obtaining robust estimates of the parameter for a first-order autoregressive time series xk The observations are y k = z k + v k, and two models are considered: Model IO, with v k ≡ 0, x k possibly non-Gaussian, and Model AO, with v k nonzero and possibly quite large a small fraction of the time, and x k Gaussian. A class of generalized M-estimates is proposed which has attractive mean-squared-error robustness properties towards both IO and AO type deviations from the Gaussian model.
249 citations
TL;DR: Algorithms for the simple class of equations considered, a generalization of the Range Kutta algorithm, to integrate numerically nonlinear stochastic differential equations (SDEs) with additive, Gaussian, white noise are extended.
Abstract: In a previous paper, a method was presented to integrate numerically nonlinear stochastic differential equations (SDEs) with additive, Gaussian, white noise. The method, a generalization of the Range Kutta algorithm, extrapolates from one point to the next applying functional evaluations at stochastically determined points. This paper extends (and at one point corrects) algorithms for the simple class of equations considered in the previous paper. In addition, the method is expanded to treat vector SDEs, equations with time-dependent functions, and SDEs higher than first order. The parameters for several explicit integration schemes are displayed.
188 citations
TL;DR: In this paper, the problem of nonlinear self focusing of Gaussian laser beams is reformulated in terms of a variational principle, and expressions are obtained for the equilibrium radii and nonlinear frequency shifts of stationary self-trapped laser beams.
Abstract: The problem of nonlinear self‐focusing of Gaussian laser beams is reformulated in terms of a variational principle. By means of approximating Gaussian functions, expressions are obtained for the equilibrium radii and nonlinear frequency shifts of stationary self‐trapped laser beams. The nonsteady propagation is given an illuminating form in terms of a potential function description. The analysis confirms the recent results obtained by moment theory as opposed to those based on paraxial ray approximations.
181 citations
TL;DR: The results show that adsorbed molecules do in fact “remember” the rigidity they possessed in solution and that the Gaussian hypothesis is well verified.
Abstract: Information on spatial correlation in the tangent direction along electron microscope images of filamentous molecule is shown to be obtainable by the analysis of statistical fluctuations in curvature, yielding an absolute measure of the persistence parameter amicro. The relationship of amicro, a local, microscopic parameter, to the persistence length introduced by Kratky and Porod is discussed. The hypotheses underlying the assumed theoretical model concern (1) the shape of the angle distribution, assumed to be Gaussian; (2) the passage from a three- to a two-dimensional situation, which is supposed to occur by deformation of the flexible chain in a manner that preserves the memory of the spatial correlation in orientation (except for the blocking of one degree of freedom); and (3) the adsorption conditions, which should meet the equilibrium requirement as closely as possible. The analytical method has been checked on computer simulated “Gaussian” molecules: the study of the simulated sample was essential in solving the problems connected with minimum statistics requirements and the effect of the reading error. Experimental images obtained for T2 DNA fragments at different ionic strengths by Kleinschmidt's adsorption technique have been analyzed by means of an automatic flying spot digitizer, the “Precision Encoder and Pattern Recognition.” The results show that adsorbed molecules do in fact “remember” the rigidity they possessed in solution and that the Gaussian hypothesis is well verified. Consequently, the slopes of log cosθ(l) or θ2(l) may be used indifferently in the estimate of amicro. The dependence of this parameter on ionic strength in the range explored shows the expected behavior.
154 citations
TL;DR: In this article, the inverse Langevin expression for the force-extension relation for the single chain of n randomly-jointed links is used to make use of the affine deformatiom of chain vector lengths.
Abstract: The theory makes use of the inverse Langevin expression for the force-extension relation for the single chain of n randomly-jointed links. An affine deformatiom of chain vector lengths is assumed on subjection of the network to pure homogeneous strain. The equations for the principal stresses are solved by numerical computation, using the Gauss quadrature method for evaluation of the relevant integrals. Numerical data are given for the whole range of principal extension ratios in biaxial strain for n = 25 and 100, and also for the particular case of simple extension for n = 16, 25, 36, 64 and 100. The results account in a semi-quantitative manner for an important feature of experimental biaxial-strain data which has only recently been observed, and which is not accounted for by previous network theories.
119 citations
TL;DR: In this paper, it was shown that nodal surfaces on which the tangential component of the electric field vector is zero can be regarded as defining mirror shapes for an open resonator supporting this Gaussian standing-wave pattern.
Abstract: Using the complex source-point method, we deduce simple approximate formulae for the six rectangular components of the electromagnetic field of a Gaussian beam. It is shown that in the standing-wave pattern formed by two oppositely directed Gaussian beams, nodal surfaces exist on which the tangential component of the electric field vector is zero. These surfaces can be regarded as defining mirror shapes for an open resonator supporting this Gaussian standing-wave pattern. These mirror shapes are nearly, but not exactly, spherical. The resonant frequencies for the fundamental transverse mode of such a resonator have been found as a function of the geometry and the axial mode number. Using a simple perturbation method the resonant frequency of an open resonator with spherical mirrors has been found. The result, though approximate is accurate to within ( kw 0 ) –6 in contrast to the commonly quoted result, which is only accurate to within ( kw 0 ) –4 , where k is the phase coefficient for TEM waves in free space and w 0 is the scale radius of the Gaussian beam at the beam waist.
95 citations
TL;DR: Stochastic simulation of hourly global radiation carried out with Auto-Regressive Moving Average and Factor Analysis techniques is found unable to describe the statistical features of time sequences.
TL;DR: In this article, the HE11 fields of weakly guiding fibers with graded refractive-index profiles are nearly identical to the fields of a step-index fiber with dimensionless frequency V¯ and radius ρ¯.
Abstract: The HE11 fields of weakly guiding fibers with graded refractive-index profiles are nearly identical to the fields of a step-index fiber with dimensionless frequency V¯ and radius ρ¯, where V¯ and ρ¯ are found by an elementary variational method. The results are remarkably accurate, with errors of a fraction of a percent at most, so that a simple, closed-form expression for the HE11 fields of graded fibers is now available. The step-fiber approximation is more widely applicable than the Gaussian field approximation which is inadequate for small V and for describing the evanescent field. However, the variational procedure also complements the Gaussian field approximation by providing analytical expressions for the spot size and propagation constant in several profiles of interest.
TL;DR: The electron momentum distribution for molecular hydrogen is calculated from an explicitly correlated Gaussian wave function corresponding to the total energy of -1.174 42 a.u. and accounting for 99.9% of the correlation energy as discussed by the authors.
Abstract: The electron momentum distribution for molecular hydrogen is calculated from an explicitly correlated Gaussian wave function corresponding to the total energy of -1.174 42 a.u. and accounting for 99.9% of the correlation energy. The high-accuracy Compton profile obtained confirms the results of recent high-energy electron-impact spectroscopy measurements of Lee, but disagrees in the region of very small momentum with earlier x-ray Compton-scattering data.
TL;DR: In this paper, a survey of the literature on observed distributions of velocities and velocity derivatives from turbulent fields with high Reynolds number is given, and the applicability of the proposed distributions and processes for modelling turbulent velocity fields is discussed.
Abstract: A new class of distributions, generalizing the Gaussian, the hyperbolic, and the so-called exponential power distributions, is introduced and studied to some extent. In particular, the possibilities are discussed of representing the distributions as mixtures of Gaussian distributions and of constructing a certain kind of stationary stochastic processes whose one dimensional distributions are of the type considered. A brief survey is given of the literature on observed distributions of velocities and velocity derivatives from turbulent fields with high Reynolds number and the applicability of the proposed distributions and processes for modelling turbulent velocity fields is discussed.
01 Jan 1979
TL;DR: For the multidimensional analogue of the Gaussian formula, the authors need at least the knowledge of lower bounds for the number of nodes and informations on the strictness of the estimates.
Abstract: One of the convenient properties of the Gaussian formulae is, that among all quadrature formulae of a fixed degree the Gaussian formula has the minimal number of nodes. For its multidimensional analogue, we need at least the knowledge of lower bounds for the number of nodes and informations on the strictness of the estimates.
TL;DR: For a general vector linear time series model, this article proved the strong consistency and asymptotic normality of parameter estimates obtained by maximizing a particular time domain approximation to a Gaussian likelihood, although they do not assume that the observations are necessarily normally distributed.
Abstract: For a general vector linear time series model we prove the strong consistency and asymptotic normality of parameter estimates obtained by maximizing a particular time domain approximation to a Gaussian likelihood, although we do not assume that the observations are necessarily normally distributed. To solve the normal equations we set up a constrained Gauss-Newton iteration and obtain the properties of the iterates when the sample size is large. In particular we show that the iterates are efficient when the iteration begins with a VN- consistent estimator. We obtain similar results to the above for a frequency domain approximation to a Gaussian likelihood. We use the asymptotic estimation theory to obtain the asymptotic distribution of several familiar test statistics for testing nonlinear equality constraints.
TL;DR: In this paper, a general method for obtaining rational approximations to formal power series is defined and studied based on approximate quadrature formulas and an application to e − t is studied and a method for Laplace transform inversion is proposed.
TL;DR: For quadratic functions with long range memory (not instantaneous) with a normalization nα, 0 <α < 1/2 where n is the sample size.
Abstract: Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been nα, with 1/2<α<1 where n is the sample size. Here examples of limit theorems are given for quadratic functions with long range memory (not instantaneous) with a normalization nα, 0<α<1/2.
TL;DR: In this paper, an accurate potential surface of 68 grid points for the H3+ system was calculated at the level of a full configuration expansion using a large basis set of 63 contracted Gaussian functions.
Abstract: An accurate potential surface of 68 grid points for the H3+ system is given. The surface was calculated at the level of a full configuration expansion using a large basis set of 63 contracted Gaussian functions. This surface should be suitable for extremely accurate predictions of the vibration–rotation spectrum of H3+.
TL;DR: In this article, the authors proposed two estimators of 0, say 0, 2, by minimizing two criteria D,(-), D,( ) respectively, which measure the nearness of a parametric family of spectral densities to a Gaussian stationary process with the true spectral density g(x).
Abstract: In fitting a certain parametric family of spectral densities f,(x) to a Gaussian stationary process with the true spectral density g(x), we propose two estimators of 0, say 0,, 2, by minimizing two criteria D,(-), D,( ) respectively, which measure the nearness of fo(x) to g(x). Then we investigate some asymptotic behavior of 0,, 2, with respect to efficiency and robustness. GAUSSIAN STATIONARY PROCESS; SPECTRAL DENSITY; PERIODOGRAM; ASYMPTOTIC EFFICIENCY; ROBUSTNESS
TL;DR: An analysis of Gaussian beam effects in LDA systems in order to quantify their influence on the Doppler signal shows that both axial and lateral frequency gradients can exist in the probe volume of improperly aligned L DA systems.
Abstract: The unique properties of Gaussian beams must be considered when designing and using laser Doppler anemometer systems. This paper presents an analysis of Gaussian beam effects in LDA systems in order to quantify their influence on the Doppler signal. The analytical results, which are verified by experiment, show that both axial and lateral frequency gradients can exist in the probe volume of improperly aligned LDA systems. The effects of lens aberrations and of optical path changes due to the insertion of planar optical elements in the beam are also considered.
01 Dec 1979
TL;DR: In this article, a class of robust smoothing and interpolator algorithms for vector Markov processes in additive non-Gaussian noise is introduced. And a continuity theorem is also presented which lends support to the intuitive notion that the conditional density in question will be nearly Gaussian in a strong sense when the additive noise is nearly GNN in a comparatively weak sense.
Abstract: A class of robust smoother and interpolator algorithms is introduced. The motivation for these smoothers and interpolators is a theorem concerning approximate conditional-mean smoothers for vector Markov processes in additive non-Gaussian noise. This theorem is the smoothing analog of Masreliez’s approximate non-Gaussian filter theorem (IEEE-Auto. Control, AC-20, 1975). The theorem presented here relies on the assumption that a certain conditional density is Gaussian, just as does Masreliez’s result. This assumption will rarely, if ever, be satisfied exactly. Thus a continuity theorem is also presented which lends support to the intuitive notion that the conditional density in question will be nearly Gaussian in a strong sense when the additive noise is nearly Gaussian in a comparatively weak sense. Approaches for implementing the robust smoothers and interpolators is discussed and an application to a real data set is presented.
TL;DR: The Wiener-Lee-Schetzen scheme of using Gaussian white noise to test a nonlinear dynamical system is extended and the deviation of the kernels obtained with the ternary truncation as compared to the Wiener kernels obtained by cross correlating with the same Gaussian as was used for the stimulus.
Abstract: The Wiener-Lee-Schetzen scheme of using Gaussian white noise to test a nonlinear dynamical system is extended in two ways 1) An arbitrary non-Ganssian white noise stationary signal can be used as the test stimulus 2) An arbitrary function of this stimulus can then be used as the analyzing function for cross correlating with the response to obtain the kernels characterizing the system Closed form expressions are given for the generalized orthogonal basis functions The generalized kernels are expanded in terms of Volterra kernels and Wiener kernels The expansion coefficients are closely related to the cumulants of the stimulus probability distribution These results are applied to the special case of a Gaussian stimulus and a three-level analysis function For this case a detailed analysis is Lade of the magnitude of the deviation of the kernels obtained with the ternary truncation as compared to the Wiener kernels obtained by cross correlating with the same Gaussian as was used for the stimulus The deviations are found to be quite small
TL;DR: It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation.
Abstract: The problem of quantizing two-dimensional Gaussian random variables is considered. It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation. Applications of the results to a transform coding scheme known as spectral phase coding are discussed.
TL;DR: In this article, the statistical properties of the amplitude and the intensity of a monochromatic speckle pattern as well as the behaviour of the spatial derivatives of these quantities are studied theoretically.
Abstract: The statistical properties of the amplitude and the intensity of a monochromatic speckle pattern as well as the behaviour of the spatial derivatives of these quantities are studied theoretically. Under the assumption of circular complex gaussian statistics general expressions are derived for the distribution density of the spatial derivative of the amplitude and the intensity of the speckle field. The spatial derivative of the real and the imaginary part of the amplitude is jointly gaussian distributed, whereas the distribution density of the spatial derivative of the intensity turns out to be of simple Laplacian form. Explicit formulas are given for speckle patterns produced by uniformly diffusing screens. Furthermore, the spatial density of level crossings of the intensity is investigated.
TL;DR: In this article, a Gaussian basis (5s, 3p, 2d) is used to calculate the CI basis superposition error, and the effect of various changes in the basis and the addition of f orbitals is investigated.
Abstract: Configuration interaction calculations, involving all single and double excitations, are used to calculate the He2 potential as a test of a method of calculating the CI basis superposition error. The method is also investigated using model calculations with a limited basis. For the main He2 calculations a gaussian basis (5s, 3p, 2d) is employed. This basis includes functions appropriate to the description of both intra and interatomic correlation. The effect of various changes in the basis and the effect of the addition of f orbitals is investigated. Qualitative agreement with earlier calculations which avoided the superposition error is obtained.
TL;DR: In this paper, the authors compared measured hurricane-generated realizations by means of chi-square goodness-of-fit measures computed from the Gram-Charlier probability distribution in which the statistical measures of skewness and the excess of kurtosis are determined from the measured hurricane generated realizations.
Abstract: Digital realizations of unidirectional nonlinear random seas correct to second order in an ocean of finite depth are simulated from three types of two-parameter theoretical spectra and are compared with measured hurricane-generated realizations by means of chi-square goodness-of-fit measures computed from the Gram-Charlier probability distribution in which the statistical measures of skewness and the excess of kurtosis are determined from the measured hurricane-generated realizations. The finite Fourier transform (FFT) algorithm is shown to be an efficient method for nonlinear simulations since the FFT coefficients are complex and, therefore, capable of retaining the nonlinear random phase interactions. The second-order nonlinear simulations demonstrate improved third-order and fourth-order statistical moments compared to the linear Gaussian simulations. Previous comparisons with measured wave forces on cylindrical pilings have demonstrated improvements in the statistics of random wave force predictions computed by digital filter methods as a result of these improved nonlinear random sea simulations.
TL;DR: The true nonlinear functional dependence of a physical parameter on the local light intensity can be obtained and an exact formula is derived for the decomposition of Gaussian spatially averaged experimental data.
Abstract: We derive an exact formula for the decomposition of Gaussian spatially averaged experimental data. The true nonlinear functional dependence of a physical parameter on the local light intensity can be obtained.
TL;DR: In this paper, a formalism has been developed so that spherical gaussian basis sets can be used for the calculation of relativistic Hartree-Fock wavefunctions for atomic systems.
TL;DR: In this article, it was shown that the partition function for ferromagnetic plane rotators in a complex external field is bounded below in modulus by its value at μ = 0.
Abstract: The partition function for ferromagnetic plane rotators in a complex external field μ, with ¦Im μ¦ ⩽ ¦Re μ ¦, is bounded below in modulus by its value at μ=0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and μ=0), the associated “Gaussian inequalities” imply that all truncated correlation functions decay at least as the two-point function.