Showing papers on "Gaussian published in 1981"

Statistical Properties of the Generalized Inverse Gaussian Distribution

Abstract: 1 Introduction.- 2 Basic properties.- 2.1 Moments and cumulants.- 3 Related distributions.- 3.1 Normal approximations.- 3.2 Powers and logarithms of generalized inverse Gaussian variates.- 3.3 Products and quotients of generalized inverse Gaussian variates.- 3.4 A generalized inverse Gaussian Markov process.- 3.5 The generalized hyperbolic distribution.- 4 Maximum likelihood estimation.- 4.1 Estimation for fixed ?.- 4.2 On the asymptotic distribution of the maximum likelihood estimate for fixed ?.- 4.3 The partially maximized log-likelihood for ?, estimation of ?.- 4.4 Estimation of ? when ? and ? are fixed.- 4.5 Estimation of ? when ? and ?>0 are fixed.- 5 Inference.- 5.1 Distribution results.- 5.2 Inference about ?.- 5.3 Inference about ?.- 5.4 One-way analysis of variance.- 5.5 A regression model.- 6 The hazard function. Lifetime models..- 6.1 Description of the hazard function.- 7 Examples.- 7.1 Failures of airconditioning equipment.- 7.2 Pulses along a nerve fibre.- 7.3 Traffic data.- 7.4 Repair time data.- 7.5 Fracture toughness of MIG welds.- References.- List of symbols.

Robustness results in linear-quadratic Gaussian based multivariable control designs

Abstract: The robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria. Available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity. In particular, robustness results associated with multivariable control systems designed on the basis of linear-quadratic (LQ) and the linear-quadratic Gaussian (LQG) design methodologies are presented.

Risk-sensitive linear/quadratic/gaussian control

Abstract: The conventional linear/quadratic/Gaussian assumptions are modified in that minimisation of the expectation of cost G defined by (2) is replaced by minimisation of the criterion function (5). The scalar –θ is a measure of risk-aversion. It is shown that modified versions of certainty equivalence and the separation theorem still hold, that optimal control is still linear Markov, and state estimate generated by a version of the Kalman filter. There are also various new features, remarked upon in Sections 5 and 7. The paper generalises earlier work of Jacobson.

Alternative methods to smooth the Earth's gravity field

Abstract: Convolutions on the sphere with corresponding convolution theorems are developed for one and two dimensional functions. Some of these results are used in a study of isotropic smoothing operators or filters. Well known filters in Fourier spectral analysis, such as the rectangular, Gaussian, and Hanning filters, are adapted for data on a sphere. The low-pass filter most often used on gravity data is the rectangular (or Pellinen) filter. However, its spectrum has relatively large sidelobes; and therefore, this filter passes a considerable part of the upper end of the gravity spectrum. The spherical adaptations of the Gaussian and Hanning filters are more efficient in suppressing the high-frequency components of the gravity field since their frequency response functions are strongly field since their frequency response functions are strongly tapered at the high frequencies with no, or small, sidelobes. Formulas are given for practical implementation of these new filters.

Nonlinear system modeling based on the Wiener theory

Abstract: This paper is a tutorial of nonlinear system modeling methods which are based on the Wiener theory of nonlinear systems. The basic concepts that underlie the Wiener theory are discussed and illustrated. Various modeling methods are presented by which a non-linear system can be modeled using either white Gaussian, nonwhite Gaussian, or certain non-Gaussian inputs. The experimental error in determining the Wiener model is discussed in terms of a new concept called measurement stability. Since attempts are being made to apply these modeling methods to diverse areas of study, this paper is written to be comprehensible by nonspecialists in system theory

A response spectrum method for random vibration analysis of mdf systems

Abstract: A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.

Multiple Wiener–Itô Integrals

Abstract: Here we introduce the multiple Wiener–ito integrals with respect to a Gaussian random spectral measure and prove some important results about them.

A Gaussian expression to describe ϕ(ρz) curves for quantitative electron probe microanalysis

Abstract: It is shown that the depth distribution of X-ray production in the electron microprobe can be accurately described by a Gaussian modified at the sample surface by a transient. Assuming that the Gaussian is a consequence of the electrons undergoing a random walk in the sample and that the transient corresponds with the change from collimated to random electron trajectories enables theoretical values for the amplitudes and coefficients of the two functions to be predicted. There is excellent agreement between the predicted and the observed values. The Lenard coefficient and the electron range can both be estimated from the coefficient in the Gaussian. It is suggested that for microprobe work, Lenard's law for electron absorption be replaced by this revised law. Both the absorption and atomic number corrections of the traditional ZAF approach are replaced by the new function.

The choice of Gaussian basis sets for molecular electronic structure calculations

Abstract: The choice of Gaussian type basis sets for electronic structure calculations of molecules is discussed in detail for treatments on the SCF and Cl level. This article is organized in the following sections : I. Introduction, II. Mathematical foundation of the LCAO-MO method, III. Basis sets of first and second row atoms in SCF calculations, IV. Transition metals, V. Beyond-Hartree-Fock calculations, VI. Summary.Detailed proposals are made for the choice of basis sets at various levels of computational expense.

A systematic preparation of new contracted Gaussian‐type orbital sets. V. From Na through Ca

Abstract: Minimal compact contracted Gaussian basis sets are constructed for the atoms from Na to Ca. They give satisfactory valence shell orbital energies, although they are minimal-type basis sets. Split-type basis sets are also derived from the minimal Gaussian basis sets in order to enhance the flexibility of the basis sets for molecular calculations.

Bonding between transition metal atoms. Abinitio effective potential calculations of Cu2

Abstract: Valence Hartree–Fock calculations using an ab inito effective potential and a Gaussian basis set of triple zeta quality followed by extensive CI’s have been carried out for several states of the copper atom and for the ground state of diatomic copper. Correlation effects are determined to obtain a satisfactory agreement with experimental data for Cu2(re = 2.25 vs 2.22 A; ωe = 265 vs 266 cm−1; De = 15 550 vs 16 500 cm−1). The most striking effect is the 0.1 A shortening of the bond length induced by the d correlation energy. Important basis set superposition errors are shown to be possible (especially in all electron calculations) and the use of six Gaussian primitives for the d orbitals was required to avoid them.

Modern development of statistical methods

Abstract: Publisher Summary This chapter discusses the use of the minimum akaike information criterion estimation (MAICE) procedure and its conceptual generalization, the entropy maximization principle, in relation to the problem of stochastic system identification. The most important contribution of MAICE is the clarification of the importance of modeling. The systematic approach to the parameter and structure estimation realized by MAICE makes it almost unnecessary to spend researchers' time for the search of ad hoc procedures. The explicit use of likelihood in MAICE makes it possible to provide clear-cut answers to problems which so far have been treated rather unsystematically. A real system may have various possible choices for its model. A Gaussian AR model can approximate a stationary Gaussian process arbitrarily closely by increasing the order. As the maximum likelihood computation for this model reduces, at least approximately, to the least squares computation, this explains why the so-called least squares method can generally be useful for the identification of stochastic systems.

Detection of weak signals in non-Gaussian noise

Abstract: A locally optimum detector structure is derived for the detection of weak signals in non-Gaussian environments. Optimum performance is obtained by employing a zero-memory nonlinearity prior to the matched filter that would be optimum for detecting the signal were the noise Gaussian. The asymptotic detection performance of the locally optimum detector under non-Gaussian conditions is derived and compared with that for the corresponding detector optimized for operations in Gaussian noise. Numerical results for the asymptotic detection performance are shown for signal detection in noise environments of practical interest.

A non-gaussian model for random surfaces

Abstract: The central concern of this paper is to develop for rough (two-dimensional, metallic) surfaces a model other than the Gaussian one usually used. An analysis, via the notion of `upcrossing characteristics', of some new data on abraded stainless steel, as well as a new look at some old data, indicates the need for such a model. The model adopted is of a form that gives $\chi ^{2}$-type marginal height distributions for the surface. After the new model has been introduced and motivated, its properties are investigated in some detail. In particular, the properties of the surface and its profiles at local maxima are studied by examining, for example, the height distribution and the surface curvature at such points. Phenomena are observed that are notably, qualitatively, different to what happens in the Gaussian model. Although the model introduced here is motivated by problems in the study of metallic surfaces, we believe it to be useful in other areas. Consequently, those sections of the paper that investigate the properties of the model are written so as to be independent of the original motivation. The paper also reintroduces, in an applied setting, the idea of examining surfaces via their upcrossing characteristics.

Gaussian basis sets for the transition metals of the first and second series

Abstract: Gaussian basis sets consisting of, respectively, (13s, 7p, 5d) and (14s, 8p, 7d) Gaussian functions have been optimized for the transition metal atoms of the first and second series. The optimization criteria and the applicability of these atomic sets for molecular calculations are discussed.

A systematic preparation of new contracted Gaussian‐type orbital sets. VI. Ab initio calculation on molecules containing Na through Cl

Abstract: Compact contracted Gaussian basis sets introduced in the preceding article are tested for ab initio molecular calculations on molecules containing third-row atoms (Na through Cl). It is found that the effect of splitting valence orbitals is essential for these molecules and addition of polarization functions to split basis sets can yield computed geometries, spectroscopic constants, and atomization energies in close agreement with the result of near Hartree–Fock calculations.

Sampling theorem for the complex spectrogram, and Gabor's expansion in Gaussian elementary signals

Abstract: The complex spectrogram of a signal is defined as the Fourier transform of the product of the signal and the shifted and complex conjugated version of a so-called window function; it is thus a function of time and frequency, simultaneously, from which the signal can be reconstructed uniquely. It is shown that the complex spectrogram is completely determined by its values on the points of a certain time-frequency lattice. This lattice is exactly the one suggested by Gabor in 1946; it arose in connection with Gabor's suggestion to expand a signal into a discrete set of Gaussian elementary signals. Such an expansion is a special case of the more general expansion of a signal into a discrete set of properly shifted and modulated window functions. It is shown that this expansion exists. Furthermore, a set of functions is constructed, which is bi-orthonormal to the set of shifted and modulated window functions. With the help of this bi-orthonormal set of functions, the expansion coefficients can be determined easily.

Limit theorems for Fourier transforms of functionals of Gaussian sequences

Abstract: Limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence.

Thermodynamics and Statistical Analysis of Gaussian Random Fields

Abstract: Gaussian fields are considered as Gibbsian fields Thermodynamic functions are calculated for them and the variational principle is proved As an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for Gaussian Markov fields

Lower bounds on the localization errors of a moving source observed by a passive array

Abstract: The Cramer-Rao inequality is used to set absolute bounds on the accuracy of location and velocity estimates obtainable by observing the signal radiated from a moving acoustic source at an array of stationary sensors. The source radiates a zero mean Gaussian random process and the observations are made in a background of spatially incoherent Gaussian noise. Results are first obtained for the error covariance matrix of a set of parameters characterizing the time variable differential delays observed at various sensor pairs. These are then translated into bounds on the error covariance matrix of a set of parameters describing source location and track. Numerical results are presented for the specific case of a source moving in a straight line course at constant velocity.

Limit theorems for non-linear functionals of Gaussian sequences

Abstract: We prove limit theorems for sums of non-linear functionals of Gaussian sequences. In certain cases we obtain a non-Gaussian limit with a norming factor n α , 0<α<1/2. The class of functionals we are investigating is a natural enlargement of the class investigated by M. Rosenblatt in [7]. We prove our results by refining the method of the paper [3].

Separable approximation for exchange interactions in electron-molecule scattering

Abstract: We have formulated and applied a separable approximation for treating the nonlocal exchange interactions that arise in electron-molecule collision problems. A separable representation of the exchange terms in the electron-molecule interaction potential is obtained by projection onto a finite set of Cartesian Gaussian functions and is combined with a single-center expansion of the direct potential terms that accurately treats the long-range, multipolar forces. An integral-equation method is used to obtain a solution of the set of coupled equations obtained in a body-frame formulation of the collision problem. The method is illustrated by application to low-energy e/sup -/-H/sub 2/ and e/sup -/-LiH scattering in the static-exchange approximation.

Holographic filter that transforms a Gaussian into a uniform beam

Abstract: A simple method of constructing a holographic filter is described which transforms a Gaussian into a uniform beam and conserves 30% of the beam power.

Molecular wave functions in momentum space

Abstract: Momentum-space calculations exhibit two kinds of advantages over position space: First, the numerical solution of Hartree-Fock equation is feasible without expansion of the wave functions in a particular basis. Equations only exhibit one avoidable singularity even for the multicenter case. Several mathematical techniques are presented, including standard fast Fourier-transform (FFT) techniques and numerical calculation of the involved convolutions. Second, momentum representation contributes in an original way to a better understanding of several physical problems arising in quantum chemistry. The two-body density matrix involving the electronic correlation are examined in both position and momentum space. If an expansion in Gaussian functions is used, momentum space renders feasible the obtainment of a multidimensional fully correlated wave function, starting from the Hartree-Fock solution.

Non-Gaussian scattering by a random phase screen

Abstract: A theoretical investigation of non-Gaussian scattering by a smoothly varying deep random phase screen is presented. New analytical results, valid for arbitrary illuminated area, are derived for the contrast of the intensity pattern in the Fraunhofer region and the effect of two scale sizes in the screen is calculated.

First-order intensity and phase statistics of Gaussian speckle produced in the diffraction region.

Abstract: General expressions for first-order statistics of Gaussian speckle produced in the diffraction region are derived for an arbitrary profile of the illuminating beam with a plane wave front. The statistical properties of the complex amplitude, the intensity, and the phase of speckles are studied under illumination of a Gaussian beam. The joint probability density function of the speckle field characterized by an equiprobability density ellipse is investigated in some detail with an intimate relation to the correlation coefficient between the real and imaginary parts of the complex speckle amplitude. This correlation coefficient is found to affect appreciably the statistics of speckles produced especially in the near-field diffraction region when a standard deviation of the random phase variation produced by a diffuse object under illumination is relatively small and the number of independent scatterers contributing to the formation of speckles is small.

Adaptive Gaussian filtering and local frequency estimates using local curvature analysis

Abstract: This paper presents an adaptive filtering technique for smoothing noisy sampled data Due to the adaptive nature of the process, distortion of the information content is significantly reduced Each point of the smoothed output is the result of a central convolution of the noisy data with a Gaussian Gaussians of different width are used to produce each point of the smoothed output The width of each Gaussian is selected, following local curvature estimates of the data, so that the smoothed points contain a nearly constant and acceptable error resulting from the smoothing process Since each Gaussian has its half-power frequency equivalent, it is possible to infer the system of narrowest bandwidth that can be tolerated in transmitting the signal The rationale used to determine the convolving Ganssians will be developed here along with brief discussions of applications

Depth resolution factor of a static gaussian ion beam

Abstract: The dependence of the depth resolution on the contribution of a Gaussian crater shape and the finite width of a Gaussian excitation beam (and/or Gaussian acceptance function) is considered. The results show that for Auger electron spectroscopy depth profiling, the contribution of the beam shapes to the depth resolution can, in most cases, be neglected. With X-ray photoelectron spectroscopy and secondary ion mass spectroscopy depth profiling, care has to be taken to obtain a well resolved depth profile.