Showing papers on "Gaussian published in 1985"

Classification of textures using Gaussian Markov random fields

Abstract: The problem of texture classification arises in several disciplines such as remote sensing, computer vision, and image analysis. In this paper we present two feature extraction methods for the classification of textures using two-dimensional (2-D) Markov random field (MRF) models. It is assumed that the given M × M texture is generated by a Gaussian MRF model. In the first method, the least square (LS) estimates of model parameters are used as features. In the second method, using the notion of sufficient statistics, it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification. Simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem.

Convergence analysis of LMS filters with uncorrelated Gaussian data

Abstract: Statistical analysis of the least mean-squares (LMS) adaptive algorithm with uncorrelated Gaussian data is presented. Exact analytical expressions for the steady-state mean-square error (mse) and the performance degradation due to weight vector misadjustment are derived. Necessary and sufficient conditions for the convergence of the algorithm to the optimal (Wiener) solution within a finite variance are derived. It is found that the adaptive coefficient μ, which controls the rate of convergence of the algorithm, must be restricted to an interval significantly smaller than the domain commonly stated in the literature. The outcome of this paper, therefore, places fundamental limitations on the mse performance and rate of convergence of the LMS adaptive scheme.

Mixture autoregressive hidden Markov models for speech signals

Abstract: In this paper a signal modeling technique based upon finite mixture autoregressive probabilistic functions of Markov chains is developed and applied to the problem of speech recognition, particularly speaker-independent recognition of isolated digits. Two types of mixture probability densities are investigated: finite mixtures of Gaussian autoregressive densities (GAM) and nearest-neighbor partitioned finite mixtures of Gaussian autoregressive densities (PGAM). In the former (GAM), the observation density in each Markov state is simply a (stochastically constrained) weighted sum of Gaussian autoregressive densities, while in the latter (PGAM) it involves nearest-neighbor decoding which in effect, defines a set of partitions on the observation space. In this paper we discuss the signal modeling methodology and give experimental results on speaker independent recognition of isolated digits. We also discuss the potential use of the modeling technique for other applications.

Exact solutions of the wave equation with complex source locations

Abstract: New exact solutions of the homogeneous, free‐space wave equation are obtained. They originate from complex source points moving at a constant rate parallel to the real axis of propagation and, therefore, they maintain a Gaussian profile as they propagate. Finite energy pulses can be constructed from these Gaussian pulses by superposition.

CLT and other limit theorems for functionals of Gaussian processes

Abstract: Conditions for the CLT for non-linear functionals of stationary Gaussian sequences are discussed, with special references to the borderline between the CLT and the non-CLT. Examples of the non-CLT for such functionals with the norming factor $$\sqrt N $$ are given.

Periodic autoregressive‐moving average (parma) modeling with applications to water resources

Abstract: Results involving correlation properties and parameter estimation for autoregressive-moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum-likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.

The first-passage density of a continuous gaussian process to a general boundary

Abstract: Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.

A strategy for time dependent quantum mechanical calculations using a Gaussian wave packet representation of the wave function

Abstract: We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrodinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical‐like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.

Non-Gaussian reflectivity, entropy, and deconvolution

Abstract: Standard deconvolution techniques assume that the wavelet is minimum phase but generally make no assumptions about the amplitude distribution of the primary reflection coefficient sequence. For a white reflection sequence the assumption of a Gaussian distribution means that recovery of the true phase of the wavelet is impossible; however, a non‐Gaussian distribution in theory allows recovery of the phase. It is generally recognized that primary reflection coefficients typically have a non‐Gaussian amplitude distribution. Deconvolution techniques that assume whiteness but seek to exploit the non‐Gaussianity include Wiggins’ minimum entropy deconvolution (MED), Claerbout’s parsimonious deconvolution, and Gray’s variable norm deconvolution. These methods do not assume minimum phase. The deconvolution filter is defined by the maximization of a function called the objective. I examine these and other MED‐type deconvolution techniques. Maximizing the objective by setting derivatives to zero results in most case...

ARMA Representation of Random Processes

Abstract: Auto-regressive moving-average (ARMA) models of the same order for AR and MA components are used for the characterization and simulation of stationary Gaussian multivariate random processes with zero mean. The coefficient matrices of the ARMA models are determined so that the simulated process will have the prescribed correlation function matrix. To accomplish this, the two-stage least squares method is used. The ARMA representation thus established permits one, in principle, to generate sample functions of infinite length and with such a speed and computational mode that even real time generations of the sample functions can be easily achieved. The numerical example indicates that the sample functions generated by the method presented herein reproduce the prescribed correlation function matrix extremely well despite the fact that these sample functions are all very long. This is seen from the closeness between the analytically prescribed auto- and cross-correlation functions and the corresponding sample correlations computed from the generated sample functions.

New approach to the statistical properties of energy levels.

Abstract: The joint distribution of energy eigenvalues of a Hamiltonian is derived by means of the usual statistical laws of classical many-body systems. It makes a transition from the Poisson type to the Gaussian type depending on the value of a single parameter characteristic of the Hamiltonian.

Identification of linear systems with input and output noise: the Koopmans-Levin method

Abstract: The Koopmans-Levin (KL) method of parameter estimation of discrete-time linear systems with input and output noise is based on the spectral decomposition of a covariance matrix, which gives approximately maximum likelihood estimates (MLE) if the noise is white Gaussian. In the paper, three robust algorithms, namely the batch method, the sequential updating of the batch solution and the sequential square-root estimation using an information matrix, are developed, based on the singular-value decomposition of matrices. Coding of these algorithms is relatively straightforward using matrix routines available in standard program libraries. The procedures and the properties of the methods are illustrated using published examples.

The first-passage density of a continuous gaussian process

Abstract: Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.

The Role of AM-to-PM Conversion in Memoryless Nonlinear Systems

Abstract: A generalized proof is presented that AM-to-PM conversion can only degrade, never improve, the intermodulation-noise performance of memoryless nonlinear systems with random input signals having even probability density functions, and a measure of degradation is defined. It is also shown for such signals that AM-to-PM conversion causes a deterministic constant phase shift to be added to the argument of the signal component at the output but has no other effect on its phase. This class of inputs includes one or the sum of several PSK signals, as well as large ensembles that can be modeled as Gaussian noise. The latter are dealt with by using Bussgang's theorem on input-output cross correlation. In the proof, Bussgang's theorem is extended to the complex case, to include phase as well as amplitude nonlinearities, yielding a complex version of the theorem. For Gaussian inputs it is shown that the undistorted signal and the intermodulation noise at the output of such systems are uncorrelated.

Large-size Gaussian mode in unstable resonators using Gaussian mirrors

Abstract: Gaussian modes with large sections have been experimentally produced in Cassegrain resonators using Gaussian reflectivity convex couplers. The far field of the beam, which was coupled through a Gaussian coupler, was found to be free from secondary rings.

The Estimation of Higher-Order Continuous Time Autoregressive Models

Abstract: A method is presented for computing maximum likelihood, or Gaussian, estimators of the structural parameters in a continuous time system of higherorder stochastic differential equations. It is argued that it is computationally efficient in the standard case of exact observations made at equally spaced intervals. Furthermore it can be applied in situations where the observations are at unequally spaced intervals, some observations are missing and/or the endogenous variables are subject to measurement error. The method is based on a state space representation and the use of the Kalman–Bucy filter. It is shown how the Kalman-Bucy filter can be modified to deal with flows as well as stocks.

Stochastic Response of Nonlinear Dynamic Systems Based on a Non-Gaussian Closure

Abstract: The random response of nonlinear dynamic systems involving stochastic coefficients is examined. A non-Gaussian closure scheme is outlined and employed to resolve an observed contradiction of the results obtained by other techniques. The method is applied to a system possessing nonlinear damping. The stationary response is obtained by numerical integration of the closed differential equations of the moments (up to fourth order). An interesting feature of the numerical results reveals the existence of a jump in the response statistic functions. This new feature may be attributed to the fact that the non-Gaussian closure more adequately models the nonlinearity, and thus results in characteristics that are similar to those of deterministic nonlinear systems. The results are compared with solutions derived by the Gaussian closure and stochastic averaging method.

Optimal control of jump-linear gaussian systems†

Abstract: This paper investigates the problem of controlling a discrete-time linear system with jump parameters. A review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems. Dynamic programming has been applied by many researchers and it was observed that no closed-form analytical solution could be constructed because of the ‘dual’ aspects of the controller. The main contribution of the present work is an algorithm, suitable for computer implementation, for the optimal dual control. The construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite-dimensional optimization procedure to obtain the optimal control as a function of the statistics. This is achieved by first developing a suitable recursive realization of a ‘filter’ which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations. ...

Comparison of Gaussian Conditional Mean and Kriging Estimation in the Geostatistical Solution of the Inverse Problem

Abstract: Two separate applications of the geostatistical solution to the inverse problem in groundwater modeling are presented. Both applications estimate the transmissivity field for a two-dimensional model of a confined aquifer under steady flow conditions. The estimates are based on point observations of transmissivity and hydraulic head and also on a model of the aquifer which includes prescribed head boundaries, leakage, and steady state pumping. The model used to describe the spatial variability of the log-transmissivity describes large-scale fluctuations through a linear mean or drift intermediate and small-scale fluctuations through a two-parameter covariance function. The first application presented estimates the log-transmissivities using Gaussian conditional mean estimation. The second application uses an extended form of cokriging. The two methods are compared and their relative merits discussed. The extended cokriging application is applied to the Jordan Aquifer of Iowa. A comparison is also made between the conditional mean application and an analytical approach.

The estimation of parameters in nonstationary higher-order continuous-time dynamic models

Abstract: This paper is concerned with derivation of a new efficient algorithm for computing the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models and with its application in the estimation of these parameters. The algorithm completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data. It is used as the basis for an iterative procedure in which the structural parameters and the initial state vector are estimated alternately.

Large deviations for stationary Gaussian processes

Abstract: In their previous work on large deviations the authors always assumed the base process to be Markovian whereas here they consider the base process to be stationary Gaussian. Similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process. A rather explicit formula for the entropy involved is also obtained.

Rate-distortion performance of DPCM schemes for autoregressive sources

Abstract: An analysis of the rate-distortion performance of differential pulse code modulation (DPCM) schemes operating on discrete-time auto-regressive processes is presented. The approach uses an iterative algorithm for the design of the predictive quantizer subject to an entropy constraint on the output sequence. At each stage the iterative algorithm optimizes the quantizer structure, given the probability distribution of the prediction error, while simultaneously updating the distribution of the resulting prediction error. Different orthogonal expansions specifically matched to the source are used to express the prediction error density. A complete description of the algorithm, including convergence and uniqueness properties, is given. Results are presented for rate-distortion performance of the optimum DPCM scheme for first-order Gauss-Markov and Laplace-Markov sources, including comparisons with the corresponding rate-distortion bounds. Furthermore, asymptotic formulas indicating the high-rate performance of these schemes are developed for both first-order Gaussian and Laplacian autoregressive sources.

Gaussian Estimation for Correlated Binomial Data

Abstract: SUMMARY The mean/variance structure of correlated binomial data is discussed, and Whittle's (1961) Gaussian estimation is suggested as a useful general method in this context. Comparisons are made with other monrent methods in current use.

Non-Gaussianclosure techniques for stationary random vibration

Abstract: The classical method of statistical linearization when applied to a non-linear oscillator excited by stationary wide-band random excitation, can be considered as a procedure in which the unknown parameters in a Gaussian distribution are evaluated by means of moment identities derived from the dynamic equation of the oscillator. A systematic extension of this procedure is the method of non-Gaussian closure in which an increasing number of moment identities are used to evaluate additional parameters in a family of non-Gaussian response distributions. The method is described and illustrated by means of examples. Attention is given to the choice of representations of non-Gaussian distributions and to techniques for generating independent moment identities directly from the differential equation of the non-linear oscillator. Some shortcomings of the method are pointed out.

A complete characterization of minimax and maximin encoder- decoder policies for communication channels with incomplete statistical description

Abstract: The problem is considered of transmitting a sequence of independent and identically distributed Gaussian random variables over a channel whose statistical description is incomplete. The channel is modeled as one that is conditionally Gaussian, with the unknown part being controlled by a so-called "jammer" who may have access to the input to the encoder and operates under a given power constraint. By adopting a game-theoretic approach, a complete set of solutions is obtained (encoder and decoder mappings, and least-favorable distributions for the channel noise) for this statistical decision problem, under two different sets of conditions, depending on whether the encoder mapping is deterministic or stochastic. In the latter case, existence of a mixed saddle-point solution can be verified when a side channel of a specific nature is available between the transmitter and the receiver. In the former case, however, only minimax and maximin solutions can be derived.

The Shape of Doppler Spectra from Precipitation

Abstract: From October 1982 through May 1983 an extensive weather clutter registration program was executed near the Dutch coast. Coherent echo series of 2 s were obtained from a cluster of adjacent antenna pencil beams every 10 or 15 min., mainly between 16:00 and 08:30 h and on the weekends. The beam cluster was pointed toward the intensity maximum of the clutter volume. The radar operated at 5650 MHz. Spectra with 10 Hz Doppler resolution have been computed by averaging over 19 discrete Fourier transforms of overlapping and tapered subseries of 200 echo vectors. To quantify the deviation from a Gaussian shape a spectral variability is defined which is computed for every estimated spectrum. It is found that the deviation from Gaussian is considerable in about one-fourth of the spectra. A selection of "typical worst case" spectra is presented.