Showing papers on "Gaussian process published in 1985"

Some Random Series of Functions

Abstract: 1. A few tools from probability theory 2. Random series in a Banach space 3. Random series in a Hilbert space 4. Random Taylor series 5. Random Fourier series 6. A bound for random trigonometric polynomials 7. Conditions on coefficients for regularity 8. Conditions on coefficients for irregularity 9. Random point masses on the circle 10. A few geometric notions 11. Random translates and coverings 12. Gaussian variables and Gaussian series 13. Gaussian Taylor series 14. Gaussian Fourier series 15. Boundedness and continuity for Gaussian processes 16. The Brownian motion 17. Brownian images in harmonic analysis 18. Fractional Brownian images and level sets.

On the Gaussian interference channel

Abstract: The determination of the capacity region of the Gaussian interference channel remains an open problem. The channel model is described and cases that have been solved by other researchers are summarized. The optimality is established of two extreme points in the achievable region of the general Gaussian interference channel. It is proved that the class of degraded Gaussian interference channels is equivalent to the class of Z (one-sided) interference channels.

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.

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.

Realization Theory for Multivariate Stationary Gaussian Processes

Abstract: This paper collects in one place a comprehensive theory of stochastic realization for continuous-time stationary Gaussian vector processes which in various pieces has appeared in a number of our earlier papers. It begins with an abstract state space theory, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy functions. Finally, differential-equations type stochastic realizations are constructed. The theory is coordinate-free, and it accommodates infinite-dimensional representations, minimality and other systems-theoretical concepts being defined by subspace inclusion rather than by dimension. We have strived for conceptual completeness rather than generality, and the same framework can be used for other types of stochastic realization problems.

Adaptive Optical Target Detection Using Correlated Images

Abstract: A method for target detection that achieves clutter rejection by the use of multiple observations of the same target scene is developed. Multiple scene observations can be obtained by processing separate frequency bands of the same target scene or by recursively processing sequential observations in time. Optimal detection algorithms are developed, based on the assumption that the image intensity can be modeled as a variable mean spatial Gaussian process. Several fast detection algorithms are derived which make use of the fact that the covariance matrices of many optical and infrared (IR) images can be accurately approximated by diagonal matrices. These algorithms provide efficient solutions to the problem of processing multiple correlated scenes or multiple sequential imaging. Computer simulations based on actual optical and IR image data were used for checking the theoretical results. The new detection algorithms achieved performance improvement in detection signal-to-noise ratio of up to 10 dB over conventional target correlation methods.

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.

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.

Analysis of Pairwise Pivoting in Gaussian Elimination

Abstract: The method of Gaussian elimination using triangularization by elementary stabilized matrices constructed by pairwise pivoting is analyzed. It is shown that a variant of this scheme which is suitable for implementation on a paralle computer is numerically stable although the bound is larger than the one for the standard partial pivoting algorithm.

Transformation Between Continuous and Discrete Representations of Images: A Perceptual Approach

Abstract: The transformation of image signals between the continuous domain of the real world and the discrete domain of modern data processing can have a significant effect on quality and efficiency. Analysis based on perceptual rather than mathematical considerations has been carried out. A series of experiments based on the analysis has shown that substantial improvement over usual techniques can be achieved by the use of a cascade of a presharpening filter combined with Gaussian presampling and interpolation filters. The resulting ``sharpened Gaussian'' filter, although not exactly circularly symmetrical, gives a high degree of isotropy. Each element in the cascade is separable, so that computational efficiency is high. A favorable tradeoff is achieved among sharpness, smoothness, and the effects of aliasing. Subjective testing, in comparison with other commonly used filters, has shown the clear superiority of this filter.

Mathematics of Kalman-Bucy filtering

Abstract: 1. Elements of Probability Theory.- 1.1 Probability and Probability Spaces.- 1.1.1 Measurable Spaces, Measurable Mappings and Measure Spaces.- 1.1.2 Probability Spaces.- 1.2 Random Variables and "Almost Sure" Properties.- 1.2.1 Mathematical Expectations.- 1.2.2 Probability Distribution and Density Functions.- 1.2.3 Characteristic Function.- 1.2.4 Examples.- 1.3 Random Vectors.- 1.3.1 Stochastic Independence.- 1.3.2 The Gaussian N Vector and Gaussian Manifolds.- 1.4 Stochastic Processes.- 1.4.1 The Hilbert Space L2(?).- 1.4.2 Second-Order Processes.- 1.4.3 The Gaussian Process.- 1.4.4 Brownian Motion, the Wiener-Levy Process and White Noise.- 2. Calculus in Mean Square.- 2.1 Convergence in Mean Square.- 2.2 Continuity in Mean Square.- 2.3 Differentiability in Mean Square.- 2.3.1 Supplementary Exercises.- 2.4 Integration in Mean Square.- 2.4.1 Some Elementary Properties.- 2.4.2 A Condition for Existence.- 2.4.3 A Strong Condition for Existence.- 2.4.4 A Weak Condition for Existence.- 2.4.5 Supplementary Exercises.- 2.5 Mean-Square Calculus of Random N Vectors.- 2.5.1 Conditions for Existence.- 2.6 The Wiener-Levy Process.- 2.6.1 The General Wiener-Levy N Vector.- 2.6.2 Supplementary Exercises.- 2.7 Mean-Square Calculus and Gaussian Distributions.- 2.8 Mean-Square Calculus and Sample Calculus.- 2.8.1 Supplementary Exercise.- 3. The Stochastic Dynamic System.- 3.1 System Description.- 3.2 Uniqueness and Existence of m.s. Solution to (3.3).- 3.2.1 The Banach Space L2N(?).- 3.2.2 Uniqueness.- 3.2.3 The Homogeneous System.- 3.2.4 The Inhomogeneous System.- 3.2.5 Supplementary Exercises.- 3.3 A Discussion of System Representation.- 4. The Kalman-Bucy Filter.- 4.1 Some Preliminaries.- 4.1.1 Supplementary Exercise.- 4.2 Some Aspects of L2 ([a, b]).- 4.2.1 Supplementary Exercise.- 4.3 Mean-Square Integrals Continued.- 4.4 Least-Squares Approximation in Euclidean Space.- 4.4.1 Supplementary Exercises.- 4.5 A Representation of Elements of H (Z, t).- 4.5.1 Supplementary Exercises.- 4.6 The Wiener-Hopf Equation.- 4.6.1 The Integral Equation (4.106).- 4.7 Kalman-Bucy Filter and the Riccati Equation.- 4.7.1 Recursion Formula and the Riccati Equation.- 4.7.2 Supplementary Exercise.- 5. A Theorem by Liptser and Shiryayev.- 5.1 Discussion on Observation Noise.- 5.2 A Theorem of Liptser and Shiryayev.- Appendix: Solutions to Selected Exercises.- References.

New Ananlytical Approach For Long-Term Generation Expansion Planning Based on Maximum Principle And Gaussian distribution Function

Abstract: This paper presents a new approach for the optimal long-range generation planning. A completely new analytical approach for the production costing model and reliability measure is developed under the assumption of Gaussian probabilistic distribution of random load fluctuations and plant outages. The long-range generation investment problem is formulated as an optimal control problem to determine optimally the annual investment in new generation capacity. The Hamiltonian minimization is performed by using a version of the gradient projection method.

Second-order equivalence of rectangular and exponential windows in least-squares estimation of Gaussian autoregressive processes

Abstract: Exact least-squares algorithms for autoregressive signals can be made to track time-varying parameters by using sliding windows on the data. Two common choices for such windows are the exponential one and the rectangular one. In this paper, it is shown that for Gaussian autoregressive processes, the two types of windows yield approximately the same mean-square error if the windows' parameters are properly chosen.

A geometric approach to the singular filtering problem

Abstract: We consider the least-squares filtering problem for a stationary Gaussian process when the observation is not fully corrupted by white noise, the so-called "singular" case. An optimal estimator is constructed consisting of an integrating part, which is, as in the regular case, computed from a spectral factorization or an equivalent matrix problem, and a differentiating part whose parameters are computed from a single matrix equation. This improves on older results which either work under restrictive assumptions, or describe the solution only as the result of some nested algorithm.

Harmonizable stable processes on groups: spectral, ergodic and interpolation properties

Abstract: This work extends to symmetric α-stable (SαS) processes, 1<α<2, which are Fourier transforms of independently scattered random measures on locally compact Abelian groups, some of the basic results known for processes with finite second moments and for Gaussian processes. Analytic conditions for subordination of left (right) stationarily related processes and a weak law of large numbers are obtained. The main results deal with the interpolation problem. Characterization of minimal and interpolable processes on discrete groups are derived. Also formulas for the interpolator and the corresponding interpolation error are given. This yields a solution of the interpolation problem for the considered class of stable processes in this general setting.

Behaviour of χ2 processes at extrema

Abstract: We study certain aspects of the sample path behaviour of X2 processes; in particular, problems related to the behaviour of these processes at their local extrema. Emphasis is placed on behaviour that is qualitatively different to that observed for Gaussian processes, rather than on phenomena common to both classes of processes, such as previously studied (global) extremal type results. reasons for this. The first is the ubiquitous central limit theorem, which applies to stochastic processes just as it applies to simple random variables. The second, less appealing but perhaps more important, is the relative ease with which Gaussian processes can be studied analytically. In an attempt to extend the modeller's box of tricks beyond the Gaussian case, while at the same time not losing all its mathematical tractability, a considerable amount of interest has recently been shown in the so-called 'X2 process', precisely defined below. Thus, for example, Sharpe (1978) and later Lindgren (1980a,b) studied the level-crossing behaviour of these processes, and, in particular, the asymptotic behaviour of their (high-level) extremes. In Adler (1981) and Adler and Firman (1981) it was argued that these processes, as well as their random field versions, provide excellent models for the modelling of (microscopically) rough surfaces, whose height distributions are markedly skew, and whose distribu- tions (as processes) do not seem to be readily transformable to Gaussian. Furthermore, in the latter paper, a number of distributional properties of x2

Probabilistic Methods in the Mechanics of Solids and Structures

Abstract: Structural engineering makes a giant stride forward with this symposium. This book represents an international sym­ posium held in Stockholm honoring Prof. W. Weibull. It drew participants from 21 countries with a total of 55 invited papers. There were 7 scientific sessions. Most of our studies in school revolved about deterministic happenings. Random loading becomes more and more important when we equate events in our world with physical and mathematical reasoning. Probabilistic failure model and stochastic fatigue crack growth play important roles in the safety of nuclear structures, random loading due to wind turbulence, earthquake spectra, offshore structural design, and aircraft design. The initial paper summarizes Prof. Weibull's life and ac­ complishments including a mention of Weibull distribution used in fatigue studies. The initial session deals with extreme value theory (EVT). The first paper reviewed applications of extreme value theory to material behavior and indicates the importance of the accompanying statistical theory. The next paper generalizes Weibull distribution in matrix form. The following papers proclaim (a) extreme value theory as com­ patible to random field theory, (b) incomplete samples can be transformed to a mixed-exponential survival for a more com­ prehensive Weibull distribution, (c) EVT in endurance testing of ball and roller bearings, (d) use of the Slepian model identi­ fying the behavior of Gaussian noise near or between its zero crossing, (e) plastic movement of SDOF elastic plastic oscillator subjected to stationary Gaussian process excitation, and (/) generalized Hermitian polynomial used in the pertur­ bation method for nonlinear random vibrations. The final paper reports on Weibull distribution in large earthquake modelling. Session 2 reports on fatigue crack growth. Beginning with an analysis of stochastic equation models of crack growth, it continues with stochastic models of fatigue crack growth and propagation, prediction of crack growth under spectrum loading employing a cycle-by-cycle technique and fatigue life distribution in a random loading sense with interacting failures. The final paper considers a new approach to fatigue crack propagation under random loading and the use of Monte Carlo simulation in correlating micro fracture process and fatigue crack propagation. This session indicates that more experimental work is required in crack propagation to reach the same level as crack initiation. Session 3 comprises a total of 13 papers and dwells upon probabilistic failure models. The initial paper proposes a generalized probabilistic model for the fatigue life and reliability prediction of a structure containing noninteracting cracks and employs Poisson random sets. The next group of papers focuses upon (a) structural reliability treated as a single stochastic model, (Jb) fatigue life and reliability estimates of mineral pipe lines using probability theory, (c) Weibull-based equation employed in predicting the failure of a stressed brittle material, (d) a preliminary suggestion that specimens failed in proof tests may be used to estimate the parameters of the reliability function for the survivors, and (e) brittle material design using three parametric Weibull distributions. The next set of papers considers ring-in-ring tests on strength of cladded glass using the three-parameter Weibull distribution, prob­ abilistic analysis of plastic plates utilizing yield-line theory plus lower bound reliability analysis of plastic plates evaluated by inclusion-exc lusion theory. The final set of papers ques­ tions the possibility of flaws in the present probabilistic models for unidirectional fibrous composites, strength rela­ tions for cracks in unidirectiona l long fiber composites employing Weibull distribution and a stochastic approach in studying the fracture and fatigue of concrete. Session 4 delves into probabilistic fracture mechanics. The initial paper estimates the probable failure of PWR pressure vessels employing Weibull distribution and updates the wellknown Marshall A(a) and B(a) functions. The final set of papers covers probabilistic assessment of structures with weld defects, probability of fracture in main coolant pipes of PWR, and statistical modelling of shaft predictions (as measured by Charpy impact curve) in the reference temperature of pressure vessel welds. The authors suggested that U.S. Nuclear Regulatory Guide 1.99 be updated to remove excessive conservatism.

An asymptotic bound for the tail of the distribution of the maximum of a gaussian process

Abstract: Let X(t ), t E T, be a real Gaussian process with mean 0, continuous covariance function, and continuous sample paths, where T is a closed cube in 1. The main result is a new bound for the probability P (maxT X(t ) > u), for u > 0. It is obtained by an extension of the original Fernique inequality. This bound is asymptotically smaller, for u oo, than the bound that can be obtained directly from the original inequality.

Efficient methods of estimate correlation functions of Gaussian processes and their performance analysis

Abstract: New efficient methods to estimate crosscorrelation functions of Gaussian signals are studied. In these methods, the "covariance property" of the Gaussian distribution is utilized such that the correlation estimates can be computed with only additions. To evaluate the performances of the methods, exact expressions for the bias and variance of these estimators are formulated and utilized in comparing these methods with the conventional correlation estimator. As a result, we point out that these new methods can give estimates which are comparable to the conventional approach.

Viterbi algorithm detector for Bernoulli-Gaussian processes

Abstract: This paper shows how to apply the Viterbi algorithm to detect randomly located impulses which have Gaussian distributed amplitudes. Our detector can deal with cases of severely overlapping wavelets. Experimental results and comparisons to Kormylo and Mendel's [12] single-most-likely-replacement detector are provided, using synthetic data.

Estimation of noisy telegraph processes: Nonlinear filtering versus nonlinear smoothing (Corresp.)

Abstract: In the estimation problem of a two-state stationary Markov process with Gaussian white noise added, the optimal smoother is a two-filter smoother. In a special case, the performance of the optimal nonlinear filter and smoother is evaluated analytically. Some asymptotic results are also derived.

Stable non-Gaussian distributions in scientometrics

Abstract: A mathematical treatment is given for the family of scientometric laws (usually referred to as the Zipf-Pareto law) that have been described byPrice and do not conform with the usual “Gaussian” view of empirical distributions. An analysis of the Zipf-Pareto law in relationship with stable non Gaussian distributions. An analysis of the Zipf-Pareto law in relationship with stable non Gaussian distributions reveals, in particular, that the truncated Cauchy distribution asymptotically coincides with Lotka's law, the most well-known frequency form of the Zipf-Pareto law. The mathematical theory of stable non Gaussian distributions, as applied to the analysis of the Zipf-Pareto law, leads to several conclusions on the mechanism of their genesis, the specific methods of processing empirical data, etc. The use of non-Gaussian processes in scientometric models suggests that this approach may result in a general mathematical theory describing the distribution of science related variables.

Time correlation statistics of the LMS adaptive algorithm weights

Abstract: The transient and steady-state weight correlation statistics of both the real and complex LMS adaptive filters are obtained when the inputs are independent samples from real and circularly Gaussian processes, respectively. A matrix relationship is derived between the covariance matrix of the weight vector at two different times and the covariance matrix of the weights at one time. These expressions show that the weight fluctuations have the same time constants as the mean behavior of the LMS algorithm itself (i.e., the weights are correlated over the same number of iterations that it takes for the algorithm to converge to the Wiener weights for stationary inputs).

The 'boxcar' method for the analysis of non-Gaussian random signals

Abstract: The authors describe a method of analysis of random signals containing non-Gaussian features such as pulses or oscillatory bursts; the method involves detecting a pulse locating it in time and classifying it in height and then averaging the signal over a fixed time interval centred on the pulse and over many occurrences of pulses in each class. They show that the technique retains information which is lost in the usual two-point correlation function measurements: this information is that contained in the non-random phase relationships between different Fourier modes, which are necessarily entailed by the non-Gaussian distribution. They also show that in suitable cases the technique allows the direct detection of some kinds of non-linear effect. They illustrate the method with examples from fluctuation measurements in magnetically confined plasmas.

A fast maximum-likelihood estimation and detection algorithm for Bernoulli-Gaussian processes

Abstract: We derive and implement a maximum-likelihood detection and estimation algorithm based on the same channel and statistical models used by Kormylo and Mendel [1], that leads to less computations than the approach presented by Chi, Mendel and Hampson [2]. We introduce a single generalized likelihood function and we develop the Multiple-Most-Likely Replacement (MMLR) detector. This detector is computationally faster compared with the Single-Most-Likely Replacement (SMLR) detector developed by Kormylo and Mendel [3]. We demonstrate good performance of our algorithm for a synthetic data example.