Second-order Distributions Research Articles

Periodic hotspot distribution and small-scale convection in the upper mantle

Hotspots are commonly attributed to mantle plumes and to a first order, the hotspot distribution is correlated with long-wavelength geoid anomalies and with higher temperatures in the lower mantle, suggesting that the plumes are generated in the lower mantle. I show that the second-order distribution is periodic with a typical spacing of 800–1000 km. Such a spacing would be too narrow for plumes to ascend directly from the lower mantle, but is close to the thickness of the upper mantle. The alignment of convection cells in the upper mantle seems to control the periodicity. To reconcile the first-a and second-order distributions, assuming limited mass fluxes across the 670-km boundary, a lateral deflection of plumes as they breach the transition zone from below is required.

Quantum statistical nature of the chemical bond.: Part I. Theoretical framework of population analysis and application to SCF closed shell wavefunctions

It is well known that the reduced density operator of order q is the essential tool for describing the electronic distribution in many-electron extended systems such as crystalline and molecular systems. In the present work this operator was used to generate a rigorous and general model of population analysis for particles ( q=1), “pairons” ( q=2), or in a more general sense, q-ons. For all q, it is possible to define population analyses of linear and non-linear structures, with regard to the mathematical shape of the generating functional of the particular analysis. If q=1, one of the linear types of analysis is the Mulliken type which is derived from a different basis from the original derivation, and from the non-linear type the Armstrong, Perkins and Stewart type (APS ) may be derived rigorously from physical principles, thereby allowing generalization of these definitions to more sophisticated wavefunctions. We define, as a byproduct, the bond multiplicity or degree of bonding, and the active, inactive and gross or proper populations in atoms within this last type of analysis which we call the statistical population analysis (SPA), in order to make explicit its nature. Furthermore, we apply SPA to the SCF closed shell formalism for first-order ( q=1) and second-order ( q=2) distributions. The numerical results are presented.

Small- x behavior of parton distribution functions in the next-to-leading order QCD parton model

The systematic application of the QCD-based parton model to the quantitative analysis of high energy processes requires reliable parton distributions including second-order QCD evolution effects — both for consistency and for accuracy, especially in exceptional kinematic regimes such as the small- x region. We discuss qualitative expectations and present quantitative results on the behavior of second-order evolved parton distribution in this region which is particularly important for studying physical processes at future accelerators. Direct comparisons with recently available second-order parton distributions are made in order to check consistency for future studies of high energy processes. Also presented are quantitative results on the scheme-dependence of parton distributions in the next-to-loading order approximation.

On the use of Gibbs Markov chain models in the analysis of images based on second-order pairwise interactive distributions

The paper investigates parameter estimation for Markov random fields and for hidden Markov random fields, where noisy data are available. EM algorithms are described and an approximate procedure is developed based on row-by-row relaxation and analysis. Numerical illustrations are provided.

Multiplicity distributions and Bose-Einstein correlations in high-energy multiparticle production in the presence of squeezed coherent states.

We generalize existing quantum-statistical approaches to multiparticle production processes in high-energy physics by studying the effects of squeezed states on second-order Bose-Einstein correlations and multiplicity distributions. The most important and surprising result is the appearance of oscillations in the multiplicity distributions. These oscillations as well as related effects of antibunching and enhanced bunching are investigated for various mixtures of squeezed coherent and chaotic distributions. The experimental situation is briefly discussed.

Memoryless discrete-time detection of a constant signal in m-dependent noise

The problem of designing memoryless detectors for known signals in stationary m-dependent noise processes is considered. Applying the criterion of asymptotic relative efficiency, the optimal such detector is shown to be characterized by the solution to a Fredholm integral equation whose kernel depends only on the second-order probability distributions of the noise. General expressions are derived for this solution and for the asymptotic efficiency of the optimal detector relative to other memoryless detectors. To illustrate the analysis, specific results are given for the particular case where the noise process is derived by memoryless nonlinear transformation of a Gaussian process. In addition, an extension of the analytical results to the more general case of \phi -mixing noise processes is discussed.

Inability of humans to discriminate between visual textures that agree in second-order statistics-revisited.

In an earlier study by Julesz (1962) pairs of random textures were generated side-by-side using a Markov process with different third-order joint-probability distributions but identical first- and second-order distributions. Such texture pairs could not be discriminated from each other by the human visual system without scrutiny. Unfortunately, Markov processes are inherently one-dimensional while the general processes underlying visual texture discrimination are two-dimensional. Here three new methods are introduced that generate two-dimensional non-Markovian textures with different third-order but identical first- and second-order statistics. All three methods generate texture pairs that cannot be discriminated from each other. The lack of texture discrimination is the more astonishing since the individual elements that form the texture pair are clearly perceived as being very different. However, a counterexample was found that yields discrimination although the texture pair has approximately identical second-order statistics. This case can be explained by assuming that early feature extractors do some preprocessing. These new demonstrations give support to a model of texture discrimination in which the stimulus is first analyzed by local feature extractors that can detect only simple features such as dots and edges of given sizes and orientations. Then the outputs of these simple extractors are evaluated by a global processor that can compute only second- or first-order statistics (that is can compare at most two such outputs).

Application of Rice's Exceedance Statistics to Atmospheric Turbulence

Discrepancies produced by the application of Rice's exceedance statistics to atmospheric turbulence are examined. First- and second-order densities from several data sources have been measured for this purpose. Particular care was paid to each selection of turbulence that provides stationary mean and variance over the entire segment. Results show that even for a stationary segment of turbulence, the process is still highly non-Gaussian, in spite of a Gaussian appearance for its first-order distribution. Data also indicate strongly non-Gaussian second-order distributions. It is therefore concluded that even stationary atmospheric turbulence with a normal first-order distribution cannot be considered a Gaussian process, and consequently the application of Rice's exceedance statistics should be approached with caution.

On a mean-square approximation problem (Corresp.)

This correspondence is concerned with the problem of choosing an approximation for the random variable obtained by operating on a stochastic process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> with a zero-memory non-linearity followed by a linear transformation. It is desired to approximate the nonlinearity with a simpler function and two different criteria are considered for selecting this approximation, viz., the mean-square error obtained before and after the linear transformation. A sufficient condition is presented-for an approximation to simultaneously minimize both error criteria; in addition, it is shown that if the two minima coincide for every linear transformation and every nonlinearity, this condition is also necessary. The condition appears as a restriction on the class of approximating functions and is related to the second-order distributions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> . For several processes of interest, this restriction is satisfied when the approximating functions are polynomials, the most notable example being the Gaussian process.

Approximating discrete probability distributions with dependence trees

A method is presented to approximate optimally an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional discrete probability distribution by a product of second-order distributions, or the distribution of the first-order tree dependence. The problem is to find an optimum set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - 1</tex> first order dependence relationship among the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> variables. It is shown that the procedure derived in this paper yields an approximation of a minimum difference in information. It is further shown that when this procedure is applied to empirical observations from an unknown distribution of tree dependence, the procedure is the maximum-likelihood estimate of the distribution.

Elliptically symmetric distributions

Elliptically symmetric distributions are second-order distributions with probability densities whose contours of equal height are ellipses. This class includes the Gaussian and sine-wave distributions and others which can be generated from certain first-order distributions. Members of this class have several desirable features for the description of the second-order statistics of the transformation of a random signal by an instantaneous nonlinear device. In particular, they are separable in Nuttall's sense, so that the output of the device may be described in terms of equivalent gain and distortion. These distributions can also simplify the evaluation of the output autovariance because of their similarity to the Gaussian distribution. For a certain class of functions, elliptically symmetric distributions yield averages which are simply proportional to those obtained with a Gaussian distribution. Furthermore, these distributions satisfy a relation analogous to Price's theorem for Gaussian distributions. Finally, a certain subclass of these distributions can be expanded in the series representation studied by Barrett and Lampard.

Second-order forces and pressure distributions on three-dimensional bodies by reverse-flow integral method

The general second-order reverse-flow relation derived by Glarke has been applied to fusiform bodies and to combinations of fusiform bodies and quasi-cylindrical wings. It is first shown that the second-order drag of an isolated fusiform body can be expressed in terms of the firstorder forward and reverse flow over the body provided the angle of incidence is not too large. The innovation of a reverse flow which is not simply antiparallel to the forward-flow at infinity is introduced and then used to derive separate integral statements on the three rectangular components of the second-order velocity perturbation on the surface of an isolated fusiform body. Two of these integral statements are inverted to yield information on the second-order surface pressure distribution. The inversions effected are valid for unrestricted incidence. The uninclined body is covered as a special case. By considering the perturbation field of a wing-body combination as a sum of isolated body and wing fields and an interference field, the leading contribution of the interference field to the second-order drag is obtained from the reverse-flow relation. The result presumes that the second-order fields of the isolated body and wing are known in advance. The reverse flow due to a two-dimensional flat-plate wing is then used to achieve an integral for the leading second-order contribution to the lift on a wing-body combination. This integral is used to calculate the lift contribution on the conical wing-body configuration for cone half-angles dc of 5° and 10°. The numerical results are compared to those obtained by direct application of Van Dyke's second-order cone potential. The difference ranges from 4%, for [(M2 — 1)1/2 tanSc] = 0.2, to 16%, for [(M2 - 1) 1/2 tanSc] = 0.8.

On Polynomial Expansions of Second-Order Distributions

Previous article Next article On Polynomial Expansions of Second-Order DistributionsE. Wong and J. B. ThomasE. Wong and J. B. Thomashttps://doi.org/10.1137/0110038PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. F. Barrett and , D. G. Lampard, An expansion for some second-order probability distributions and its application to noise problems, IRE Transactions on Information Theory, IT-1 (1955), 10–15 10.1109/TIT.1955.1055122 CrossrefISIGoogle Scholar[2] K. S. Miller, , R. I. Bernstein and , L. E. Blumenson, Rayleigh processes, Quart. Appl. Math., 16 (1958), 137–145, See, in particular, reference to private communication from D. G. Lampard MR0094862 0082.34503 CrossrefGoogle Scholar[3] Ming Chen Wang and , G. E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys., 17 (1945), 323–342 10.1103/RevModPhys.17.323 MR0013266 0063.08172 CrossrefGoogle Scholar[4] A. A. Andronov, , L. S. Pontryagin and , A. A. Witt, On the statistical investigation of dynamic systems, J. Exp. and Th. Phys. (U. S. S. R.), 3 (1933), 165– Google Scholar[5] William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley and Sons, Inc., New York, 1957xv+461, Chapter XIV MR0088081 0077.12201 Google Scholar[6] Harald Cramér, Mathematical Methods of Statistics, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946, 248–249 MR0016588 0063.01014 Google Scholar[7] Dunham Jackson, Fourier Series and Orthogonal Polynomials, Carus Monograph Series, no. 6, Mathematical Association of America, Oberlin, Ohio, 1941, 142–148 MR0005912 0060.16910 Google Scholar[8] W. P. Elderton, Frequency Curves and Correlation, Cambridge University Press, Cambridge, England, 1938 Google Scholar[9] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J., 24 (1945), 46–156 MR0011918 0063.06487 CrossrefISIGoogle Scholar[10] E. Wong, Masters Thesis, Vector stochastic processes in problems of communication theory, Ph.D. Dissertation, Department of Electrical Engineering, Princeton University, 1959 Google Scholar[11] Samuel Karlin and , James McGregor, Classical diffusion processes and total positivity, J. Math. Anal. Appl., 1 (1960), 163–183 10.1016/0022-247X(60)90020-2 MR0121844 0101.11102 CrossrefGoogle Scholar[12] A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1955 Google Scholar[13] J. J. Bussgang, Cross-correlation function of amplitude-distorted Gaussian signals, Tech. Rep., 216, Res. Lab. of Electronics, M.I.T., Cambridge, Mass., 1952 Google Scholar[14] J. L. Brown, Jr., On a cross-correlation property for stationary random processes, IRE Trans. on Information Theory, IT-3 (1957), 28–31 10.1109/TIT.1957.1057390 CrossrefISIGoogle Scholar[15] Roy Leipnik, The effect of instantaneous nonlinear devices on cross-correlation, IRE Trans., IT-4 (1958), 73–76 10.1109/TIT.1958.1057445 MR0147321 0109.11102 CrossrefISIGoogle Scholar[16] L. A. Zadeh, Classification of nonlinear filters, IRE WESCON Conv. Rec., (1957), Google Scholar[17] David Middleton, An introduction to statistical communication theory, International Series in Pure and Applied Physics, McGraw-Hill Book Co., Inc., New York, 1960xix+1140, Chapter X MR0118561 Google Scholar[18] S. O. Rice, Distribution of the duration of fades in radio transmission: Gaussian noise model, Bell System Tech. J., 37 (1958), 581–635 MR0094269 CrossrefISIGoogle Scholar[19] D. A. Darling and , A. J. F. Siegert, A systematic approach to a class of problems in the theory of noise and other random phenomena—Part I, IRE Trans. on Information Theory, IT-3 (1957), 32–37 10.1109/TIT.1957.1057392 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Diagonal Expansion for the 2 Dirichlet Probability Density FunctionP. A. LeeSIAM Journal on Applied Mathematics, Vol. 21, No. 1 | 12 July 2006AbstractPDF (767 KB)A Note on Positive Dependence and the Structure of Bivariate DistributionsD. R. JensenSIAM Journal on Applied Mathematics, Vol. 20, No. 4 | 12 July 2006AbstractPDF (429 KB)A Diagonal Expansion in Gegenbauer Polynomials for a Class of Second-Order Probability DensitiesJ. A. McFaddenSIAM Journal on Applied Mathematics, Vol. 14, No. 6 | 1 August 2006AbstractPDF (353 KB) Volume 10, Issue 3| 1962Journal of the Society for Industrial and Applied Mathematics History Submitted:25 April 1961Accepted:24 January 1962Published online:13 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0110038Article page range:pp. 507-516ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

A criterion for the diagonal expansion of a second-order probability distribution in orthogonal polynomials

In a recent paper, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Barrett and Lampard introduced an expansion for second-order probability distributions which expresses such a distribution as a double series involving orthogonal polynomials associated with the corresponding first-order probability distributions. Several interesting consequences were derived for the class, A, consisting of all second-order distributions having a diagonal expansion of the form

An expansion for some second-order probability distributions and its application to noise problems

In this paper it is shown that, in general, second-order probability distributions may be expanded in a certain double series involving orthogonal polynomials associated with the corresponding first-order probability distributions. Attention is restricted to those second-order probability distributions which lead to a "diagonal" form for this expansion. When such distributions are joint probability distributions for samples taken from a pair of time series, some interesting results can be demonstrated. For example, it is shown that if one of the time series undergoes an amplitude distortion in a time-varying "instantaneous" nonlinear device, the covariance function after distortion is simply proportional to that before distortion. Some simple results concerning conditional expectations are given and an extension of a theorem, due to Doob, on stationary Markov processes is presented. The relation between the "diagonal" expansion used in this paper and the Mercer expansion of the kernel of a certain linear homogeneous integral equation, is pointed out and in conclusion explicit expansions are given for three specific examples.