Showing papers on "Probability density function published in 1983"

On the Joint Distribution of Wave Periods and Amplitudes in a Random Wave Field

Abstract: A theoretical probability density is derived for the joint distribution of wave periods and amplitudes which has the following properties: (1) the distribution is asymmetric, in accordance with observation; (2) it depends only on three lowest moments m 0 , m 1 , m 2 of the spectral density function. It is therefore independent of the fourth moment m 4 , which previously was used to define the spectral width (Cavanie et al . 1976). In the present model the width is defined by the lower-order parameter v = ( m 0 m 2 / m 2 1 - 1) ½ . The distribution agrees quite well with wave data taken in the North Atlantic (Chakrabarti & Cooley 1977) and with other data from the Sea of Japan (Goda 1978). Among the features predicted is that the total distri­bution of wave heights is slightly non-Rayleigh, and that the interquartile range of the conditional wave period distribution tends to zero as the wave amplitude diminishes. The analytic expressions are simpler than those derived previously, and may be useful in handling real statistical data.

Detection of non-Gaussian signals by frequency domain Kurtosis estimation

Abstract: The detection of signals by power spectrum density (PSD) estimation is a well known and often employed method. The PSD estimate is essentially a second order measure which is not sensitive to the statistical nature of the signals. For non-Gaussian signals, a frequency domain Kurtosis (FDK) estimate supplements the PSD estimate and in some cases of practical importance is superior as a detection statistic. Non-Gaussian signals occur in underwater acoustics due to multipath and frequency modulation effects. Specifically, we have considered sinusoidal and narrowband Gaussian signals which, when propagated through fading or multipath environments, are received as non-Gaussian in terms of a frequency domain Kurtosis estimate. The environment is modeled by introducing amplitude probability density distributions which are due to the fading or multipath conditions. Several distributions were considered previously which included Rayleigh and log-normal, both of which have been experimentally verified to exist in the ocean. The asymptotic probability of detection for a randomly occurring signal is derived for the PSD and FDK. A simulation comparing the probability of detection for the FDK and PSD is also included.

Unsaturated flow in spatially variable fields: 2. Application of water flow models to various fields

Abstract: The method of modeling water flow during infiltration and redistribution formulated in part 1 of this study has been applied to compute expectations and variances of a few water flow variables and of effective hydraulic properties. Two spatially variable soils with different degrees of variation have been investigated. The expectations and variances are obtained by using the statistical averaging procedure and probability density function (pdf) of saturated hydraulic conductivity Ks. The stationarity hypotheses and the requirement that the integral scale of Ks is much smaller than the length scale characterizing the field in the x, y plane have been adopted. An approximate solution which assumes the concept of a wetting front and uniform water content is used for the statistical averaging procedure. A comparison of these results with data computed by a more accurate numerical solution to Richard's equation shows that the approximate simplified models lead to a quite accurate value of the expectations and variances of the flow variables when the field is sufficiently heterogeneous. It is suggested that in spatially variable fields, stochastic modeling represents the actual flow phenomena more realistically and provides the main statistical moments (mean, variances) by using simplified flow models which can be used with confidence in applications. The field effective hydraulic properties have been defined and derived by using approximate models. It is shown that effective properties may be meaningful only under very restricted and special conditions, such as steady gravitational flow. They do not exist in the general case of infiltration-redistribution. It is concluded that the traditional deterministic approach for solving the flow equations cannot be justified for solving flow problems in spatially variable fields.

Probability density estimation from sampled data

Abstract: For broad classes of deterministic and random sampling schemes \{t_{k}\} we establish the consistency and asymptotic expressions for the bias and covariance of discrete-time estimates f_{n}(x) for the marginal probability density function f(x) of continuous-time processes X(t) . The effect of the sampling scheme and the sampling rate on the performance of the estimates is studied. The results are established for continuous-time processes X(t) satisfying various asymptotic independence-uncorrelatedness conditions.

Nonstationary probabilistic target and clutter scattering models

Abstract: A probabilistic model for nonstationary and/or nonhomogeneous clutter and target scattering is proposed and developed. The first-order probability density of the scattered power is treated as the expected value of a conditional density that is a function of random parameters. The family of gamma densities is a general solution for the density function of the intensity reflected by objects comprised of several scatterers and is selected as the conditional density. In the general case, the gamma density is a function of two parameters: the mean and the inverse of the normalized variance. Assuming various distributions for a random mean, expressions for the first-order density of the scattered power are derived and used to explain previous experimental and theoretical results. An example of detection performance for nonstationary target fluctuation based on the developed model is also presented.

Statistical properties of the asymmetric random telegraph signal, with applications to single-channel analysis

Abstract: The asymmetric random telegraph signal has unequal transition probabilities in the up and down directions. This signal, with and without filtering by a single-time-constant filter, is used in modeling ionic channels in nerve membrane. Formulas are derived for the distribution of number of jumps in an interval, probability density functions of pulse duration, autocorrelation, spectral density, Kolmogorov and Fokker-Planck-Kolmogorov equations, transition-probability functions, and the steady-state probability density function for the filtered signal. This collection of results includes both previously published formulas, here presented in most general form and in uniform notation, and some new results not previously published. These formulas are of use in analyzing records of single channels in nerve membrane.

Saddlepoint approximations for estimating equations

Abstract: SUMMARY The saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation. Two different approaches are available, one of which is shown to be equivalent to the technique of Field & Hampel (1982). Two recent formulae for the tail probability, due to Lugannani & Rice (1980) and to Robinson (1982) respectively, which are uniformly accurate over the whole range of the estimator, are compared numerically with the exact results and those computed by Field & Hampel. They are found to be of comparable accuracy while avoiding the use of numerical integration. The most accurate is that of Lugannani & Rice. Hampel (1974) introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation. It is an example of what he called 'small sample asymptotics' where high accuracy is achieved for quite small sample sizes n, even down to single figures. In the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density. The distribution function was obtained by a second numerical integration, which could then be used to renormalize both the density and the distribution function. Field & Hampel (1982) develop the technique in detail and compare its performance with that of other approximation methods. As Hampel had pointed out, his approach is closely related to the saddlepoint method of Daniels (1954) which was applied to sample means and ratios of means. Following the private communication referred to by Field & Hampel (1982, p. 31) it was realized that the first numerical integration was unnecessary and that Hampel's approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation. The purpose of the present paper is to extend the use of the saddlepoint method to estimating equations. There are two distinct ways of doing this which lead to different approximations of similar accuracy. One appears to be more convenient for approxi- mating to tail probabilities by numerical integration, and was used to compute the saddlepoint approximations quoted in Field & Hampel's Tables 1 and 2; The other gives the density directly in a form equivalent to their equation (4 3), which is then integrated

A non-Gaussian statistical model for surface elevation of nonlinear random wave fields

Abstract: Probability density function of the surface elevation of a nonlinear random wave field is obtained. The wave model is based on the Stokes expansion carried to the third order for both deep water waves and waves in finite depth. The amplitude and phase of the first-order component of the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively. The probability density function for the deep water case was found to depend on two parameters: the root-mean-square surface elevation and the significant slope. For water of finite depth, an additional parameter, the nondimensional depth, is also required. An important difference between the present result and the Gram-Charlier representation is that the present probability density functions are always nonnegative. It is also found that the 'constant' term in the Stokes expansion, usually neglected in deterministic studies, plays an important role in determining the details of the density function. The results compare well with laboratory and field experiment data.

Microstructure of two‐phase random media. II. The Mayer–Montroll and Kirkwood–Salsburg hierarchies

Abstract: It is shown that the Mayer–Montroll (MM) and Kirkwood–Salsburg (KS) hierarchies of equilibrium statistical mechanics for a binary mixture under certain limits become equations for the n‐point matrix probability functions Sn associated with two‐phase random media. The MM representation proves to be identical to the Sn expression derived by us in a previous paper, whereas the KS representation is different and new. These results are shown to illuminate our understanding of the Sn from both a physical and quantitative point of view. In particular rigorous upper and lower bounds on the Sn are obtained for a two‐phase medium formed so as to be in a state of thermal equilibrium. For such a medium consisting of impenetrable‐sphere inclusions in a matrix, a new exact expression is also given for Sn in terms of a two‐body probability distribution function ρ2 as well as new expressions for S3 in terms of ρ2 and ρ3, a three‐body distribution function. Physical insight into the nature of these results is given by ext...

The effect of Gaussian particle‐pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence

Abstract: The consequences for the theory of concentration fluctuations of different proposals by L. F. Richardson and G. K. Batchelor about the rate of separation of pairs of marked particles (giving rise to non-Gaussian and Gaussian particle separation probability density functions respectively) are explored via two Lagrangian Monte Carlo models. The models are applied to an instantaneous one-dimensional cloud source (which is approximately equivalent to a continuous line source) and in many respects give similar results for concentration fluctuations. The crucial difference is that whereas the non-Gaussian model predicts, in agreement with observation, that at large time the fluctuations remain the same order of magnitude as the mean field (the ratio depending only on the source size), the Gaussian model incorrectly predicts that fluctuations ultimately vanish compared with the mean field. The reason for the failure of the Gaussian model is explored by partitioning the total fluctuations into contributions due to the variation of the distribution of material within the cloud (i.e. in coordinates relative to the centre-of-mass) and due to motions of the cloud as a whole (meandering). It is shown that at large time the Gaussian model accounts only for the meandering contribution to fluctuations, and by smoothing out all internal structure of the doud eliminates relative fluctuations.

Estimation of Low-Frequency Wind Stress Fluctuations over the Open Ocean

Abstract: A simple, approximate formula for mean wind stress is given in terms of the mean and variance of the wind fluctuations over the averaging period. The formula is nonlinear with respect to the mean wind speed. The formula is tested using 3 h wind observations from eight North Atlantic Ocean Weather Ships. Mean wind stress is calculated 1) by vector averaging the 3 h wind stresses and 2) by applying the approximate formula. For an averaging period of 4 months the two methods agree to within ±0.025 Pa, 95% of the time. For an averaging period of 1 month the approximate formula slightly overestimates the stress. This is due to skewness in the probability density function of the observed 3 h wind fluctuations. An expression for the modification of the mean stress due to skewness is given. A straightforward method is described for the estimation of vector mean wind and variance fields, and thus mean stress fields, over the open ocean. To cheek the method, the long-term stress field of the North Atlantic...

Random processes with specified spectral density and first-order probability density

Abstract: The procedure for generating a Gaussian process with a specified spectral density is well known. It is harder to generate a process with a specified spectral density and a specified first-order probability distribution. In this paper we explore, by simulation, the possibility of generating a process with such a dual specification by passing a Gaussian process with an appropriately chosen spectral density through an appropriately chosen zero-memory nonlinearity. Several applications are cited where such a dual specification is desirable.

Statistical description of multimodal atomic probability densities

Abstract: A general method for describing multimodal atomic densities is presented. It is based on series expansions of a harmonic Gaussian probability density function. The most suitable expansion is of the Gram-Charlier type; its Fourier transform can be easily inserted in a structure factor equation. This statistical method yields a satisfactory fit to the data and allows for a better interpretation of the fit parameters than sophisticated split-atom models. The method is especially useful for weakly resolved modes and allows a better distinction between disorder and anharmonic motion than in conventional Fourier syntheses. Calculations on CsPbCl3, ice Ih and RbAg4I5 are presented to show the strengths and the limitations of this method.

Dispersal by randomly varying currents

Abstract: The long-term oceanic dispersal of persistent contaminants is approached as a problem in turbulent diffusion, with tidal, wind-driven, and other variable currents relegated to turbulence. The mean advection velocity in this problem is typically small compared with the r.m.s. fluctuation. Therefore, close to a continuous concentrated source, puffs of contaminant of all ages are present and have significant effects. Old puffs, i.e. those released a long time previously, give rise to a background concentration field. Young puffs affect the local contaminant concentration p.d.f. according to the probability of their presence, quantified by the visitation frequency.The behaviour of young puffs is governed by variable advection and may be described approximately in terms of probability distributions obtainable from current-meter data. The visitation frequency can be calculated from the distribution of escape probability density, a Lagrangian equivalent of flux. A long-term effect of variable advection is the distribution of the contaminant over an ‘extended’ source, which serves as a starting point for the random walk of old puffs. The conventional approach of using the diffusion equation to describe this random walk is therefore valid as a description of the near-source background concentration field, provided that the extended source is used in place of the physical source.A sample calculation for a typical open coastal case shows that the background concentration plume becomes wide compared with source dimensions, of order KH/U, with KH horizontal (eddy and shear) diffusivity, U mean advection velocity. The near-source value of the background concentration is correspondingly low, of order m/hKn, with m the mass-release rate and h the water depth. Visitation frequencies calculated with the aid of current statistics drop rapidly with distance from the source, especially in the cross-shore direction. The typical cross-shore diameter of the extended source region is a few kilometres.

An easy way to calculate power spectra for digital FM

Abstract: A general method for numerical calculation of power spectra for digital FM signals is developed. Arbitrary baseband pulse shape, the modulation index and the number of levels of the data can be used. The probability density function of the statistically independent data symbols can also be chosen arbitrarily. With this method the autocorrelation function is first calculated and then numerically Fourier transformed, yielding the power spectrum. The time required to calculate a power spectrum on a digital computer is extremely small, and the calculations are simple and easy to use.

A new method for unfolding sphere size distributions

Abstract: SUMMARY Spherical particles are embedded in an opaque matrix. Circular profiles with radii X1,…, Xn are then observed from a cross-section. To estimate the distribution of the particle sizes a two-stage method is proposed. The probability density of the profile radii is statistically estimated by a smooth, piecewise quartic polynomial, and then an inversion formula is used. The method has some advantages over existing techniques in that the estimate is continuous, is quite straightforward computationally and it involves a less subjective choice of statistical parameters. In the case of several types of distributions, the procedure performed well for simulated data.

Performance Analysis for General PN-Spread-Spectrum Acquisition Techniques

Abstract: This paper presents a unified approach for computing the probability density function (pdf) of the acquisition time for pseudonoise (PN) search algorithms. This approach can be applied to arbitrary search strategies and a priori distributions of the code phase. Furthermore, it is shown that the mean and the variance of the acquisition time can be obtained directly without computing the pdf first.

Equilibrium between radiation and matter for classical relativistic multiperiodic systems. Derivation of Maxwell-Boltzmann distribution from Rayleigh-Jeans spectrum

Abstract: The motion of a charged pointlike relativistic particle under the action of a given force field plus a random electromagnetic radiation is studied. It is assumed that the given force field alone should produce a multiply periodic motion, which is perturbed by the action of both the random radiation and the reaction damping. The random radiation is represented by a stochastic process and an equation is obtained for the equilibrium probability density of the particle in phase space. In the particular case of a random radiation with Rayleigh-Jeans spectrum, it is shown that the stationary solution, corresponding to radiation-matter equilibrium, is given by the Maxwell-Boltzmann distribution.

Smoothing algorithms for nonlinear finite-dimensional systems

Abstract: Systems are considered where the state evolves either as a diffusion process or as a finitestate Markov process, and the measurement process consists either of a nonlinear function of the state with additive white noise or as a counting process with intensity dependent on the state. Fixed interval smooting is considered, and the first main result obtained expresses a smoothing probability or a probability density symmetrically in terms of forward filtered, reverse-time filterd and unfiltered quantities; an associated result replaces the unfiltered and reverse-time filtered qauantities by a likelihood function. Then stochastic differential equationsare obtained for the evolution of the reverse-time filtered probability or probability density and the reverse-time likelihood function. Lastly, a partial differential equation is obtained linking smoothed and forward filterd probabilities or probability densities; in all instances considered, this equation is not driven by any measurement process. The different...

Statistical properties of ray directions in a monochromatic speckle pattern

Abstract: The statistical properties of the spatial derivatives of the phase of a monochromatic speckle pattern are studied. Initially, a one-dimensional probability density function for the derivative of the phase is obtained and compared with the solution for the analogous problem concerning instantaneous frequency of narrow-band Gaussian noise. Subsequently, a two-dimensional probability density function is derived that depends on the two first and three second spatial moments of the illumination intensity distribution of the scattering object. Some sample intensity distributions are considered for which explicit expressions for the probability density function are given.

On the errors involved in computing the empirical characteristic function

Abstract: This paper considers discretisation errors involved in using the Fast Fourier Transform to compute the empirical characteristic function efficiently. A simple improvement to the usual histogram discretisation scheme is shown to reduce the mean square error considerably, as the grid size tends to zero. Simulation results show that the improvement is just as good in practical cases. The theoretical results are applied to the efficient calculation of kernel density estimates, described in Silverman (1982).

Stochastic Solution Method of the Master Equation and the Model Boltzmann Equation

Abstract: Here is described the stochastic solution method of two integro-differential equations for probability density; one is the master equation appearing in the theory of stochastic processes and the other is the Kac model of the Boltzmann equation. The basic idea of the method is in that a set of mutually independent random variables sampled from the probability density can take the place of the role of the probability density function. A few examples are calculated. When the exact solution is known, it is ascertained that the solution obtained from the stochastic solution method agrees with the exact solution.

Reproducible modal noise measurements in system design and analysis

Abstract: The need to predict modal-noise performance limitations in optical-fiber systems has led to noise characterization techniques based on probability density functions. These have been used for evaluation of both components and systems, and are found to give meaningful and repeatable data. The modal-noise probability density function (PDF) can be generated from several individual readings which provide information on the probability of any noise level by extrapolation.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

Abstract: In this paper we deal with the Wiener–Hermite expansion of a process generated by an Ito stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.