Showing papers on "Probability density function published in 1973"

On Some Global Measures of the Deviations of Density Function Estimates

Abstract: We consider density estimates of the usual type generated by a weight function Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved

Angle Detection on Digital Curves

Abstract: A simple parallel procedure for selecting significant curvature maxima and minima on a digital curve is described.

A representation theorem and its applications to spherically-invariant random processes

Abstract: The n th-order characteristic functions (cf) of spherically-invariant random processes (sirp) with zero means are defined as cf, which are functions of n th-order quadratic forms of arbitrary positive definite matrices p . Every n th-order spherically-invariant characteristic function (sicf) is represented as a weighted Lebesgue-Stieltjes integral transform of an arbitrary univariate probability distribution function F(\cdot) on [0,\infty) . Furthermore, every n th-order sicf has a corresponding spherically-invariant probability density (sipd). Then we show that every n th-order sicf (or sipd) is a random mixture of a n th-order Gaussian cf [or probability density]. The randomization is performed on u^2 \rho , where u is a random variable (tv) specified by the F(\cdot) function. Examples of sirp are given. Relations to previously known results are discussed. Various expectation properties of Gaussian random processes are valid for sirp. Related conditional expectation, mean-square estimation, semMndependence, martingale, and closure properties are given. Finally, the form of the unit threshold likelihood ratio receiver in the detection of a known deterministic signal in additive sirp noise is shown to be a correlation receiver or a matched filter. The associated false-alarm and detection probabilities are expressed in closed forms.

Spatial correlation in arbitrary noise fields with application to ambient sea noise

Abstract: Explicit expressions are developed for the cross‐spectral density between pairs of sensors and the wavenumber‐frequency spectrum projected onto the line joining these sensors, when they are placed in arbitrary positions in noise fields which are described by an arbitrary directional distribution of uncorrelated plane waves. The approach is to expand the directional distribution in spatial harmonics. Each harmonic leads to a corresponding term, with the same coefficient, in series representations for the cross‐spectral density function and the wave‐number‐frequency spectrum. The approach particularly attractive when the field can be adquately represented by a relatively small number of harmonics. Both two‐ and three‐dimensional fields are considered. The approach is applied to the vertical directionality of ambient sea noise and is related to some existing models of ambient sea noise. Some new models are presented. Results are compared with reported experimental data and found to be in good agreement over a wide frequency range.

Man-Made Noise in Urban Environments and Transportation Systems: Models and Measurements

Abstract: Analytically tractable statistical-physical models of man-made noise environments have been constructed [1]-[3]. These permit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation systems and urban environments generally. This paper presents a summary of some of the principal analytical results obtained to date [1], [4], and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. First-order probability density functions (pdf's) and probability distributions (pd's) are obtained explicitly; (higher order pdf's and pd's may also be found by similar methods) [2]. These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distributions, beam patterns, and specific system parameters, as long as the interference is narrow-band following (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

Probabilities for Highest Wave in Hurricane

Abstract: The function, which is called the extremal Rayleigh distribution, is found to be an accurate approximation to the probability distribution function for the maximum pro a hurricane with time-varying intensity A formula is developed for the maximum wave distribution function in time-varying storms and computed for a large number of historical hurricanes The extremal Rayleigh function was found to be a very accurate approximation in all cases Two parameters are shown to be interpretable in terms of an equivalent uniform intensity storm as the reciprocal of the logarithm square wave height and the logarithm of the number of waves in the storm, respectively Data from Hurricane Carla was analyzed to determine the parameters which best described the probability behavior of the larger waves in each 20 min record Statistical formulas and tables are given to facilitate probability use of the extremal Rayleigh probability law in coastal engineering and hurricane research

Rank Tests for One Sample, Two Samples, and $k$ samples Without the Assumption of a Continuous Distribution Function

Abstract: The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is developed with no assumptions concerning the continuous or discrete nature of the underlying distribution function. Conditional rank tests, given the vector of ties, are shown to be similar, and the locally most powerful conditional rank test is given. The asymptotic distribution of linear rank statistics is given under the null hypotheses of randomness and symmetry (which includes the one-sample problem), and under contiguous alternatives. Three methods of assigning scores, the average scores, midranks, and randomized ranks methods, are discussed and briefly compared.

General remarks concerning theories dealing with scattering and diffraction in random media

Abstract: The theories to be considered in this review deal mostly with the effect of random media on the propagation of electromagnetic or other waves traversing them. The purpose aimed at thus consists of finding connections between the statistical properties of these waves and the given statistical properties of the medium itself. The characteristics of this medium are to be fixed by one or more wave equations or by equivalent dispersion relations. The waves may be investigated as such, considering the wave function Ψ(r, t) of one or more associated scalar or vector components, or, more generally, the theory may refer to related quantities, e.g., the squared modulus |Ψ|2 which fixes an energy density or, as in quantum mechanics, a probability density.

Response of the impact damper to stationary random excitation

Abstract: The response of an impact damper system to an excitation with approximately white‐power spectral density and Gaussian probability distribution is determined, using two independent methods: digital computer and electronic‐analog techniques. Results are given for mean‐squared level, power spectral density, probability density, probability distribution, and amplitude probability density of the response. The impact damper is found to be a practical and efficient device for reducing the response amplitude of systems subjected to random excitation.

Man-Made Noise in Urban Environments and Transportation Systems: Models and Measurements

Abstract: Analytically tractable statistical-physical models of man- made noise environments have been constructed (l) -(3). These per- mit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation sys- tems and urban environments generally. This paper presents a sum- mary of some of the principal analytical results obtained to date (l) , (4), and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. First- order probability density functions (pdfs) and probability distributions (pd's) are obtained explicitly; (higher order pdfs and pd's may also be found by similar methods) (2). These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distniutions, beam patterns, and spe- cific system parameters, as long as the interference is narrow-band fol- lowing (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

Statistical properties of two sine waves in Gaussian noise

Abstract: A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf). The analysis hinges on expanding the functions of interest in a way that allows computation by means of recursive relations. Specifically, all the expansions are expressed in terms of sums of products of Gaussian hypergeometric functions and Laguerre polynomials. Computer results obtained on a CDC 6600 are presented. If a and b are the amplitudes of the two sine waves, normalized to the rms noise level, the expansions presented are useful up to values of a,b of about 17 dB, in double precision on the CDC 6600. A different approximation is also given for the case of very high SNR.

Conditional sampling and other measurements in a plane turbulent wake

Abstract: A series of hot-wire measurements were carried out in a plane wake to investigate the structure of the tubulence boundary and the relation of its instantaneous position to the behaviour in the core of the flow. The principal measured quantities were as follows: mean velocity profile; intermittency factor; burst rate; mean of the longitudinal component of velocity conditioned upon various specified interface positions; autocorrelation of the intermittency signal; probability densities at the half-intermittency point for the time between bursts and the duration of a burst; probability density for the longitudinal velocity component and its time derivative at various points across the wake; probability density at the half-intermittency point for the same quantities in the turbulent and irrotational zones separately. The present measurements appear to indicate that the turbulence boundary in the wake resembles that in other plane flows more closely than has been supposed hitherto.

Random geometric series and intersymbol interference

Abstract: An interesting and long-standing problem in probability theory is surveyed, and its applications to analyzing the effects of intersymbol interference in digital transmission systems are discussed Old and new results are presented which enable one to obtain analytical forms for the probability density function of the intersymbol interference in special eases The probability of error is then computed and compared with some popular upper bounds

First-order Probability Densities of Laser Speckle Patterns Observed through Finite-size Scanning Apertures

Abstract: The first-order probability density function of laser speckle patterns observed through finite size apertures is theoretically studied. An exact solution is obtained through the use of a two-dimens...

Population mixture models and clustering algorithms

Abstract: The problem of clustering individuals is considered within the context of a mixture of distributions. A modification of the usual approach to population mixtures is employed. As usual, a parametric family of distributions is considered, a set of parameter values being associated with each population. In addition, with each observation is associated an identification parameter, Indicating from which population the observation arose. Theresulting likelihood function is interpreted in terms of the conditional probability density of a sample from a mixture of populations, given the identification parameter of each observation. Clustering algorithms are obtained by applying a method of iterated maximum likelihood to this like-lihood function.

The Mean Residual Lifetime Function

Abstract: Necessary and sufficient conditions for a function to be a mean residual lifetime function of a random variable with finite mean are given.

Density functions of the bivariate chi-square distribution

Abstract: The purpose of this paper is to present a practical approach to solving simultaneous testing and estimation problems faced by the experimenter. New forms of the bivariate Chi-square distribution are introduced which afford straightforward computer programming. An approximation which reduces the general type of dependency to a specific form is suggested, reinforced by strong theoretical justification. Finally, the bivariate Chi-square density function is used to find the density function of a linear combination of independent Chi-square random variables

Nonparametric estimation of the mode of a multivariate density

Abstract: The problem of estimating the mode of a probability density function is a matter of both theoretical and practical interest. This problem was first considered by Parzen (1962) in the univariate situation. He has shown that under certain regularity conditions the estimate of the population mode obtained by maximizing an estimate of the true probability density function is consistent and asymptotically normal. The strongest result in this direction is due to Nadaraya (1965) who has proved that under certain regularity conditions the estimate con verges to the theoretical mode with probability one. In this paper we consider the problem of estimating the mode of an unknown multivariate probability density function. We establish conditions under which our estimate of the . population mode is strongly consistent and asymptotically normal.

Analysis of the 5 min Oscillatory Photospheric Motion. I: A Problem in Waveform Classification

Abstract: Four Mt. Wilson measurements (T>4 h) of the photospheric motion at one point on the Sun are shown to have the characteristics of a narrow-band random process. The motion is shown to have a characteristic correlation time of 23 min and a mean power spectrum that is a smooth, single-peaked function centered at 3.4 mHz. In order to make this classification we use the analytic signal to estimate the amplitude, phase, and frequency as functions of time. The power spectrum analysis differs from the common approaches in that it uses the theoretical expression for the mean spectrum for a sequence of random pulses. Because of the random nature of the motion, we doubt the existence of more than one eigenfrequency characteristic of the photosphere as a whole. Likewise, any description of the observed motion in terms of simple deterministic functions will be inadequate for the data used here.

Finding the approximate angular probability density function of wave arrival by using a directional antenna

Abstract: The first-order approximate angular probability density function of wave arrival can be obtained by plotting the mean-square value of signal level versus the pointing angle of a mobile radio directional antenna during an entire test run. The accuracy of this first-order approximation is dependent on the beamwidth of the antenna. As the beamwidth of the antenna becomes narrower, the approximate density function becomes more accurate. There are also higher order approximations for obtaining a more accurate angular probability density function without narrowing the directional antenna beamwidth. These higher order approximations are related to the moments of the signal strength received by the same directional antenna used in obtaining the first-order approximation. All of these are theoretical predictions. An experimental verification using a first-order approximate angular probability density function in finding theoretical level crossing rates of a mobile radio signal is given. The theoretical level crossing rates are in fairly good agreement with the experimental ones as long as the angular probability density function is not far from a uniform distribution.

Wave-Related Fluctuations in the Airflow Above Natural Waves

Abstract: Simultaneous observations of wave heights and velocity fluctuations at two levels above the waves are analyzed to examine properties of the wave-related fluctuations in the allow. Results are obtained from spectral and joint probability density function, conditional mean function (JPDF-CMF) analyses. Results are examined with respect to predictions from potential flow theory and recent theoretical formulations for wind-wave coupling. Of interest are recent formulations which allow interaction between the wave-induced motion and turbulence in the airflow, the so-called “turbulence” models. Cospectral results exhibit features which are predicted by theoretical formulations with regard to height variations of the wave-related momentum transfer. These features include the oscillatory variations predicted by recent turbulence models and also enhanced transfer at both levels as predicted by the quasi-laminar model. JPDF-CMF analyses are used to obtain phase-amplitude information for those variables exa...

Isotropic random flights

Abstract: The probability density of distribution function of the sum of N isotropic random vectors is studied for the general case in which the probability density of the lengths of the individual vectors vanishes identically outside a finite interval. The probability density function is expressed in a Fourier sine series whose coefficients are the sampled values of the characteristic function. Typical numerical calculations are summarized in graphical form for the case where the lengths of the vectors obey a rectangular probability density. Sampling expansions are also developed for the moments and the distribution function.

The probability density function for the output of a cross-correlator with bandpass inputs

Abstract: A general expression is derived for the probability density function of the output of a cross-correlator, the inputs of which are assumed to consist of deterministic signals with a common frequency plus correlated stationary Gaussian noises. The correlator is composed of input bandpass filters, a multiplier, and postmultiplier low-pass filter. The general expression for the density function is later reduced to some special cases for comparison with previous results. Prior treatments of the problem, however, have omitted the case of correlated noise inputs except under more restrictive conditions.