Showing papers on "Probability density function published in 1974"

Probabilistic analysis of power flows

Abstract: The paper describes and examines a technique which permits the power-flow problem in a power system to be analysed probabilistically instead of using normal deterministic methods. All the nodal loads and generation are defined as random variables and the power flow in each line is computed in terms of a probability density function. The expected values and standard deviation of each power flow are also calculated, and, in addition, the overall balance of power in the system is determined in terms of a density function. The purpose of this analysis is to account for the errors and statistical variations known to exist in the operation and planning of systems within one solution. This enables the power-flow problem to be treated objectively and allows quantitative assessment of reliability and security. The paper compares the results obtained probabilistically with those that would be obtained deterministically, and shows the much wider range of information gained in this type of analysis.

Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria

Abstract: The expected value of a multiplicative performance criterion, represented by the exponential of a quadratic function of the state and control variables, is minimized subject to a discrete stochastic linear system with additive Gaussian measurement and process noise. This cost function, which is a generalization of the mean quadratic cost criterion, allows a degree of shaping of the probability density function of the quadratic cost criterion. In general, the control law depends upon a gain matrix which operates linearly on the smoothed history of the state vector from the initial to the current time. This gain matrix explicitly includes the covariance of the estimation errors of the entire state history. The separation theorem holds although the certainty equivalence principle does not. Two special cases are of importance. The first occurs when only the terminal state is costed. A feedback control law, linear in the current estimate of the state, results where the feedback gains are functionally dependent upon the error covariance of the current state estimate. The second occurs if all the intermediate states are costed but there is no process noise except for an initial condition uncertainty. A feedback law results which depends not only upon the current dynamical state estimate but also on an additional vector which is path dependent.

Buffer Behavior with Poisson Arrival and Geometric Output Processes

Abstract: A discrete buffered system with infinite buffer size, Poisson arrival process, and the output channel available only periodically according to a geometric density function is analyzed. Under the assumption that a stochastic equilibrium is reached, the Z transform of the density function of the buffer occupancy is obtained as the result of this study.

Admissible Scoring Systems for Continuous Distributions

Abstract: ABSTRACT The defining property of an admissible scoring system is that any individual perceives himself as maximizing his expected score by reporting his true subjective distribution. The use of admissible scoring systems as a measure of probabilistic forecasts is becoming increasingly well-known in those.cases where the forecast is a discrete distributicn over a finite number of alternatives. Most serious forecastS which are made in the real world seem to be forecasts of quantitieL rather than choices between a finite number of alternatives. In such cases as this, it seems much more natural to ask the forecaster to specify a continuous probability distribution which represents his expectations rather than trying to re-cast a basically continuous process into a discrete one. To construct an admissible scoring system for a continuous distribution, a collection of possiblid bets can be postulated on a continuous variable, and an admissible scoring system can be constructed as the net pay-off to a forecaster who takes all bets (and only those bets). which appear favorable on the basis of his reported distribution. Mathematical models for this and alternative systems are presentedlAuthor/BW)

Some Effects of Temporal Coherence on the First Order Statistics of Speckle

Abstract: Moments and the first order probability density function of the intensity in a speckle pattern are considered as a function of the spectral bandwidth of the incident light. It is shown that the sta...

Turbulent pressure field beneath a hydraulic jump

Abstract: Certain statistical characteristics of the fluctaation of pressure at the bottom of a classical hydraulicjump are estimated using analog and digital modes of analysis of random data. The results indicate a relatively higher intensity of pressure fluctuation because of the stage of development of the incoming supercritical stream. The evaluated characteristics are, in general, similar in the investigated range of initial Froude number. The structure of the pressure field is discussed in terms of moments of the probability density histograms, shape of the autocorrelation and the spectral density functions and the estimated timemicroscales.

Diffusive random process approximation in certain nonstationary statistical problems of physics

Abstract: The review considers, on the basis of a unified approach, the problem of Brownian motion in nonlinear dynamic systems, including a linear oscillator acted upon by random forces, parametric resonance in an oscillating system with random parameters, turbulent diffusion of particles in a random-velocity field, and diffusion of rays in a medium with random inhomogeneities of the refractive index. The same method is used to consider also more complicated problems such as equilibrium hydrodynamic fluctuations in an ideal gas, description of hydrodynamic turbulence by the method of random forces, and propagation of light in a medium with random inhomogeneities. The method used to treat these problems consists of constructing equations for the probability density of the system or for its statistical moments, using as the small parameter the ratio of the characteristic time of the random actions to the time constant of the system (in many problems, the role of the time is played by one of the spatial coordinates). The first-order approximation of the method is equivalent to replacement of the real correlation function of the action by a δ function; this yields equations for the characteristics in closed form. The method makes it possible to determine also higher approximations in terms of the aforementioned first-order small parameter.

Nonlinear Panel Response by a Monte Carlo Approach

Abstract: The vibration of clamped and simply supported elastic panels due to subsonic and supersonic turbulent boundary-layer flows is investigated by a Monte Carlo technique. The resulting generalized random forces are simulated numerically from boundary-layer turbulence spectra, and the response analysis is performed in the time domain. The mutual interaction between panel motion and external and/or internal airflow is included. Response studies are performed with respect to rms response, probability structure, peak distribution, threshold crossing and spectral density. The effect on the response statistics of in-plane loading, static pressure differential, and cavity pressure is investigated.

An Always-Convergent Numerical Scheme for a Random Locational Equilibrium Problem

Abstract: A probabilistic extension of the classical Weber problem is studied. N destinations in the plane, $P_j ,j = 1, \cdots ,N$, are given as random variables with specified probability density functions, and the problem is to find the location of the point P which minimizes the expected sum of the Euclidean distances $\overline {PP_j } $ . Under mild assumptions on the density functions, the objective function is shown to be strictly convex and the minimum unique. An iterative scheme for finding P is shown to be a descent method which is globally convergent, and the iteration is shown to be locally linear. Finally, numerical examples using bivariate normal density functions are given.

Estimation of the Distribution Function of a Continuous Type Random Variable through Randomized Response

Abstract: In 1965 Stanley Warner [8] illustrated a technique whereby one could estimate from a sample the proportion of persons in a population possessing some characteristic X, without pointedly asking the question, “Do you possess characteristic X?”. The present article uses a variation of these ideas suggested by Warner in a later paper [7] to estimate the distribution function of a continuous-type random variable. The technique is illustrated by estimating an income distribution from the responses of a sample of 500 individuals. The potential use of devices of this type in maintaining confidentiality of existing data files is apparent.

Statistical properties of a sum of sinusoids and Gaussian noise and its generalization to higher dimensions

Abstract: This paper investigates the statistical properties of the sum, S, of an n-dimensional Gaussian random vector, N, plus the sum of M vectors, X 1 , …, X M , having random amplitudes and independent arbitrary orientations in n-dimensional space. We derive expressions for the probability density function (p.d.f.) and distribution function (d.f.) of S and of its length, |S|, as series expansions involving only the moments of |X i |, i = 1, …, M. In addition, we find the p.d.f. and d.f. of the projection of S onto 1-dimensional space. Our results are generalizations of the n = 2-dimensional problem of finding the statistical properties of a sum of constant-amplitude sinusoids having independent uniformly distributed phase angles plus Gaussian noise. The latter problem has been treated by Rice1 and Esposito and Wilson,2 but our results can also deal with sinusoids having random amplitudes. When n = 3, our findings treat, in the presence of a Gaussian vector, the classical problem of “random flights” dating back to Rayleigh. Some calculations for the 2- and 3-dimensional problem are presented, and an application to coherent phase-shift-keying communications systems is discussed.

Some estimates of the probability density of a stochastic integral

Abstract: Estimates in Lp are derived for probability densities of stochastic integrals. An example is presented which shows that for some values of p such estimates are not attainable. The method of proving these estimates is based on a study of Bellman's nonlinear equations and the properties of λ-convex functions.Bibliography: 12 items.

The probability density function for the output of an analog cross-correlator with correlated bandpass inputs

Abstract: The probability density function (pdf) for the output of an analog cross-correlator with correlated bandpass inputs is derived. The pdf is derived by a "direct method" without resorting to the "characteristic function method," which usually requires contour integrations in a complex plane for inversion operations. The correlator consists of bandpass filters, a multiplier, and a zonal low-pass filter. We treat the general situation in which the two inputs are narrow-band signals of unequal power and of different phases. The bandpass input noises are assumed to be correlated and may have different powers. In the Appendix, another derivation for the pdf is given in the special case of equal power correlated noise. This derivation is based on the fact that the correlator output random variable is the difference of two independent noncentral chi-square variables of two degrees of freedom. We show that the two expressions for the pdf (one from the direct method and the other from the characteristic function method) are indeed equivalent. Finally, we discuss two major areas of application.

Arrival Time Estimation by Adaptive Thresholding

Abstract: Arrival time estimation by adaptive thresholding is described. The probability density of arrival time is derived for differentiable Markov processes. The special case of additive, stationary noise is given particular attention. A direct derivation of the probability density of arrival time for pulses with sharply rising edges is given for arbitrary noise. The results are applied to the Gaussian and Rice distributions. Comparison with the Cramer-Rao bound of estimation theory indicates the asymptotic optimality of adaptive thresholding in the latter two cases.

On some properties of stochastic information processes in neurons and neuron populations - Mathematical model approach

Abstract: The information in the nervous spike trains and its processing by neural units are discussed. In these problems, our attention is focused on the stochastic properties of neurons and neuron populations. There are three subjects in this paper, which are the spontaneous type neuron, the forced type neuron and the reciprocal inhibitory pairs. 1. The spontaneous type neuron produces spikes without excitatory inputs. The mathematical model has the following assumptions. The neuron potential (NP) has the fluctuation and obeys the Ornstein-Uhlenbeck process, because the N P is not so perfectly random as that of the Wiener process but has an attraction to the rest value. The threshold varies exponentially and the NP has the constant lower limit. When the NP reaches the threshold, the neuron fires and the NP is reset to a certain position. After a firing, an absolute refractory period exists. In discussing the stochastic properties of neurons, the transition probability density function and the first passage time density function are the important quantities, which are governed by the Kolmogorov's equations. Although they can be set up easily, we can rarely obtain the analytical solutions in time domain. Moreover, they cover only simple properties. Hence the numerical analysis is performed and a good deal of fair results are obtained and discussed. 2. The forced type neuron has input pulse trains which are assumed to be based on the Poisson process. Other assumptions and methods are almost the same as above except the diffusion approximation of the stochastic process. In this case, we encounter the inhomogeneous process due to the pulse-frequency-modulation, whose first passage time density reveals the multimodal distribution. The numerical analysis is also tried, and the output spike interval density is further discussed in the case of the periodic modulation. 3. Two types of reciprocal inhibitory pairs are discussed. The first type has two excitatory driving inputs which are mutually independent. The second type has one common excitatory input but it advances in two ways, one of which has a time lag. The neuron dynamics is the same as that of the forced type neuron and each neuron has an identical structure. The inputs are assumed to be based on the Poisson process and the inhibition occurs when the companion neuron fires. In this case, the equations of the probability density functions are not obtained. Hence the computer simulation is tried and it is observed that the stochastic rhythm emerges in spite of the temporally homogeneous inputs. Furthermore, the case of inhomogeneous inputs is discussed.

Error probabilities in binary angle modulation

Abstract: It is shown that many signal-detection problems involving binary phase and frequency modulation can be solved by considering the problem as the convolution of two independent phase distributions. When the two distributions are similar and each represents the probability density function of the phase of a sinusoid in noise, the probability of error is described in terms of modified Bessel functions by the expression obtained recently by Jain and Blachman. The probability of error can also be expressed in terms of Rice's Ie function. By appropriate definition of the signal-to-noise ratios, this general expression can be used to determine the error probability in a number of cases of practical interest, such as detection of a hard-limited phase-shift keyed (PSK) signal, PSK detection with noisy reference, DPSK detection, frequency-shift keying (FSK) detection, binary FM with discriminator detection, and binary pulse-position modulation (PPM).

Capacity-achieving input distributions for some amplitude-limited channels with additive noise (Corresp.)

Abstract: An additive noise channel wherein the noise is described by a piecewise constant probability density is shown to reduce to a discrete channel by means of an explicit construction. In addition, conditions are found which describe a class of continuous amplitude-limited channels for which the capacity-achieving input distribution is binary.

Critical Relaxation of a Non-Linear Fokker-Planck Equation

Abstract: The Fokker-Planck equation whose equilibrium solution is given by Wilson's probability density is considered. The conditional probability of the temporal process generated through this equation is expressed in the forrri of a path integral. With the aid of renormalization group transformations applied to this path probability density, the dynamical critical expo­ nent characterizing asymptotic critical behavior of its long wavelength mode is determined to order e' (E=4 -d) in agreement with that of Halperin, Hohenberg and Ma.

Output Probability Density Functions for Cross Correlators Utilizing Sampling Techniques

Abstract: Sampling techniques provide a practical means of obtaining cross-correlation functions. In this paper, the correlation function is described by sums of the form Z = \begin{equation*}Z = \Sigma^{N}_{j=1}X_{j}Y_{j}\end{equation*}. A general expression is derived for the probability density function of the random variable Z under the condition that Xj and Yj are stationary, jointly Gaussian random processes with nonzero means and unit variances.

The Output Pdf of a Polarity Coincidence Correlation Detector

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 clipped sine waves of similar frequency plus uncorrelated, stationary Gaussian noise. The correlator output is shown to be a piecewise linear function of the random phase difference between the two input processes; hence, the density function for the correlator output is obtained by a relatively simple transformationfrom the probability density function of the random phase difference.

Joint density functions for digital spectra

Abstract: An analysis is presented for the joint probability density functions of power spectra of random processes computed under various conditions of data/frequency smoothing Spectral correlation functions are determined which illustrate the effect of smoothing on adjacent spectral ordinates The technique of segment averaging of spectra is explored and bivariate probability density functions established for correlated spectral ordinates These distributions are shown to be generalizations of the well known Rician probability density functions Finally, the case of spectra obtained as moving average values is considered and expressions are derived for the spectral correlation functions in terms of the parameter of the smoothing sequences and the averaging length It is shown that spectra obtained in this manner introduce a distortion of the spectral correlation function Several computed results are included to illustrate the analysis