Uniform Probability Density Research Articles

When a Random Walk of Fixed Length can Lead Uniformly Anywhere Inside a Hypersphere

A variation of the Pearson-Rayleigh random walk in which the steps are i.i.d. random vectors of exponential length and uniform orientation is considered. Conditioned on the total path length, the probability density function of the position of the walker after n steps is determined analytically in one and two dimensions. It is shown that in two dimensions n = 3 marks a critical transition point in the behavior of the random walk. By taking less than three steps and walking a total length l, one is more likely to end the walk near the boundary of the disc of radius l, while by taking more than three steps one is more likely to end near the origin. Somehow surprisingly, by taking exactly three steps one can end uniformly anywhere inside the disc of radius l. This means that conditioned on l, the sum of three vectors of exponential length and uniform direction has a uniform probability density.

Color image histogram equalization by absolute discounting back-off

A novel color image histogram equalization approach is proposed that exploits the correlation between color components and it is enhanced by a multi-level smoothing technique borrowed from statistical language engineering. Multi-level smoothing aims at dealing efficiently with the problem of unseen color values, either considered independently or in combination with others. It is applied here to the HSI color space for the probability of intensity and the probability of saturation given the intensity, while the hue is left unchanged. Moreover, the proposed approach is extended by an empirical technique, which is based on a hue preserving non-linear transformation, in order to eliminate the gamut problem. This is the second method proposed in the paper. The equalized images by the two methods are compared to those produced by other well-known methods. The better quality of the images equalized by the proposed methods is judged in terms of their visual appeal and objective figures of merit, such as the entropy and the Kullback–Leibler divergence estimates between the resulting color histogram and the multivariate uniform probability density function.

Stochastic heat transfer enhancement in a grooved channel

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$ , which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

Double-couple earthquake focal mechanism: random rotation and display

SUMMARY This paper addresses two problems: the random rotation of double-couple (DC) earthquake sources and the display of earthquake focal mechanisms. We consider several equivalent representations for DC sources and their properties and provide mathematical expressions for their mutual transformation. Obviously, a 3-D rotation of any object is more intricate than a 2-D rotation. Any rotation of a DC source is further complicated by its symmetry properties. Applying statistical tests to a DC distribution often requires one to compare it to a completely (or uniform) random DC pattern. We review several methods for obtaining random distribution of DC orientation; some of these seemingly natural techniques yield an incorrect result. The DC random rotation problem is closely connected to displays of focal mechanisms. In such displays, a strike or an azimuth of a focal mechanism can be neglected; hence, we are confronted with mapping a 2-D distribution onto a flat surface. We review different methods for such displays and discuss more specifically how to project a random focal mechanism distribution on a flat 2-D display with uniform probability density. Such displays can be used to analyse earthquake patterns statistically in various tectonic regions.

Improved mixing representation in Emanuel's convection scheme

AbstractRecent empirical and modelling studies suggest that mid‐tropospheric relative humidity (RH) is an important controlling factor of deep atmospheric convection, which appears to be underestimated in present cumulus parametrizations. This indicates the possible presence of shortcomings in the way that entrainment is represented in such parametrizations. This matter was explored in the European Cloud Systems project (EUROCS) by means of an idealized humidity experiment in which the main controlling parameter is RH. In the latter study, cloud‐resolving model (CRM) experiments suggested that a shallow/deep convection transition occurs when RH crosses a threshold value that ranges from about RH=50% to RH=60%.In this paper, we seek to increase the responsiveness of Emanuel's convection scheme to RH, and to reproduce the threshold behaviour of the idealized humidity case, by replacing the original uniform probability density function (PDF) for mixing fractions by a more flexible two‐parameter bell‐shaped function that allows a wider range of behaviour. The main result is that the parameters of this PDF can be tuned to allow a regime transition to occur near a threshold value of RH≈55%. In contrast to CRM results, however, this transition is between two different regimes of deep convection rather than between a shallow and deep regime. Possible ways to obtain a shallow‐to‐deep transition with Emanuel's scheme are discussed. Copyright © 2004 Royal Meteorological Society

Attractor coverage time, time dimension, and its relation to capacity dimension

We have studied the time to complete coverage of a chaotic attractor by cells as a function of the cell size. This dependence is described with good precision by a power law. The system attractor is characterized by a new quantity called the time dimension by analogy with the capacity dimension. A relation between the two values has been studied. For a homogeneous chaotic attractor characterized by a uniform invariant probability density distribution, the two dimensions coincide. In the case of an inhomogeneous chaotic attractor, the time dimension is greater than the capacity one.

Contrasting probabilistic and anti-optimization approaches in an applied mechanics problem

Probabilistic and non-probabilistic, anti-optimization analyses of uncertainty are contrasted in this study. Specifically, the comparison of these two competing approaches is conducted for an uniform column, with initial geometric imperfection, subjected to an impact axial load. The reliability of the column is derived for the cases when the initial imperfections posses either (a) uniform probability density, (b) truncated exponential density or (c) generic truncated probability density. The problem is also analyzed in the context of an interval analysis. It is shown that in, the most important near-unity reliability range these two approaches tend to each other. Since the interval analysis constitutes a much simpler procedure than the probabilistic approach, it is argued that the former is advantageous over the latter in some circumstances.

Reconstruction of time-averaged temperature of non-axisymmetric turbulent unconfined sooting flame by inverse radiation analysis

A multi-wavelength inversion method is extended to reconstruct the time-averaged temperature distribution in non-axisymmetric turbulent unconfined sooting flame by the multi-wavelength measured data of low time-resolution outgoing emission and transmission radiation intensities. Gaussian, β and uniform distribution probability density functions (PDF) are used to simulate the turbulent fluctuation of temperature, respectively. The reconstruction of time-averaged temperature consists of three steps. First, the time-averaged spectral absorption coefficient is retrieved from the time-averaged transmissivity data by an algebraic reconstruction technique. Then, the time-averaged blackbody spectral radiation intensity is estimated from the outgoing spectral emission radiation intensities. Finally, the time-averaged temperature is approximately reconstructed from the multi-wavelength time-averaged spectral emission radiation data by the least-squares method. Noisy input data have been used to test the performance of the proposed inversion method. The results show that the time-averaged temperature distribution can be estimated with good accuracy, even with noisy input data. The accuracy of the estimation decreases with the increase of turbulent fluctuation intensity of temperature and the effects of assumed PDF on the reconstruction of temperature are small.

State of polarisation bias in aerial fibres

The state of polarisation was measured for 7 days in an aerial fibre during the wintertime. A biased probability density was found. This is contrary to the generally accepted uniform probability density. A Maxwellian probability density function for differential group delay at a fixed wavelength will thus not occur when measured on this fibre for a median time period.

Fuzzy systems with overlapping Gaussian concepts: Approximation properties in Sobolev norms

In this paper the approximating capabilities of fuzzy systems with overlapping Gaussian concepts are considered. The target function is assumed to be sampled either on a regular gird or according to a uniform probability density. By exploiting a connection with Radial Basis Functions approximators, a new method for the computation of the system coefficients is provided, showing that it guarantees uniform approximation of the derivatives of the target function.

Blind multiuser separation of instantaneous mixture algorithm based on geometrical concepts

In this paper, we propose a new and simple algorithm for blind separation of sources based on geometrical concepts. This algorithm deals with the instantaneous mixtures and does not require the estimation of high-order statistics (HOS). The proposed algorithm can separate sources that belong to two different categories: (1) Signals with uniform or close to uniform probability density function (PDF). (2) Unimodal PDF (including Pascal and Gamma) as well as some real signals such as speech or music. Unfortunately, the actual version of the algorithm cannot deal with the mixing of signals from both categories. Finally, the experimental results show good performances and that the separation can be carried out in a very short time (a few seconds, using Matematica as the programming language and our ultra 30 Sun station).

Analytical propagation of uncertainties through fault trees

Abstract A method is presented which enables one to propagate uncertainties described by uniform probability density functions through fault trees. The approach is analytical. It is based on calculating the expected value and the variance of the top event probability. These two parameters are then equated with the corresponding ones of a beta-distribution. An example calculation comparing the analytically calculated beta-pdf (probability density function) with the top event pdf obtained using the Monte-Carlo method shows excellent agreement at a much lower expense of computing time.

An attempt to model distributions of machined component dimensions in production

In this study, normal, log-normal, triangular, uniform, Weibull, Erlang and unit beta probability density functions are tried to represent the behaviour of frequency distributions of workpiece dimensions collected from various manufacturing firms. Among the distribution functions, the unit beta distribution function is found to be the best fit using the chi-square test of fit. An attempt is made for the adoption of the unit beta model to x-bar charts of quality control in manufacturing. In this direction, upper and lower control limits (UCL and LCL) of x-bar control charts of dimension measurements are estimated for the beta model, and the observed differences between the beta and normal model control limits are discussed for the measurement sets.

A minimalist probabilistic description of root zone soil water

The probabilistic response of depth‐integrated soil water to given climatic forcing can be described readily using an existing supply‐demand‐storage model. An apparently complex interaction of numerous soil, climate, and plant controls can be reduced to a relatively simple expression for the equilibrium probability density function of soil water as a function of only two dimensionless parameters. These are the index of dryness (ratio of mean potential evaporation to mean precipitation) and a dimensionless storage capacity (active root zone soil water capacity divided by mean storm depth). The first parameter is mainly controlled by climate, with surface albedo playing a subsidiary role in determining net radiation. The second is a composite of soil (through moisture retention characteristics), vegetation (through rooting characteristics), and climate (mean storm depth). This minimalist analysis captures many essential features of a more general probabilistic analysis, but with a considerable reduction in complexity and consequent elucidation of the critical controls on soil water variability. In particular, it is shown that (1) the dependence of mean soil water on the index of dryness approaches a step function in the limit of large soil water capacity; (2) soil water variance is usually maximized when the index of dryness equals 1, and the width of the peak varies inversely with dimensionless storage capacity; (3) soil water has a uniform probability density function when the index of dryness is 1 and the dimensionless storage capacity is large; and (4) the soil water probability density function is bimodal if and only if the index of dryness is <1, but this bimodality is pronounced only for artificially small values of the dimensionless storage capacity.

Nonlinear techniques for Mode C climb/descent rate estimation in ATC systems

Secondary surveillance radars (SSRs) currently form the main data source for target tracking in air traffic control systems. Altitude and climb/descent rate estimation is then based on Mode C altitude reports obtained from transponder replies from the aircraft. Accurate climb/descent rates are essential for the performance of short-term conflict alert (STCA) systems where prediction times of up to 2 min may be required. This is however difficult to accomplish because of the 100 ft quantization of the Mode C reports which limits the applicability of standard Kalman filtering techniques. To overcome these problems two new nonlinear filters have been derived. They perform recursive state estimation when only coarsely quantized measurements are available. The first method follows from a solution to the Fokker-Planck equation, and is therefore optimal. A linear trend model with uniform prior probability density function (pdf) forms the model assumptions. Together with properties of the quantizer this allows an exact and recursive updating of the pdf using only the corners of a convex polygon. The second method is obtained by modifications of the extended Kalman filter (EKF), and is therefore more widely applicable, at the price of being suboptimal.

Antiplane coherent scattering from a slab containing a random distribution of cavities

The scattering of antiplane waves from a slab region containing a random distribution of cylindrical cavities is investigated. The cavities have a uniform probability density in the slab region and...

In-plane waves in an elastic solid containing a cracked slab region

Propagation of longitudinal and transverse waves in an elastic solid that contains a cracked slab region is investigated. The cracks have a uniform probability density in the slab region, are parallel to the boundaries of the slab, and the solid is uncracked on either side of the slab. The waves are normally incident on the cracks. It is shown that the resulting average total motion in the solid is governed by a pair of coupled integral equations. These equations are solved under the special assumption that the average exciting motion near a fixed crack is equal to the average total motion. In this case, one finds that in the cracked region, where multiple scattering occurs, there is a forward motion and a backward motion. The two motions have identical frequency-dependent velocity and attenuation, for which simple closed-form formulae are obtained. Simple formulae are also obtained for the wave amplitudes outside the slab. Numerical results corresponding to the velocity, attenuation, reflection amplitude, and transmission amplitude are presented for several values of crack density and slab thickness.

Anti-Optimization Versus Probability in an Applied Mechanics Problem: Vector Uncertainty

In this study probabilistic and nonprobabilistic anti-optimization approaches are contrasted to evaluate their relative advantages and disadvantages while solving a mechanical problem in presence of vector uncertainty. The different cases that are analyzed in probabilistic setting that deal with either uniform or generic probability density functions for the uncertain variables varying in a rectangular domain. This case has been compared with interval analysis, a particular case of anti-optimization. The presence of a convex, smooth boundary of the uncertain domain has been also considered for comparing results obtained with these two alternative methods. It is shown that in case of vector uncertainty the anti-optimization method yields the same solution for the design problem as is provided by means of more complex probabilistic considerations. [S0021-8936(00)03103-2]

New detailed numerical procedures for calculating risk measures in hazardous materials transportation

Abstract Historical evidence has shown that accidents due to hazardous material releases during transportation can lead to consequences as heavy as those created by releases occurring at fixed plants and therefore quantified risk analysis has to be performed for transportation networks too. The calculation of risk measures, like individual and societal risk, due to transportation networks has considerable complexity and needs a great computational effort. For this application two new detailed procedures, named T rans I n and T rans S oc , for the evaluation of individual and societal risk, have been introduced, which can take into account, at the same time, different transportation modes, hazardous materials, meteorological conditions and seasonal situations, a non uniform wind probability density distribution and an accurate description of the indoor and outdoor population both on-route and off-route. In this paper, firstly, the arrangement of the input data is explained, and then the fundamentals of the methodology are described, often focusing attention on mathematical ways to perform operations saving accuracy without increasing calculation time.

LOCALIZATION OF LONGITUDINAL WAVES IN BI-PERIODIC ELASTIC STRUCTURES WITH DISORDER

A formulation for studying the effect of random variation in the transfer matrix on the attenuation behavior of disordered one-dimensional bi-periodic layered structures is developed. This formulation, however, can be used for both stochastically and deterministically disordered systems. The mean and variance of localization factors for the disordered systems are numerically evaluated under the assumption that the source of disorder is the variance in the Young's modulus of the first layer of a set, and this variation is modelled by a random variable with uniform probability density function. In the calculations for the mean localization factor and its variance for two layered systems with different elasticity properties are considered for four different levels of disorder. The results presented in the current work is used to explain how the attenuation zones of a perfect system expand into adjacent propagation zones due to disorder. It is found that while the existence of disorder affects the structures of all propagation zones, its significance is most predominant in the first propagation zones of both systems. However, as the frequency increases, the disorder level and the effect of elastic coupling become less significant. Therefore, the wave components corresponding to higher propagation zones penetrate deeper into a structure. It is observed that the right boundary of the propagation zones is the mean localization factor asymptote. The behavior of the mean Lyapunov curves near the right propagation zone boundaries explains how the disorder level governs the expansion of attenuation zones into propagation zones. This expansion is strongest in the first propagation zone. Since the power density of motion induced by an external force is high in the first propagation zone, the structure and size of first propagation zone are of a particular interest in transient analysis. The structures of the variance curves exhibit similar behavior to those for the mean localization factor.