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This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable X with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of X as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable X and characterize type- 2 and type- 3 real-valued random variables. We prove that a necessary condition for X to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type. [ABSTRACT FROM AUTHOR]

The relationship between intensity-duration-frequency of rainfall is a significant tool for estimating flood discharge. According to the sparsely available rain gauge stations and the development of technology, it is possible to use satellite rainfall data with different temporal and spatial resolutions. PERSIANN rainfall data with a time resolution of 1 and 6 hours were used in this research. Also, the spatial resolution of these data is 0.04 x 0.04 degrees. Rainfall data from synoptic stations around the Kan basin were also used. Three common continuous probability distributions of Gamble, Pearson type 3, and Log Pearson type 3 with return periods of 2, 5, 10, 25, 50, and 100 years were investigated to calculate and check the IDF curve. In general, the precipitation intensity obtained from Gumble's method was more than Pearson Type 3's method. Log Pearson type 3 distribution did not provide acceptable results in this research. The two interpolation methods of inverse distance weighting and empirical Bayesian kriging were used to generalize the frequency intensity curves to the entire Kan basin. The results showed little difference between these two methods, except for Pearson type 3 probability distribution.

In the case where the square of an eigenfunction F with respect to an eigenvalue of Markov generator L can be expressed as a sum of eigenfunctions, we find the largest number excluding zero among the eigenvalues in the terms of the sum. Using this number, we obtain an improved bound of the fourth moment theorem for Markov diffusion generators. To see how this number depends on an improved bound, we give some examples of eigenfunctions of the diffusion generators L such as Ornstein–Uhlenbeck, Jacobi, and Romanovski–Routh. [ABSTRACT FROM AUTHOR]

Two multi-parameter distributions, namely the Pearson type IV and metalog distributions, are discussed and suggested as alternatives to the normal distribution for modelling path delay data that determines the maximum clock frequency (FMAX) of a microprocessor or other digital circuit. These distributions outperform the normal distribution in goodness-of-fit statistics for simulated path delay data derived from a fabricated microcontroller, with the six-term metalog distribution offering the best fit. Furthermore, 99.7% confidence intervals are calculated for some extreme quantiles on each dataset using the previous distributions. Considering the six-term metalog distribution estimates as the golden standard, the relative errors in single paths vary between 4 and 14% for the normal distribution. Finally, the within-die (WID) variation maximum critical path delay distribution for multiple critical paths is derived under the assumption of independence between the paths. Its density function is then used to compute different maximum delays for varying numbers of critical paths, assuming each path has one of the previous distributions with the metalog estimates as the golden standard. For 100 paths, the relative errors are at most 14% for the normal distribution. With 1000 and 10,000 paths, the corresponding errors extend up to 16 and 19%, respectively. [ABSTRACT FROM AUTHOR]

In the case where the square of an eigenfunction F with respect to an eigenvalue of Markov generator L can be expressed as a sum of eigenfunctions, we find the largest number excluding zero among the eigenvalues in the terms of the sum. Using this number, we obtain an improved bound of the fourth moment theorem for Markov diffusion generators. To see how this number depends on an improved bound, we give some examples of eigenfunctions of the diffusion generators L such as Ornstein–Uhlenbeck, Jacobi, and Romanovski–Routh.

Two multi-parameter distributions, namely the Pearson type IV and metalog distributions, are discussed and suggested as alternatives to the normal distribution for modelling path delay data that determines the maximum clock frequency (FMAX) of a microprocessor or other digital circuit. These distributions outperform the normal distribution in goodness-of-fit statistics for simulated path delay data derived from a fabricated microcontroller, with the six-term metalog distribution offering the best fit. Furthermore, 99.7% confidence intervals are calculated for some extreme quantiles on each dataset using the previous distributions. Considering the six-term metalog distribution estimates as the golden standard, the relative errors in single paths vary between 4 and 14% for the normal distribution. Finally, the within-die (WID) variation maximum critical path delay distribution for multiple critical paths is derived under the assumption of independence between the paths. Its density function is then used to compute different maximum delays for varying numbers of critical paths, assuming each path has one of the previous distributions with the metalog estimates as the golden standard. For 100 paths, the relative errors are at most 14% for the normal distribution. With 1000 and 10,000 paths, the corresponding errors extend up to 16 and 19%, respectively.

Investments with high Sharpe ratios tend to be fixed income investments, particularly short-term bonds. Such securities dominate income investment, but not growth investment. We generalize the Sharpe ratio to interpolate smoothly between risk-averse and risk-seeking behaviors, with the traditional Sharpe ratio as a special case. The generalized ratios may be used to optimize portfolios within a lifestyle strategy or a lifecycle strategy. We demonstrate that the Sharpe ratio has an interpretation connected to value at risk (VaR), if returns are normally distributed. We study broad market indices and indicators for US stocks, US corporate bonds, gold, and Japanese stocks. Using monthly data obtained from FRED, we show that these datasets are neither normal, nor log-normal, nor stable. We argue that none of these distributions is appropriate for market data. We argue instead for using infinitely divisible distributions, and fit Pearson distributions to the data. [ABSTRACT FROM AUTHOR]

The concept of functional controllability of the milk ejection is considered, which makes it possible to predict the intensity of milk ejection in the online mode of the milking machine. The architecture of the functional controllability by intensity of milk ejection is developed. Input and output parameters of the structural-functional scheme of adaptive control of milk ejection intensity are described. An analytical model of milk ejection intensity based on Pearson's distribution is developed. The milk ejection intensity for different productivity and duration of cows milking is modelled. The microprocessor unit is designed using a single-chip microcontroller. It ensures the algorithm set by the central computer and implements a step of changing the pulsation frequency of 0.1 Hz, the ratio between the cycles of 0.25%, the phase shift step of 0.1 s. [ABSTRACT FROM AUTHOR]

Featured Application: This study proposes a method for quantifying the effect of uncertainty characteristics of renewable energy resources on power system flexibility, based on the ramping capability shortage probability index. The procedure involves scenario generation and evaluation of uncertainty characteristics. The proposed approach can provide system operators with the means to evaluate uncertainty characteristics, as well as possible strategies for a more flexible power system. This study investigates the effect of uncertainty characteristics of renewable energy resources on the flexibility of a power system. The more renewable energy resources introduced, the greater the imbalance between load and generation. Securing the flexibility of the system is becoming important to manage this situation. The degree of flexibility cannot be independent of the uncertainty of the power system. However, most existing studies on flexibility have not explicitly considered the effects of uncertainty characteristics. Therefore, this study proposes a method to quantitatively analyze the effect of uncertainty characteristics on power system flexibility. Here, the uncertainties of the power system indicate the net load forecast error, which can be represented as a probability distribution. Of the characteristics of the net load forecast error, skewness and kurtosis were considered. The net load forecast error was modeled with a Pearson distribution, which has been widely used to generate the probability density function with skewness and kurtosis. Scenarios for the forecast net load, skewness, and kurtosis were generated, and their effects on flexibility were evaluated. The simulation results for the scenarios based on a modified IEEE-RTS-96 revealed that skewness is more effective than kurtosis. The proposed method can help system operators to efficiently respond to changes in the uncertainty characteristics of renewable energy resources. [ABSTRACT FROM AUTHOR]

The performance of a free-space optical (FSO) communications link suffers from the deleterious effects of weather conditions and atmospheric turbulence. In order to better estimate the reliability and availability of an FSO link, a suitable distribution needs to be employed. The accuracy of this model depends strongly on the atmospheric turbulence strength which causes the scintillation effect. To this end, a variety of probability density functions were utilized to model the optical channel according to the strength of the refractive index structure parameter. Although many theoretical models have shown satisfactory performance, in reality they can significantly differ. This work employs an information theoretic method, namely the so-called Jensen–Shannon divergence, a symmetrization of the Kullback–Leibler divergence, to measure the similarity between different probability distributions. In doing so, a large experimental dataset of received signal strength measurements from a real FSO link is utilized. Additionally, the Pearson family of continuous probability distributions is also employed to determine the best fit according to the mean, standard deviation, skewness and kurtosis of the modeled data.

The urgency of the issue is caused by needs of improving the accuracy and simplifying the approximation of experimental data laws for bilateral distribution. The main aim of the study: modernization of the Pearson method, which eliminates some of its disadvantages and simplifies the procedure for approximation of bilateral distribution laws of experimental data, taking both positive and negative values. The methods used in the study: calculations using the methods of probability theory and statistics, as well as the software MathCAD; methods of integral and differential calculus. The results: The authors have proposed the modernized Pearson distributions to approximate distribution laws of experimental data, taking positive and negative values, which can significantly simplify the procedure of approximation. Topographic classification of modernized Pearson distributions with use of coefficient of joint asymmetry and excess kurtosis instead of coefficient of excess is designed. The paper introduces the formulas for calculating numerical characteristics of the modernized Pearson distributions.

The Hilbert–Huang transform (HHT) extracts the intrinsic oscillation modes of input data, and estimates instantaneous amplitude (IA) and frequency (IF) for each mode. The HHT is applied to detection of some anomaly structures of signals as well as to analysis of signals. However, only qualitative discussions have been conducted on the applications to the detections. To make more statistically-based arguments on the application of the HHT, we investigated the probability distribution of the means of IA and IF for white Gaussian noise and found that it fits the Pearson distribution rather than the normal distribution. We defined a feature value for an anomaly detection by using the probability density function estimated on the basis of the Pearson distribution. Our method does not require different models for different lengths of the segment over which the mean is calculated, and therefore it is useful especially for the case that the length cannot be fixed. [ABSTRACT FROM AUTHOR]

In this paper, R control chart for positively skewed distributions is designed. For chosen exponential, gamma and Weibull distributions of quality characteristic, we calculated theoretical distribution of sample range and fitted corresponding Pearson distribution. Upper control limits and power of the R control chart were established, giving evidence of the goodness of fit of Pearson distributions to the theoretical distribution of sample range. For implementation of R control chart in practice, measures of sample skewness and kurtosis were compared and numerical example of construction of a proposed chart was given. [ABSTRACT FROM AUTHOR]

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable. [ABSTRACT FROM AUTHOR]

Statistical Moment (SM) based modelling is a quite straightforward approach in stochastic modelling of uncertain structures. However, the method still has deficiencies including determination of SMs of natural frequencies of vibratory structures, and it has not been tested yet for realistic structures. This study aims such verification by employing high-degree statistical moments in stochastic equations. In this respect, SM approach is applied for two different uncertainty cases. In the first case, uncertain parameters are experimentally reproduced from the batch of laminated composite beams. Then, those uncertainties are fed to SM equations used in finite element model to obtain descriptive statistical quantities (mean, variance, skewness, and kurtosis) of stochastic natural frequencies. Next, Pearson model is utilized to obtain probability density function of the natural frequency by using standardized SMs. Beside this, uncertain fundamental natural frequency of fifty samples of composite beams is measured by experimental modal tests. All SM based predictions and modal test results are also compared with numerical Monte Carlo Simulations. The latter case examines composite beams having nonnormal uncertain thickness. Since the results are in good harmony with each other, it is concluded that high-order SM based approach may effectively be used in uncertainty modelling of realistic structures. [ABSTRACT FROM AUTHOR]

Discretization is a common decision analysis technique for which many methods are described in the literature and employed in practice. The accuracy of these methods is typically judged by how well they match the mean, variance, and possibly higher moments of the underlying continuous probability distribution. Previous authors have analyzed the accuracy of differing discretization methods across a limited set of distributions drawn from particular families (e.g., the bell-shaped beta distributions). In this paper, we extend this area of research by (i) using the Pearson distribution system to consider a wide range of distribution shapes and (ii) including common, but previously unexplored, discretization methods. In addition, we propose new three-point discretizations tailored to specific distribution types that improve upon existing methods. [ABSTRACT FROM AUTHOR]

This "From the Editors" column is coauthored by new Editor-in-Chief Rakesh K. Sarin and former Editor-in-Chief L. Robin Keller. Our first two articles are on techniques for approximating probability distributions. The first article, by Robert K. Hammond and J. Eric Bickel is on "Reexamining Discrete Approximations to Continuous Distributions." Next, Luis V. Montiel and J. Eric Bickel present a method for "Approximating Joint Probability Distributions Given Partial Information." The next two articles present methods for analyzing terrorist threats or measures for terrorist protection. Kevin Ni, Daniel Faissol, Thomas Edmunds, and Richard Wheeler present a new method for "Exploitation of Ambiguous Cues to Infer Terrorist Activity." The next article is on "Passenger Profiling and Screening for Aviation Security in the Presence of Strategic Attackers" by Huseyin Cavusoglu, Young Kwark, Bin Mai, and Srinivasan Raghunathan. Our final paper is on "WholeSoldier Performance Appraisal to Support Mentoring and Personnel Decisions," by Robert A. Dees, Scott T. Nestler, and Robert Kewley. [ABSTRACT FROM AUTHOR]

In this paper, we propose to analyze the binary integration of the cell-averaging constant false-alarm rate (CA-CFAR) and order statistics constant false-alarm rate (OS-CFAR) detectors in the presence of non-Gaussian spiky clutter modeled as a Pearson distribution. We derive new closed form expressions for false alarm and detection probabilities for the CA-CFAR detector in the presence of Pearson-distributed clutter backgrounds. We first show that the use of binary integration improves the detection probabilities of the detectors considered. Secondly, the maximum of detection probability occurs for an optimum choice when the second threshold is set to be equal to M = (3/4) L. For this optimum M-out-of- L rule, the comparison analysis of the CA-CFAR and OS-CFAR binary integrators showed that the latter has better performance in homogeneous Pearson- distributed clutter. [ABSTRACT FROM AUTHOR]

The generation of multivariate probability distributions follows several approaches. Within financial applications the emphasis has mostly been on two methodologies. The first is the elliptical methodology, where the leap from univariate to multivariate has taken place by constructing density functions that are functions of quadratic forms of the marginals. The second is the copula philosophy, where the dependency structure is treated entirely separately from marginals. In financial applications one often needs to work with combinations of marginals of various distributional types, and the copula philosophy is very attractive as it copes well with heterogeneous marginals. However, with some notable exceptions, the copula approach does not normally correspond to any natural or canonical multivariate structure arising from some underlying dynamic. This paper presents an approach to multivariate distribution theory that allows for heterogeneous marginals but without the limitations of the elliptical ansatz or the arbitrariness of the copula approach. The approach is to consider multivariate distributions as arising naturally from coupled stochastic differential equations (SDEs), and in this paper we consider a first step based on the equilibrium situation. The scope of this paper is where the marginals are one of the classic Pearson family of distributions, allowing for great diversity among the marginals. For simplicity we present details for the bivariate case. Using a recently developed quantile form of the Fokker-Planck equation it is first shown how the members of the Pearson family are associated with various types of one-dimensional SDE in a transparent manner. Then appropriate two-dimensional SDEs and the associated Fokker-Planck equations are considered. In the equilibrium limit these give solutions that are natural bivariate structures. Some examples with marginals drawn from Gaussian, Student and one-sided exponential are explored. Some possible generalizations are outlined. [ABSTRACT FROM AUTHOR]

Abstract: A drought index is defined based on the inland river runoff from Hexi corridor, Gansu Province, Northwest China. The relationship between this drought index and yields of spring wheat is examined. The data used in this study include the following: monthly runoff data during 1959–2004 from the hydrological stations at Changmapu, Yingluoxia and Jiutiaoling on the three representative inland rivers in the Hexi corridor belt; the yields of spring wheat, monthly temperature and precipitation from three agro-meteorological stations in Jiuquan, Zhangye and Wuwei; and monthly precipitation data from three meteorological stations at Tole, Qilian and Menyuan. The runoff, following the Pearson type III distribution, is normalized to translate into the standard normal distribution as function of Z. Z is the variable in normalizing process. According to the characteristic of standard normal distribution of Z, a runoff-denoted drought index (Z rd hereafter) is defined. The grades of Z rd are determined by the standard normal distribution theoretical frequencies of Z. In order to validate the drought grades, runoff drought indices are compared with the atmospheric dryness indices determined by the precipitation. Results indicate that the division of drought categories based on runoff is rational and thus reliable. The Z rd, as it considers both temperature and rainfall in upstream mountains, is close to reality. Based on the relationship between the drought grade and water usage, suggestions are made for irrigation in the area. Five grades of Z rd are then translated into 4 grades of runoff irrigation drought index (Z ir). The relationship between Z ir and tendency-free yield (i.e. climate yield) of spring wheat from the Hexi Corridor irrigation zones at station Jiuquan, Zhangye and Wuwei is investigated. Results show that the Z ir represents an anti-phase trend of the climate yields of the crops. As a conclusion, Z ir can be utilized to qualitatively predict the trend of the wheat yield. An empirical yield model is created using a multi-variable regression method. [Copyright &y& Elsevier]

• SM analysis is enhanced to determine the third and fourth central statistical moments. • SM analysis is also implemented for the structures having a local variable parameter. • As an application, SMA is used to determine probability density function of eigenvalues of cracked structures. • Mathematical model of the cracked structure is constructed by Rayleigh-Ritz method. • The results are compared with Monte Carlo simulations. • The results yield that the SM analysis is a promising method for the analysis of the structures having input parameter variation. Statistical moment analysis proved its accuracy on the determination of mean and variance of free and forced vibration response for the structures having normally distributed global input and output parameters. Apart from the mean and variance, higher order standardized central statistical moments (CSMs) i.e., skewness and kurtosis, must also be evaluated for the nonnormal distributions to obtain probability density function of random variables. In this study, statistical moment analysis is enhanced to calculate higher order CSMs in order to evaluate the distribution of eigenvalues of beams. The method is applied for intact and cracked beams having variable parameters. For the intact beam case, higher order CSMs of eigenvalues are determined corresponding to the normally distributed global variable parameter, i.e., Young's modulus. The latter application of the method is on the cracked beams having a nonnormal variable local parameter, i.e., crack depth. For this case, the method is tested by modelling the crack depth by two different distributions. In this regard, firstly, expressions of CSMs are analytically derived for the mathematical operations (summation and multiplication) of two statistical variables. Then, these expressions are fed to mathematical model (constructed via Rayleigh Ritz method) of cracked beams to calculate statistical moments of eigenvalues. Next, distributions of the eigenvalues corresponding to the variable cracked depth are obtained by utilizing CMSs in Pearson distribution. The results are compared with Monte Carlo simulation and present unique advantages in the sense of computational cost for the structures having variable parameters. [ABSTRACT FROM AUTHOR]

This paper deals with the constant false alarm rate (CFAR) radar detection of targets embedded in Pearson distributed clutter. We develop new CFAR detection algorithms-notably cell averaging (CA), greatest of selection (GO) and smallest of selection SO-CFAR operating in Pearson measurements based on a non-linear compression method for spiky clutter reduction. The technique is similar to that used in non uniform quantization where a different law is used. It consists of compressing the output square law detector noisy signal with respect to a non-linear law in order to reduce the effect of impulsive noise level. Thus, it can be used as a pre-processing step to improve the performance of automatic target detection especially in lower generalised signal-to-noise ratio (GSNR). The performance characteristics of the proposed CFAR detectors are presented for different values of the compression parameter. We demonstrate, via simulation results, that the pre-processed compression procedure is computationally efficient and can significantly enhance detection performance. [ABSTRACT FROM AUTHOR]

Adhesion failure is a significant reliability concern and a major barrier to commercializing MEMS devices. In this paper, a beam-peel-based experimental setup was improved to measure the adhesion energy between a microcantilever and a substrate at different humidity levels. The microcantilever arrays were separated from the substrate and their fixed ends were directly attached to a piezoactuator to control the beam displacement with sub-nanometer accuracy. The experimental setup was sealed in a chamber with precise humidity control. An in situ interferometer was used to measure the beam deflection and crack length during the peel test. To examine the effects of surface roughness and relative humidity on adhesion energy, four different surface pairs were measured at humidity levels ranging from 40 to 92%. Before testing, the microcantilevers and substrates were scanned using an Atomic Force Microscope. Surface roughness parameters and the exact probability density function of the asperity heights were extracted and directly entered into a statistics-based model. An Extended-Maugis–Dugdale single asperity meniscus model considering both asperity deformation and solid surface interaction was coupled with the Pearson surface statistical model to develop an improved elastic asymmetrical surface meniscus model. The model compared favorably with the experimental data. [ABSTRACT FROM AUTHOR]