Type Of Probability Density Function Research Articles

Statistical Behavior of Lepton Pair Spectrum in the Drell-Yan Process and Signal from Quark-Gluon Plasma in High-Energy Collisions

We analyze the transverse momentum ( p T ) spectra of lepton pairs ( ℓ ℓ ¯ ) generated in the Drell-Yan process, as detected in proton-nucleus (pion-nucleus) and proton-(anti)proton collisions by ten collaborations over a center-of-mass energy s N N or s if in a simplified form) range from ~ 20 GeV to above 10 TeV. Three types of probability density functions (the convolution of two Lévy-Tsallis functions, the two-component Erlang distribution, and the convolution of two Hagedorn functions) are utilized to fit and analyze the p T spectra. The fit results are approximately in agreement with the collected experimental data. Consecutively, we obtained the variation law of related parameters as a function of s and invariant mass Q . In the fit procedure, a given Lévy-Tsallis (or Hagedorn) function can be regarded as the probability density function of transverse momenta contributed by a single quark ( q ) or anti-quark ( q ¯ ). The Drell-Yan process is then described by the statistical method.

Assessment of a flamelet approach to evaluating mean species mass fractions in moderately and highly turbulent premixed flames

Complex-chemistry direct numerical simulation (DNS) data obtained from lean methane-air turbulent flames are analyzed to perform a priori assessment of predictive capabilities of the flamelet approach to evaluating mean concentrations of various species in turbulent flames characterized by Karlovitz numbers Ka=6.0, 74.0, and 540. Six definitions of a combustion progress variable c are probed and two types of probability density functions (PDFs) are adapted: (i) actual PDFs extracted directly from the DNS data or (ii) presumed β-function PDFs obtained using the DNS data on the first two moments of the c-field. Results show that the mean density, the mean temperature, and the mean mass fractions of CH4, O2, H2O, CO2, CO, CH2O, CH3, and HCO are very well predicted using the temperature-based combustion progress variable cT and the actual PDF. For other considered species, the quantitative predictions are worse but still appear to be encouraging (with the exception of CH3O at Ka=540). The use of the flamelet library obtained from the equidiffusive laminar flame improves results for H2, HO2, and H2O2 at the highest Karlovitz number. Alternative definitions of the combustion progress variable perform worse and the reasons for this are explored. The use of the β-function PDF yields worse results for intermediate species such as OH, O, H, CH3, and HCO, with this PDF being significantly different from the actual PDF. Application of the flamelet approach to rates of production/consumption of various species is also addressed and implications of obtained results for modeling are discussed.

Analysis of Process Variations in W-Band GaN MMIC PAs Using Nonparametric Statistics

This article presents an analysis of the effects of amplifier process-related performance variation on the radiation patterns of a W-band active transmit antenna array. Based on measured on-wafer S-parameters for about 80 amplifier chips designed in an experimental 90-nm gallium nitride (GaN) on SiC process, a statistical analysis of the array patterns is performed. Several types of probability density functions (pdfs) are generated from measured data and compared to determine which statistical approach is most relevant. We find that a joint distribution that maintains the correlation structure between |S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> | and ∠S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> is important for an accurate analysis. The monolithic microwave integrated circuits (MMICs) are connected to the waveguide-fed horn antenna elements in the array via microstrip-to-waveguide transitions fabricated on alumina. The transitions are analyzed in full-wave simulations with fabrication tolerances included in the analysis. Finally, the MMIC and transition statistical variations are cascaded, resulting in a quantitative evaluation of spatial power combining. Given a random choice of power amplifier chips in a 4×4 array, the EIRP is shown to vary by ±3 dB at 94 GHz.

Computational Methods in Problems of Subjective Assessment of Investment Decisions Effectiveness

The paper describes the implementation of investment projects under conditions of uncertainty. In the developed mathematical model, the effectiveness of an investment project is evaluated by the NPV index. This index is considered a random variable that can be estimated by an investor to within a segment [NPV1;NPV2]. The main difficulties of the decision-making process arise when the segment [NPV1;NPV2] includes zero value. In the developed model, we use the probability density function of NPV value in the form of Pearson curves of the first type. This paper discusses in detail some particular moments, which are to be taken into consideration while choosing a specific type of probability density function of NPV.&#x0D; The main element of the proposed model is the subjective utility function. Many questions regarding the usage of this function in practice are also extensively reviewed in this article. The main requirement for successful usage of the subjective utility function in real life is a well-calculated chapter of scenario analysis of an investment project. This chapter is present in almost all modern business plans. The practical application of the developed mathematical model improves the quality of decisions concerning the investment of funds into projects.

Surface rupture in stochastic slip models

SUMMARYAs an alternative to spectral methods, stochastic self-similar slip can be produced through a composite source model by placing a power-law scaling size-frequency distribution of circular slip dislocations on a fault surface. However these models do not accurately account for observed surface rupture behaviour. We propose a modification to the composite source model that corrects this issue. The advantage of this technique is that it accommodates the use of fractal slip distributions on non-planar fault surfaces. However to mimic a surface rupture using this technique, releasing the boundary condition at the top of the fault, we observed a systematic decrease in slip at shallow depths. We propose a new strategy whereby the surface is treated like a reflector with the slip being folded back onto the fault. Two different techniques based on this principal are presented: the first is the method of images. It requires a small change to pre-existing codes and works for planar faults. The second involves the use of a multistage trilateration technique. It is applied to non-planar faults described by an unstructured mesh. The reflected slip calculated using the two techniques is near identical on a planar fault, suggesting they are equivalent. Applying this correction, where reflected slip is accounted for in the composite source model, the lack of slip at shallow depths is not observed any more and there is no systematic trend with depth. However, there are other parameters which may affect the spatial distribution of slip across the fault plane. For example, the type of probability density function used in the placement of the subevent is also important. In the case where the location of maximum slip is known to a first order, a Gaussian may be appropriate to describe the probability function. For hazard assessment studies a uniform probability density function is more suitable as it provides no underlying systematic spatial trend.

Capability indices for circular tolerance regions based on a Gaussian copula

Positional characteristics can be found in many manufactured products. This type of quality characteristics requires the use of capability indices to measure their performance regarding a circular tolerance region and the center of this region. Different indices have been proposed in the literature. However, the main assumption of these indices is that the positional characteristic follows a bivariate normal distribution. In this sense, two process capability indices are proposed in this paper. These indices are based on the Gaussian copula in the aims of allowing the marginal components to follow any type of probability density function. Thus, these copula-based indices can be applied when the normality assumption cannot be followed. The construction of the indices is based on the Cpm and Cpmk indices, given the interest that the positional characteristics to be close to the center of the tolerance region. A simulation study is carried out to analyze the performance of the proposed indices, and a case study about the position of a pin is also analyzed. It is shown in the results of both studies that the proposed indices can be used to measure the performance of positional characteristics with circular tolerance regions.

Reliability analysis of H-section steel columns under blast loading

In order to prevent structural collapse by terrorist attacks, the column behavior is highlighted since when a column is damaged consequently losing its load bearing capacity, failure will spread to the rest of the structure resulting in the collapse of the entire structure. In this paper, using the reliability theory and taking into account the uncertainties associated with blast loading and material properties, a methodology is improved for determining the damage probability of a steel column under various blast scenarios. Damage is determined according to damage index based on axial load-bearing capacity. The Monte Carlo simulation method is utilized using an explicit finite element method to obtain the failure probability. The results are described in terms of damage probability, effects of boundary conditions, effect of type of probability density function and sensitivity analyses. The results show the importance of considering the uncertainties in assessing the damage index of the columns against explosion. Also, it is found that in the flexural mode the probability of damage decreases as the support condition changes from pinned to fixed ends. Furthermore, the safe protective distances are estimated based on the concept of damage probability. Likewise, results for log-normal random variables are relatively identical with truncated normal random variables. The results of the sensitivity analyses show that damage index has the highest sensitivity to peak reflected pressure.

Fishnet model for failure probability tail of nacre-like imbricated lamellar materials

Nacre, the iridescent material of the shells of pearl oysters and abalone, consists mostly of aragonite (a form of CaCO3), a brittle constituent of relatively low strength ([Formula: see text]10 MPa). Yet it has astonishing mean tensile strength ([Formula: see text]150 MPa) and fracture energy ([Formula: see text]350 to 1,240 J/m2). The reasons have recently become well understood: (i) the nanoscale thickness ([Formula: see text]300 nm) of nacre's building blocks, the aragonite lamellae (or platelets), and (ii) the imbricated, or staggered, arrangement of these lamellea, bound by biopolymer layers only [Formula: see text]25 nm thick, occupying [Formula: see text] of volume. These properties inspire manmade biomimetic materials. For engineering applications, however, the failure probability of [Formula: see text] is generally required. To guarantee it, the type of probability density function (pdf) of strength, including its tail, must be determined. This objective, not pursued previously, is hardly achievable by experiments alone, since [Formula: see text] tests of specimens would be needed. Here we outline a statistical model of strength that resembles a fishnet pulled diagonally, captures the tail of pdf of strength and, importantly, allows analytical safety assessments of nacreous materials. The analysis shows that, in terms of safety, the imbricated lamellar structure provides a major additional advantage-∼10% strength increase at tail failure probability [Formula: see text] and a 1 to 2 orders of magnitude tail probability decrease at fixed stress. Another advantage is that a high scatter of microstructure properties diminishes the strength difference between the mean and the probability tail, compared with the weakest link model. These advantages of nacre-like materials are here justified analytically and supported by millions of Monte Carlo simulations.

A Geometrical Look at MOSPA Estimation Using Transportation Theory

It was shown in [6] that the Wasserstein distance is equivalent to the Mean Optimal Sub-Pattern Assignment (MOSPA) measure for empirical probability density functions. A more recent paper [7], extends on it by drawing new connections between the MOSPA concept, which is getting a foothold in the multi-target tracking community, and the Wasserstein distance, a metric widely used in theoretical statistics. However, the comparison between the two concepts has been overlooked. In this letter we prove that the equivalence of Wasserstein distance with the MOSPA measure holds for general types of probability density function. This non trivial result allows us to leverage one recent finding in the computational geometry literature to show that the Minimum MOPSA (MMOSPA) estimates are the centroids of additive weighted Voronoi regions with a specific choice of the weights.

A Generalized Approach for Enhanced Solar Energy Harvesting Using Stochastic Estimation of Optimum Tilt Angles: A Case Study of Bangkok City

Abstract This paper estimates the monthly, seasonal and yearly optimal tilt angles that maximize solar irradiation received on an inclined surface, thus enhancing the energy harvested. The uncertainties in global and diffuse radiations on the horizontal surface are accounted using stochastic analysis of their daily statistical measured data. The measured data is taken over a seven-year time span. The study is carried out for south-facing flat plate solar collectors at the Bangkok site, Thailand, situated in the northern hemisphere. The position of the sun at any time and location is predicted by the mathematical procedure of Julian dating. Further, four isotropic and anisotropic sky models are used to evaluate the ratio of diffuse solar radiation on an inclined surface to that on a horizontal surface. The best sky model is opted by using the ranking method of errors and is used for further analysis. Moreover, the frequency distributions of global and diffuse radiations are studied using four types of probability density functions (PDFs), viz., Weibull, lognormal, gamma and beta. Kolmogorov–Smirnov (K-S) test is used as a criterion to find the best fit among the aforementioned PDFs. The proposed optimization problem is solved using particle swarm optimization with time-varying acceleration coefficients (PSO-TVAC). The monthly tilt angles obtained are found to be varying with respect to the corresponding latitude angle, suggested by the thumb rule. The collected energy using this tilt angle over a period of one year is found to be 588 M J / m 2 $588 \,{\rm{MJ/}}{{\rm{m}}^2}$ in excess to that harvested using the latitude angle.

Heterogeneous Mixture Distributions for Modeling Multisource Extreme Rainfalls*

Abstract Frequency analysis has been widely applied to investigate the behavior and characteristics of hydrometeorological variables. Hydrometeorological variables occasionally show mixture distributions when multiple generating phenomena cause the extreme events to occur. In such cases, a mixture distribution should be applied. Past studies on mixture distributions assumed that they are drawn from the same probability density functions. In fact, many hydrometeorological variables can consist of different types of probability density functions. Research on heterogeneous mixture distributions can lead to improvements in understanding the behavior and characteristics of hydrometeorological variables and in the capacity to model them properly. In the present study heterogeneous mixture distributions are developed to model extreme hydrometeorological events. To fit heterogeneous mixture distributions, the authors present an extension of the metaheuristic maximum likelihood approach. The performance of the parameter estimation method employed was verified through simulation tests. The fits of nonmixture, homogeneous mixture, and heterogeneous mixture distributions were evaluated through the application to a real-world case study of the extreme rainfall events of South Korea. Results indicate that the heterogeneous mixture distribution is a good alternative when sources possessing dissimilar statistical characteristics influence extreme hydrometeorological variables.

A methodology for quantifying risk and likelihood of failure for carbon dioxide injection into deep saline reservoirs

Tectonically stable deep saline reservoirs are considered to be the most abundant carbon dioxide (CO2) sequestration sites. Pressure, temperature, salinity and other characteristics of these geologic formations vary at each injection site. It is essential to understand the roles of geologic and engineering factors to optimize the CO2 injection conditions and to quantify the associated risks. Factors such as the magnitude of injection-induced reservoir pressure, quantity of supercritical phase CO2 that comes in contact with caprock, and the amount of residually trapped CO2 govern the fate of CO2, and provide quantitative assessment of the storage integrity.A streamlined protocol was developed using response surface methodology to quantify the risks related to CO2 injection. The proposed methodology includes the design of simulation scenarios, selection and screening of parameters, multiple-linear regression of outcomes, and the development of probability density functions (PDF) of various potential risk factors. Multiphase numerical simulations were performed to understand the behavior of the injected CO2 and associated parameters in deep saline reservoirs with prescribed geometries and petrophysical properties. Formation thickness, formation depth, porosity, horizontal permeability, brine density, and the end-point residual CO2 saturation were the six critical parameters identified that affected important outcomes. A six-factor Box–Behnken experimental design procedure was used to establish an understanding of the sensitivity of the parameters on the important factors, and for subsequently establishing response surfaces. Closed boundary domains with different operational constraints were employed. A stepwise, sequential regression method was used to determine statistically significant coefficients of a response surface model. Monte Carlo simulations with logical distributions of input parameters were performed using the response surface coefficients. Uncorrelated and correlated porosity-permeability distributions were used to generate two types of probability density functions (PDF). PDFs of CO2 plume extent under the caprock and average reservoir overpressure after injection were generated given all of the variability in the input parameters. These results will allow initial screening of a large number of potential injection sites without detailed simulations of each.

Deterministic Approach to Estimate the Upper Bound of the Electric Field in a Reverberation Chamber

In this study, we propose a deterministic approach to estimate the upper bound of the electric field in a reverberation chamber (RC), where the field is well stirred. That approach is based on the plane wave integral representation for fields in an RC, assuming the ergodicity of the first two moments of the amplitude of the angular spectrum, which includes an energy constraint. Also, we consider further hypotheses on a sample function of the angular spectrum, such as a conditional phase and polarization, and the type of probability density function (PDF) of its amplitude. First, we calculate the maximum value of the electric field as a function of its root mean square value and of the inverse of the variation coefficient of the PDF of the considered sample function's amplitude. Then, we consider both the realistic values and the extreme value of the inverse of the variation coefficient in order to achieve the upper bound of the electric field as an extreme event of a statistical model with finite energy. Finally, we achieve an upper bound of the electric field as a function of its root mean square value only and we point out that it is consistent with respect to the results available in bibliography, both experimental and theoretical.

MAXIMUM RESPONSE ESTIMATION BY STOCHASTIC DYNAMIC RESPONSE ANALYSIS CONSIDERING UNCERTAINTIES OF NATURAL CIRCULAR FREQUENCY AND DAMPING FACTOR

This paper aims at estimating the maximum dynamic response of the structure with time-invariant random natural circular frequency and damping factor subjected to random excitations. In the stochastic response analyses, the λ-PDF probability density function and the Gegengauer polynominal approximation is used to deal with the bounded parameter uncertainties. In order to estimate maximum responses, the 4th moment and the beta distribution model are used. Two types of probability density function for parameter uncertainties are studied, and evaluation method of high-order moment is introduced in each case. A series of numerical analyses show that the proposed method estimates the maximum response quite accurately. The method is applied to calculating response spectra considering parameter uncertainties to investigate the required accuracy of the damping factor in the seismic design.

Phase statistics approach to human ventricular fibrillation

Ventricular fibrillation (VF) is known to be the most dangerous cardiac arrhythmia, frequently leading to sudden cardiac death (SCD). During VF, cardiac output drops to nil and, unless the fibrillation is promptly halted, death usually ensues within minutes. While delivering life saving electrical shocks is a method of preventing SCD, it has been recognized that some, though not many, VF episodes are self-terminating, and understanding the mechanism of spontaneous defibrillation might provide newer therapeutic options for treatment of this otherwise fatal arrhythmia. Using the phase statistics approach, recently developed to study financial and physiological time series, here, we reveal the timing characteristics of transient features of ventricular tachyarrhythmia (mostly VF) electrocardiogram (ECG) and find that there are three distinct types of probability density function (PDF) of phase distributions: uniform (UF), concave (CC), and convex (CV). Our data show that VF patients with UF or CC types of PDF have approximately the same probability of survival and nonsurvival, while VF patients with CV type PDF have zero probability of survival, implying that their VF episodes are never self-terminating. Our results suggest that detailed phase statistics of human ECG data may be a key to understanding the mechanism of spontaneous defibrillation of fatal VF.

Caracterização do regime de evapotanspiração real, em escala decendial, no estado de São Paulo

A seca agrícola está diretamente relacionada ao déficit entre o valor de evapotranspiração real (ETR) estimado em um certo período e o valor climatologicamente esperado desse parâmetro no mesmo período. Conseqüentemente, tais valores esperados de ETR são dependentes das características climáticas de cada região. Assim, esse estudo objetivou verificar, por meio do teste Kolmogorov-Smirnov, o ajuste de séries de ETR decendias no estado de São Paulo às distribuições gama, normal, log-normal e beta. O cálculo dos balanços hídricos climáticos teve como base dados diários de temperatura do ar e precipitação pluvial de 13 localidades do estado. O uso da função densidade probabilidade beta é recomendado para a caracterização da variabilidade temporal do regime de ETR e, portanto, para o desenvolvimento de um índice de seca padronizado que permita monitorar a evolução desse fenômeno em escala temporal e espacial.

Effect of spatial variation of textural layers on regional field water balance

The spatial variability of textural layers is a major factor influencing the field water and solute transport in alluvial soils. To quantify the water transport process at a regional scale accurately, one has to take the spatial variability of textural layers into account. In this paper, a recently presented Markov chain simulation model for soil textural profiles was coupled with a deterministic field water balance model to conduct a stochastic analysis of the field water balance in a 15 km2 alluvial soil region. The aim is to assess the effect of spatial variability of textural layers on the field water balance at a regional scale. By combining simulated soil textural profiles with the field water balance model, the mean values, extreme values, and probability distributions of field water balance variables were calculated. Results showed large differences in the magnitude of soil water balance variables between different profiles. The extreme difference in the water storage in 1 m depth soil during the winter wheat growth varied with time between 175 mm and 180 mm, which accounted for 86.5–135.0% of the mean of the soil water storage. This indicates that the soil water balance variables derived from only a few soil profiles are not representative for the situation of field water balance in the entire region. The simulated root water uptake showed different types of probability density functions when the soil water storage and the deficit of soil water in the field changed with time. The simulated water storage in 1 m depth soil showed obviously a lognormal distribution, but the measured data showed an approximate normal distribution. It may be heterogeneous irrigation and cropping and some other factors that induce this discrepancy.

Batch process schedule optimization under parameter volatility

A general strategy to treat the uncertainties in parameters of batch process scheduling has been developed. The strategy consists of three algorithms: flexible planning, flexible scheduling and reactive schedule adaptation. In this paper, we introduce the flexible scheduling and the reactive schedule adaptation algorithms. The flexible scheduling algorithm is based on both a Monte Carlo simulation and a simulated annealing. It can deal with multiple uncertain parameters and any type of probability density function for the uncertain parameter. We seek the flexible schedule that maximizes the expected profit, including net present values of products less raw material and processing costs, as well as due date penalties, inventory costs and setup costs. In reality, the values of uncertain parameters always change after the batch process schedule and plan are set up. Since the flexible schedule has periods of free time that can be used to accommodate uncertainties during the actual production, the reactive schedule adaptation algorithm can modify the flexible schedule in response to any change in an uncertain parameter with little or no penalty. This algorithm finds a new optimal or suboptimal solution under the new condition by using a combination of different local search methods, which are described.

Effects of NO Models on the Prediction of NO Formation in a Regenerative Furnace

A study of the effects of nitric oxide (NO) models on the prediction of NO formation in a gas-fired regenerative furnace with highly preheated air was undertaken. Three chemical kinetic processes for NO formation/depletion, i.e., thermal NO, prompt NO, and NO reburning, are included. In the thermal NO model, the sensitivity encountered when using two different approaches, namely the equilibrium approach and the partial equilibrium approach, for determining the O radical concentration was studied. The effects of the third reaction in the thermal NO mechanism, NO reduction (reburning) mechanism, and different types of probability density functions (PDFs) on the NO predictions have also been tested. The sensitivity of the excess air ratio on the NO generation rate in the furnace has been investigated. Finally, the impact of the temperature on the NO formation rate in the regenerative furnace was discussed. [S0195-0738(00)00304-6]

Understanding of Relationship Between the Average Mass Transport Rate And The Moments Of Permeability

AbstractTo estimate the transport rate of radionuclides in the geosphere, we must consider the spatial variability of permeability. However, the borehole data of permeability are limited and we can not determine the type of probability density function, though the measurement data reflect the most significant hydraulic properties about geologic media including innumerable cracks or fast flow paths. While the recent models describing radioactive nuclide transport in near/far-field have assumed a certain probability density function (typically a lognormal distribution) as a permeability distribution, we cannot always obtain sufficient measurement data to define the function. However, the available data of permeability at least give us the moments such as the arithmetic mean, the standard deviation and the skewness for the distribution.The purpose of this paper is to get an understanding of the general relationship between the average mass transport rates and the moments. Using various types of probability density functions and pseudo random-numbers, hypothetical permeability distributions are generated. With these distributions, this paper obtains the average transport rates described as the numerical impulseresponse based on the advection-dispersion model for a two-dimensional region. The calculated results show that, for the dimensionless standard deviation up to around 1, the three moments are enough to characterize the permeability distribution for the purposes of the nuclide transport prediction.In this work, for five specified probability density functions, the upper and lower bounds of skewness are derived as a function of the dimensionless arithmetic mean and standard deviation. The obtained upper and lower bounds explicitly show that the Bernoulli trials (a discrete probability density function) yield the widest range in the skewness against the standard deviation. Since the response has lower peak and longer tail as the skewness goes to the lower bound value, we can predict the shape of the breakthrough curve from the three moments of the borehole data.