Reconciling statistical models with practical experience of reverberation chambers

The three-condition version of the uniform conditional state combustion model makes use of the mixture fraction, progress variable, and normalized total enthalpy as conditioning variables to build a three-dimensional conditional manifold for chemistry. In order to map the solution in conditional space into the flow domain, the joint Probability Density Function (PDF) of the conditioning variables needs to be modeled. In simulations, presumed functions (i.e., β-PDF for the mixture fraction and progress variable and δ-PDF for total enthalpy) are often used for modeling the marginal PDFs. In this work, the measurements from the Cambridge/Sandia burner are employed to obtain the marginal PDFs for the conditioning variables at various points in the reacting domain. The measurements are then combined from all positions in space to form conditional PDFs of the normalized total enthalpy for various values of the other two variables. In the vicinity of the flame brush, the marginal PDF of the normalized total enthalpy resembles a bimodal Gaussian distribution; nonetheless, the conditional PDFs for this variable are nearly Gaussian distributions. The correlation coefficients between the conditioning variables are also investigated, and the assumption of their statistical independence is examined. To consider the association between the conditioning variables for modeling, the copula concept is introduced, and the performances of three different copulas are tested. Furthermore, the statistical moments of the conditioning variables are computed from the experimental data at different points and are utilized for modeling the joint PDF of the conditioning variables from two different approaches that are compared.

<p class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph; line-height: 120%; layout-grid-mode: char; mso-layout-grid-align: none; margin: 12.0pt 0cm 6.0pt 0cm;"><span lang="EN-US" style="font-size: 10.0pt; mso-bidi-font-size: 11.0pt; line-height: 120%; font-family: 'Times New Roman',serif;">Variability characterization is a key component in the reliability assessment of geotechnical systems, particularly in scenarios involving tailings deposits, where spatial heterogeneity can critically impact design safety. Despite advances in modeling spatial randomness, many reported cases still rely on theoretical assumptions to define appropriate statistical characterizations—such as theoretical probability density functions (PDFs) and correlation structures—which may misrepresent site-specific conditions. To bridge this gap, piezocone penetration tests (CPTu) stand out as promising tools for providing continuous measurements along a vertical profile that can be used to define statistical behavior and avoid bias. This paper presents a spatial variability characterization of a bauxite tailings deposit based on mechanical parameters derived from CPTu data. The study includes basic statistical analysis—mean (<em>μ</em>), standard deviation (</span><span lang="EN-US" style="font-size: 10.0pt; mso-bidi-font-size: 11.0pt; line-height: 120%; font-family: Symbol;">s</span><span lang="EN-US" style="font-size: 10.0pt; mso-bidi-font-size: 11.0pt; line-height: 120%; font-family: 'Times New Roman',serif;">), coefficient of variation (CV)—alongside a comparison with theoretical PDFs. Subsequently, spatial correlation is evaluated through covariance analysis and estimation of the vertical scale of fluctuation (</span><em><span lang="EN-US" style="font-size: 10.0pt; line-height: 120%; font-family: 'Times New Roman',serif;">δ</span></em><span lang="EN-US" style="font-size: 10.0pt; mso-bidi-font-size: 11.0pt; line-height: 120%; font-family: 'Times New Roman',serif;">), using a dedicated subroutine that fits theoretical autocorrelation models (TAMs). The deposit is classified as highly variable according to the I<sub>cRW</sub> index. The normal and Weibull PDFs best represent the data distributions. The vertical scales of fluctuation vary significantly: 0.01 m to 4.43 m for cone resistance (<em>q</em><sub>c</sub>), 0.01 m to 4.36 m for sleeve friction (<em>f</em><sub>s</sub>), and 0.01 m to 5.00 m for pore water pressure (<em>u</em><sub>2</sub>). These findings offer valuable input for probabilistic stability and serviceability analyses, contributing to safer and more informed geotechnical designs involving mine tailings. </span></p>

Probability density functions of the involved random variables are essential for the reliability-based design of offshore structures. The objective of present research was the derivation of probability density function (PDF) for the local joint flexibility (LJF) factor, fLJF, in two-planar tubular DK-joints commonly found in jacket-type offshore structures. A total of 162 finite element (FE) analyses were carried out on 81 FE models of DK-joints subjected to two types of axial loading. Generated FE models were validated using available experimental data, FE results, and design formulas. Based on the results of parametric FE study, a sample database was prepared for the fLJF values and density histograms were generated for respective samples based on the Freedman-Diaconis rule. Nine theoretical PDFs were fitted to the developed histograms and the maximum likelihood (ML) method was applied to evaluate the parameters of fitted PDFs. In each case, the Kolmogorov-Smirnov and chi-squared tests were used to evaluate the goodness of fit. Finally, the Inverse Gaussian model was proposed as the governing probability distribution function for the fLJF. After substituting the values of estimated parameters, two fully defined PDFs were presented for the fLJF in tubular DK-joints subjected to two types of axial loading.

The polarisation ratio (PR) of an electromagnetic wave can provide information on the scattering from natural targets or on the emission from radio sources. In this study, a dual-polarisation radar operating at simultaneous reception mode is taken into account. Statistical properties of the PR of the radar signals on the basis of complex Gaussian models with zero and non-zero means are derived, respectively. Especially for the non-zero-mean case, the probability density functions (PDFs) and moments of the amplitude distribution in a single polarimetric channel are expressed in closed forms. PDFs of the amplitude ratio and the phase difference between orthogonal polarimetric channels are also derived. Comparisons of statistical properties are presented between theoretical PDFs and simulated data, as well as real polarimetric radar data. The well fitted results demonstrate the advantages of the PR models.

A method to compute several scalar quantities of Cosmic Microwave Background maps on the sphere is presented. We consider here four type of scalars: the Hessian matrix scalars, the distortion scalars, the gradient related scalars and the curvature scalars. Such quantities are obtained directly from the spherical harmonic coefficients (alm) of the map. We also study the probability density function of these quantities for the case of a homogeneous and isotropic Gaussian field, which are functions of the power spectrum of the initial field. From these scalars it is posible to construct a new set of scalars which are independent of the power spectrum of the field. We test our results using simulations and find a good agreement between the theoretical probability density functions and those obtained from simulations. Therefore, these quantities are proposed to investigate the presence of non-Gaussian features in CMB maps. Finally, we show how to compute the scalars in presence of anisotropic noise and realistic masks.

For the wind energy assessment, it is important to know the probability density distribution of the wind speed in order to calculate the mean power of a wind. In this study wind data, for the period from January 2000 to December 2002, hourly records of speed and direction at Alexandria coast are analyzed as random monthly observations. Statistical methods are used for determination of best fit random distributions of wind speed and direction using statistical standard software that helps in evaluation of wind speed energy potential in a study area. In the present paper, different theoretical Probability Density Functions (PDF) have been tried to determine the appropriate models, including Weibull distribution and their parameters. The minimum wind power density at Alexandria found in July and August (2000-2002) were 20.41 and 21.34 w/m² and the maximum was in December and January (110.81 and 123.34 w/m²). The most important distributions of the wind speed were Weibull and Dagum and the direction followed mostly the Johnson SB model.

We analyzed the probability density function (PDF) of velocity and temperature differences in the canopy sublayer of Amazonia based on Tsallis' generalized thermostatistics theory. We show that such a theory provides an accurate framework for modeling the statistical behavior of the inertial subrange above and below the canopy. For this, we compared the experimental PDFs with the theoretically predicted ones. The data were measured during the wet season of the Large‐Scale Biosphere‐Atmosphere Experiment in Amazonia (LBA), which was carried out during the months of January–March 1999 in the southwestern part of Amazonia region. Measurements were made simultaneously at different heights in a 60 m micrometeorological tower located in the Biological Reserve of Jaru (10°04′S, 61°56′W), Brazil. The fast response wind speed measurements, sampled at 60 Hz rate, were made using three‐dimensional sonic anemometers at the heights of 66 m (above the canopy) and 21 m (below the canopy). The results showed good agreement between experimental data measured above the canopy forest and Tsallis' generalized thermostatistics theory. For below canopy data, the agreement between experimental and theoretical PDFs was fairly good, but some distortion was observed. This is probably due to some peculiar characteristics of turbulent momentum transfer process inside the forest crown. Discussion is presented to explain these results. Conclusions regarding the absence of “universal scaling” in the inertial subrange are also presented in the context of the entropic parameter of Tsallis' theory.

The boresight and atmospheric jitter errors in a satellite tracking experiment are currently estimated by matching the probability density function (PDF) of the received signal counts with a set of PDFs of the signal for several combinations of jitter and boresight errors and then the best choice of jitter and boresight error is accepted via the chi-square test. Here a technique that can estimate atmospheric beam jitter and boresight error directly in a satellite active tracking experiment using the moments of the returns off the satellites is proposed. That is, we use the theoretical PDF for the signal return from a small target and compute the corresponding theoretical PDF moments. We can then form a few equations from these moments with only two unknowns, namely, the jitter and boresight. Solving for the unknowns is then unambiguous and very rapid. The method is valid for small physical cross-section targets and has been verified by using simulation and experimental data. Extending the case to asymmetric jitter and asymmetric boresight is possible.

The AcceptReject package, available for the R programming language on the Comprehensive R Archive Network (CRAN), versioned and maintained on GitHub, offers a simple and efficient solution for generating pseudo-random observations of discrete or continuous random variables using the acceptance-rejection method. This method provides a viable alternative for generating pseudo-random observations in univariate distributions when the inverse of the cumulative distribution function is not in closed form or when suitable transformations involving random variables that we know how to generate are unknown, thereby facilitating the generation of observations for the variable of interest. The package is designed to be simple, intuitive, and efficient, allowing for the rapid generation of observations and supporting multicore parallelism on Unix-based operating systems. Some components are written using C++, and the package maximizes the acceptance probability of the generated observations, resulting in even more efficient execution. The package also allows users to explore the generated pseudo-random observations by comparing them with the theoretical probability mass function or probability density function and to inspect the underlying probability density functions that can be used in the method for generating observations of continuous random variables. This article explores the package in detail, discussing its functionalities, benefits, and practical applications, and provides various benchmarks in several scenarios.

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

Environmental noise energy can be effectively captured by an energy harvester (EH) and then converted into usable electricity for low-powered monitoring sensors in aerospace vehicles. It is important to investigate the influence of potential well parameters on the energy harvesting performance under random excitation. Therefore, this paper theoretically derives the marginal probability density function (PDF), the joint PDF, and the averaging output power of the decoupled bistable energy harvester driven by the Gaussian white noise. First, the bistable energy harvester is linearized using the statistical linearization method, which is advantageous in obtaining coarse mean square values of outputs for further analysis. Second, the dimension of Fokker–Planck–Kolmogorov (FPK) equations is reduced by the state-space-split (SSS) method. Then, the approximate marginal and joint PDF can be obtained. Third, the influence of potential well width and height on PDFs and the averaging output power are, respectively, analyzed and discussed, respectively. Finally, the theoretical derivation is successfully validated by Monte Carlo simulations. Results show that the large potential well width can improve the averaging output power of the bistable EH under random excitation.

In this paper, we present a proportionate-type normalized least mean square algorithm which operates by choosing adaptive gains at each time step in a manner designed to maximize the joint conditional probability that the next-step coefficient estimates reach their optimal values. We compare and show that the performance of the joint maximum conditional probability density function (PDF) one-step algorithm is superior to the proportionate normalized least mean square algorithm when operating on a sparse impulse response. We also show that the new algorithm is superior to a previously introduced algorithm which assumed that the conditional PDF could be represented by the product of the marginal conditional PDFs, i.e., that the weight deviations are mutually conditionally independent.

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

Generalized thermostatistics and wavelet analysis of solar wind and proton density variability