Gauss–Seidel Fixed-Point Approach for Maximum Likelihood Estimation in Epanechnikov–Burr XII Distributions

Oil recovery forecast relies on predictions of saturation fields from reservoir simulations. These predictions, in return, are dependent on the quality of the input data available, such as the reservoir permeability and porosity fields. However, because of inherent uncertainties in these input data, due to sparse and possibly inaccurate measurements, a stochastic framework is usually adopted to describe their variability. Probability distributions are then assigned to the input fields. In this stochastic context, predicting the saturation fields consists in determining the saturation probability distributions. For nonlinear immiscible two-phase flow in highly heterogeneous porous media, we propose an efficient distribution method to estimate the saturation cumulative distribution function (CDF) and probability density function (PDF). While high heterogeneity is traditionally regarded as an obstacle to designing stochastic methods, this distribution method exploits the physics of highly heterogeneous porous media. The method identifies hidden smooth random fields (RFs) that explain the saturation distribution, together with the nonlinear mapping from these hidden RFs to the saturation. We show that the statistics and distributions of these hidden RFs can be estimated very efficiently and simply, leading to a method that is computationally inexpensive compared to full Monte Carlo (MC) simulations. We provide numerical evidence of the agreement between MC-based and distribution-based saturation CDF, PDF and moments, in high heterogeneous cases such as high variance lognormal permeability fields, anisotropic layered fields and channelized systems. Finally, we show how the saturation CDF can be used for risk assessment by relying not only on the mean and standard deviation of the saturation, but primarily on saturation quantiles, obtained from probability of exceedance.

The truncated version of the two-parameter Sujatha distribution is analysed. In particular, its probability density function and distribution function are obtained. The results are applied to the initial mass function for stars, to the luminosity function for galaxies, to the number of galaxies as a function of the redshift and to the average absolute magnitude of a galaxy as a function of its redshift.

The use of aerial platforms has recently attracted significant interest in Internet of Things (IoT)-based heterogeneous communication networks to support flexible and efficient network connection. This work investigates the performance of a ground-aerial autonomous aerial vehicle (AAV)-assisted hybrid power line communication (PLC)/free-space optical (FSO) communication system employing both decode-and-forward (DF) and fixed-gain amplify-and-forward (AF) relaying protocols. It is assumed that the PLC channel gain is Log-normally distributed while the Bernoulli-Gaussian random process models the PLC channel noise. The FSO link established between the FSO transmitter and the AAV considers the combined influence of the generalized Málaga distributed atmospheric turbulence (AT), random fog, nonzero boresight pointing errors, and Angle-of-Arrival (AoA) fluctuations of the AAV. Based on this, we evaluate the cumulative distribution function (CDF) and probability density function (PDF) of the end-to-end signal-to-noise ratio (SNR). Further, capitalizing on the statistical characterization of CDF and PDF, we derive series-based expressions for average symbol error rate (ASER), outage probability (OP), and average channel capacity (ACC) utilizing both heterodyne detection (HD) and direct detection (DD) techniques. Additionally, we conduct an asymptotic analysis of ASER in terms of the coding gain and diversity order in a high SNR regime. Likewise, a detailed analysis is carried out to find the optimal value of the field-of-view (FOV) angle and average SNR to achieve minimum ASER under the combined influence of the aforementioned channel conditions using the DF relaying scheme. Furthermore, the impact of all the considered channel parameters and impairments is demonstrated through the numerical results.

This paper optimizes human resource management from the perspective of marginal utility. The utility function proposes the subtraction function of consumption, which represents the law of diminishing marginal utility. The Lagrange multiplier method solves the maximum utility problem. In human resource management, utility and marginal utility are redefined, and the probability density function and distribution function of exponential distribution are used as the index’s marginal utility and utility functions, respectively. Human resource management optimization models I and II were constructed based on the marginal utility, and the models and model parameters were solved. This paper applies the human resource management model to Company R as an example. The calculation shows that the total investment of the current index resources of Company R is 2.55 million yuan, and the total utility calculation is 0.8306. Under the optimization of human resource management based on Model I, the expected value of company R reached 0.8631 while maintaining the unchanged index resource input, an increase of 3.91%. The optimization based on Model II resulted in the index resource input being 2.3285 million yuan, with an unchanged total utility that was reduced by 8.69%.

Bayesian system reliability assessment method with maximum entropy as prior distribution is developed in the present analysis. The system reliability assessment model is established, and the method and process of system reliability assessment are put forward. The method is based on the theory of information entropy. The probability density function and distribution function of system reliability parameters are given by using maximum entropy method. Prior distribution of Bayesian analysis method is the distribution function. Likelihood function is given by a small amount of field test data. According to Bayesian methods, posterior distribution is determined by using prior distribution and likelihood function. The point estimation and interval estimation of system reliability parameters are calculated in order to evaluate the system reliability. The method solves better the problem of system reliability assessment for small sample cases, and is an optimum method for reliability assessment of large and complex systems.

Data uncertainty is ubiquitous in real-world applications, and the number of variables such as imprecise estimation, network latency, obsolete sources, and sampling errors leads to data uncertainty. It is important to carefully deal with this kind of ambiguity, or else the mining results might be inaccurate or even incorrect. We have therefore used a new method called bagging and boosting algorithms for decision tree for the classification and prediction algorithm. As a model of sequential decision problems under uncertainty, a decision tree may be used. To manage data with uncertain information, classifiers are used. To deal with data confusion, we primarily use the PDF distribution method. It is the method in which the complete data are carried out by the distributions of probability to construct a decision tree. Supply these values to the decision tree by calculating the PDF and accordingly generate the decision tree, and on the other hand, supply the bagging and boosting algorithms with PDF and then compare both outcomes. Bagging is one of the well-known techniques of the ensemble, which constructs data bags with the same class mark and the same probability of the original data collection. More complex strategies, such as boosting the distribution of the sample, rely on how difficult it is to identify each sample. Instead of abstracting unknown data by statistical derivatives, if we use the probability density function (PDF) distribution, the accuracy of a decision tree can be much better. The dataset is initially provided to the PDF generation in our project, we provide it to the bagging and boosting algorithm in the next stage instead of giving it directly to the decision tree, then, the generated dataset is given to the decision tree, and then, the classification is carried out.KeywordsBaggingBoostingNumerical uncertain dataDecision treeClassifiersEnsemblesRandom forestANN

SER performance investigation of a MPPM relay-aided FSO system with three decision thresholds over EW fading channel considering pointing errors

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.

Latin Hypercube Sampling (LHS), unlike other simulation methods such as Monte Carlo Simulation (MCS), has not enjoyed wide applications in determination of reserve despite its capability to realize same accuracy as with the latter even with much fewer simulation trials. The parameters used in these distributions are extracted using software ‘digitizer’ from different plot of values derived from measurements of reservoir and fluid properties. This work dwells on analysis of two depleted gas fields for which the ultimate recovery is known. The distribution of porosity, water saturation, reservoir thickness and reserve size are input to crystalbal software to generate cumulative distribution function (CDF) and Probability density function (PDF) of original gas in place (OGIP) for each of the fields. The accuracy of Latin Hypercube Sampling is analysed by comparing the actual ultimate recovery and the OGIP predicted by LHS. The calculated recovery factors of 68.7% and 93.3%, for water driven and depletion driven gas reservoirs respectively, compare favourably with those of known gas reservoirs. Therefore, LHS is a good estimator of gas reserve. It is recommended that similar study should be carried out on condensate and oil reservoir, particularly on Niger Delta fields.

Uncertainty analysis, which assesses the impact of the uncertainty of input random variables on performance functions, is an important and indispensable component in engineering design under uncertainty. In this paper, a simulation method based on the Saddlepoint Approximation (SPA) is proposed to estimate accurately and efficiently the distribution of a response variable. The proposed method combines both simulation and analytical techniques and involves three main steps: (1) sampling on input random variables, (2) approximating the cumulant generating function (cgf) of the response variable with its first four cumulants and (3) estimating the cumulant distribution function (cdf) and probability density function (pdf) of the response variable using the SPA. This method provides more computationally efficient solutions than the general Monte Carlo Simulation (MCS) while maintaining high accuracy. The effectiveness of the proposed method is illustrated with a mathematical example and two engineering analysis problems.

In this paper, an analytical framework is provided to analyze the energy coverage performance of unmanned aerial vehicle (UAV) energy harvesting networks with clustered user equipments (UEs). Locations of UAVs are modeled as a Poison Point Process (PPP), while locations of UEs are modeled as a Poisson Cluster Process (PCP). Two different models are considered for the line-of-sight (LOS) probability function to compare their effect on the network performance. Moreover, ultra-wideband (UWB) antennas with doughnut-shaped radiation patterns are employed in both UAVs and UEs, and the impact of practical 3D antenna radiation patterns on the network performance is also investigated. Initially, the complementary cumulative distribution function (CCDF) and probability density function (PDF) of path losses for each tier are derived. Subsequently, association probabilities with each tier are obtained. Energy coverage probability is derived for the entire network using tools from stochastic geometry. Via numerical results, we show that UAV height and antenna orientation play crucial roles on the energy coverage performance.

Representations of noncentral chi-square cumulative distribution function and probability density function are reviewed and new repre¬sentations are given. One representation of the cdf in terms of an integral is easily computed on any machine which has an accurate algorithm for computing the normal cdf.

A K-tier heterogeneous downlink millimeter wave (mmWave) cellular network with user-centric small cell deployments is studied in this paper. In particular, we consider a heterogeneous network model with user equipments (UEs) being deployed according to a Poisson Cluster Process (PCP), i.e., Thomas cluster process, where the UEs are clustered around the base stations (BSs) and the distances between UEs and the BS are modeled as Gaussian distributed. In addition, distinguishing features of mmWave communications including directional beamforming and a sophisticated path loss model incorporating both line-of-sight (LOS) and non-line-of-sight (NLOS) transmissions, are taken into account. In this paper, the complementary cumulative distribution function (CCDF) and probability density function (PDF) of the path loss are provided. Also, using tools from stochastic geometry, we derive a general expression of the signal-to-interference-plus-noise ratio (SINR) coverage probability. Our results demonstrate that coverage probability can be improved by decreasing the size of UE clusters around BSs, and interference has noticeable influence on the coverage performance of our model.

It is well known that the shear strength parameters [cohesion (c) and friction angle (ϕ)] are important parameters for evaluating deformation and stability of geotechnical structures, such as slope stability, bearing capacity of foundations, and earth pressure of retaining walls. As far as the reliability analysis of these geotechnical structures is concerned, the shear strength parameters are typically treated as uncertain parameters (Griffiths et al. 2011; Cherubini 2000; Abd Alghaffar and Dymiotis-Wellington 2007). Furthermore, it is widely accepted that c and ϕ are negatively correlated in the literature (e.g., Low 2007; Li et al. 2011; Tang et al. 2012, 2013a). To evaluate the reliability of geotechnical structures exactly, the joint cumulative distribution function (CDF) or probability density function (PDF) of shear strength parameters should be known. It is concluded by the previous studies (e.g., Low 2007; Li et al. 2011; Tang et al. 2012, 2013a) that the negative correlation between cohesion and friction angle has a significant effect on geotechnical reliability and ignoring such a correlation would lead to an overestimate of the probability of failure.

In this paper, we consider relay selection based decode-and-forward (DF) cooperative communication over free space optical (FSO) links. A relay selection scheme is considered in which source selects a relay on the basis of highest instantaneous signal-to-noise ratio (SNR) of the source-relay links. We derive the cumulative distribution function (CDF) and probability density function (PDF) of the highest instantaneous SNR of Gamma-Gamma distributed source-relay links. The average error performance of the DF protocol with relay selection over the Gamma-Gamma fading channels is derived in terms of power series. Coding gain and diversity order of the relay selection based DF scheme are analytically obtained. It is inferred by the analysis and simulation that the performance of the DF FSO system can be significantly improved by using the simple relay selection scheme considered in this paper.