Cumulative Density Function Research Articles

Characterization of MmWave Full-Duplex Cloud-Radio Access Network (C-RAN) with RRH Selection for 5G and Beyond

In this paper, a millimeter-wave (mmWave) full-duplex (FD) cloud radio access network (C-RAN) for 5G and beyond systems is studied, where the central unit is equipped with multiple antennas, and the spatially distributed remote radio heads (RRHs) operate in FD mode. In particular, we analyze a relay selection method based on end-to-end signal to interference plus noise ratio (SINR) in the proposed system. Distinguishing features of mmWave communications such as Rician small scale fading, path loss, blockage and directivity are taken into account in both fronthaul (FH) and access links. In order to have a comparison basis, the performance of the RRH selection method is also investigated in half-duplex (HD) mmWave C-RAN. Initially, the expression of the end-to-end SINR in the proposed system is derived, and then the cumulative density function (CDF) of the end-to-end SINR is derived in both FD and HD RRHs. Subsequently, we derive performance expressions such as outage probability, average rate and energy efficiency for the RRH selection method in the proposed system in both HD and FD mmWave C-RAN. Finally, the effectiveness of the RRH selection method and the accuracy of the obtained results are verified via Monte Carlo simulation results, which show that the RRH selection method in the mmWave FD C-RAN improves the performance significantly.

L-SC receiver over equally correlated TWDP channels

In this paper, we have derived novel expressions of the probability density function (PDF), the cumulative density function (CDF), and the Moment Generating Function (MGF) of the output signal to noise ratio (SNR) of the L-branch Selection Combining (L-SC) receiver in an equally correlated Two Wave Diffused Power (TWDP) fading environment. Further, we have calculated the Outage Probability and Average Symbol Error Rate (ASER) of the L-SC receiver to evaluate its performance. The numerical values have been assessed and analyzed in the results section. Verifications of the obtained results were done with the help of Monte Carlo simulation.

Intervened Exponential Distribution: Properties and Applications

This manuscript aims to study the intervention-based probability model. Statistical and reliability properties such as the expressions for, cumulative density function (CDF), mean deviations about mean and median, rth order central and non-central moments, ”generation functions” for moments have been derived. Moreover, the expression for reliability function, hazard rate, reverse hazard rate, aging intensity, mean residual life function, stress-strength reliability, and entropy metrics due to R´enyi and Shannon are also derived. Monte Carlo simulation study performance of maximum likelihood estimates (MLEs) has been carried out, followed by calculations of Average Bias (ABias), and Mean Square Error (MSE). The applicability of the model in real-life situations has been discussed by analyzing the two real-life data sets.

Probabilistic analysis of blast–obstacle interaction in a crowded internal environment

Recent terror events such as the Manchester Arena bombing and Brussels and Istanbul airport attacks featured improvised explosive devices detonated in crowded internal spaces. A blast wave that propagates in the presence of obstacles will have fundamentally different properties to those of an unimpeded blast wave. Physical processes such as reflection, diffraction, and superposition of multiple wave fronts result in highly complex and situational-dependent loading characteristics which cannot be predicted using simple tools such as those for predicting free-field blast parameters. The influence of blast–obstacle interaction within an internal environment has not yet been studied. This article uses computational fluid dynamics within a probabilistic framework to quantify the influence of obstacle density and positioning on blast loading characteristics. Two mechanisms which alter the properties of a blast wave are studied: ‘channelling’ and ‘shielding’. It is shown that channelling effects are highly localised and result in increased loading near the explosive, the effect of which increases with obstacle density. Shielding is shown to be a cumulative effect which increases with distance from the explosive, and with increasing obstacle density. Whole-domain cumulative density functions are used to derive quantitative descriptors of the loading characteristics and how they vary relative to simple benchmark cases, with a view to providing clear guidance on the development of future predictive tools.

Mixed Dual-Hop IRS-Assisted FSO-RF Communication System With H-ARQ Protocols

Intelligent reflecting surface (IRS) is an emerging key technology for the fifth-generation (5G) and beyond wireless communication systems to provide more robust and reliable communication links. In this letter, we propose a mixed dual-hop free-space optical (FSO)-radio frequency (RF) communication system that serves the end user via a decode-and-forward (DF) relay employing hybrid automatic repeat request (H-ARQ) protocols on both hops. Novel closed-form expressions of the probability density function (PDF) and cumulative density function (CDF) of the equivalent end-to-end signal-to-noise ratio (SNR) are computed for the considered system. Utilizing the obtained statistics, we derive the outage probability (OP) and packet error rate (PER) of the proposed system by considering generalized detection techniques on the source-to-relay (S-R) link with H-ARQ protocol and IRS having phase error. We obtain useful insights into the system performance through the asymptotic analysis which aids to compute the diversity gain. The derived analytical results are validated using Monte Carlo simulation.

Sec-G Class of Distributions: Properties and Applications

Although there are many continuous distributions in the literature, only a handful take advantage of the modeling power provided by trigonometric functions. To our knowledge, none of them are based on the so-called secant function, defined as the reciprocal of the cosine function. The secant function can go to large values whenever the cosine function goes to small values. The idea is to profit from this trigonometric property to modify well-known distribution tails and overall skewness features. With this in mind, in this paper, a new class of trigonometric distributions based on the secant function is introduced. It is called the Sec-G class. We discuss the main mathematical characteristics of this class, including series expansions of the corresponding cumulative distribution and probability density functions, as well as several probabilistic measures and functions. In particular, we present the moments, skewness, kurtosis, Lorenz, and Bonferroni curves, reliability coefficient, entropy measure, and order statistics. Throughout the study, emphasis is placed on the unique four-parameter continuous distribution of this class, defined with the Kumaraswamy-Weibull distribution as the baseline. The estimation of the model parameters is performed using the maximum likelihood method. We also carried out a numerical simulation study and present the results in graphic form. Three referenced datasets were analyzed, and it is proved that the proposed secant Kumaraswamy-Weibull model outperforms important competitors, including the Kumaraswamy-Weibull, Kumaraswamy-Weibull geometric, Kumaraswamy-Weibull Poisson, Kumaraswamy Burr XII, and Weibull models.

The F- Distribution with Applications to Number of Female Patients with Cancer

The F-distribution is a particular parameterization of the beta prime distribution, which is also called the beta distribution of the second kind. This considered a new four-parameter generalized version of the Fisher Snedecor distribution called Beta- F distribution is introduced. The comprehensive account of the statistical properties of the new distributions was considered. Formal expressions for the cumulative density function, moments, and moment generating function and maximum likelihood estimation as well as its Fisher information were obtained.
 This study employed a descriptive survey research design. The target population was female patients with cancer in Lagos State University teaching Hospital for complete for 36 days. The data collected was analyzed using both descriptive statistics and presented in tables, charts and figures. Some well-known distributions were fit, some existing single distributions and the proposed F distribution to these data using the method of maximum likelihood was applied using Micro-Soft Excel 2007.
 The study established that the minimum AIC and minimum statistic on kolmogorov Smirnov of the F-distribution shows that female patients with cancer data better than the other life time distributions considered. The study concluded that Fisher-distribution is better off than existing single distribution with reference to Akaike Information Criterion (AIC) been used. It is instructive to note that the F-distribution preserves the originality of the data without transformation.

EFFECTIVE CAPACITY AS A MANAGEMENT AND ASSESSMENT TOOL FOR A HYDRO-POWER STATION

Management of large power stations in a developing nation like Nigeria is difficult due to inadequate technical expertise and funding available to plant managers, engineers and technicians. Often the effectiveness of the plants is couched in mystery as measures of “reliability”, “maintainability” and in aggregate terms “availability”, which are not easily understood by funding authorities. A more practical measure that can be easily interpreted by even the most technically uninformed, for the purpose of decision making is a new metric (Effective Capacity) that precisely represents the actual ability of the plant to power loads. Effective Capacity is determined for each unit by calculating its total energy output over a specified time and averaging it to determine the effective performance of the plant in that period. This method was applied quarterly in each year to the five turbo-alternators in the Jebba Hydropower Station (JHEPS) in Nigeria using the data collected over a total period of eleven years (2004 - 2014). The computed effective capacity for each turbo-alternator was studied by plotting the quarterly effective capacity, the frequency polygon of the quarterly effective capacity and their cumulative density functions over the said period. The results showed that the machines were working less than the rated values for most part of the period which implies under-utilization and wastage of substantial revenue - generation potential, making argument for additional funding of the station.

Optimal Solution of Reactive Power Dispatch in Transmission System to Minimize Power Losses Using Sine-Cosine Algorithm

Optimal reactive power dispatch (ORPD) has a crucial impact to enhance safety, reliability, and economical operation of the electric power system. ORPD is a non-linear, non-convex and mixed variable problem, which has been solved by many researchers via different meta-heuristic algorithms during the last decade. In this work, a novel algorithm named sine-cosine algorithm (SCA) is utilized to solve ORPD problem by considering both dependent and independent control variable constraints. SCA has been tested and validated on standard 14, 30 and 57-bus power systems. To validate the superiority of proposed algorithm, the outcomes obtained through SCA are compared with recent published results attained through particle swarm optimization (PSO), modified Gaussian barebones teaching–learning based optimization (BBTLBO), ant bee colony optimization (ABCO), whale optimization algorithm (WOA) and backtracking search algorithms (BSA). The results attained using SCA show the improvement in the power losses minimization. Thus, with standard 14-bus system, the power losses are minimized from 0.04% to 4.78%. While, using standard 30-bus, the power losses are minimized from 0.4% to 3.4% and with standard 57-bus, power losses are reduced from 0.9% to 1.99%. Furthermore, a comparative analysis with 30 independent runs on the above-mentioned bus systems is performed to examine the functioning of the proposed method in terms of probability density function (PDF) and cumulative density function (CDF). For such analysis, well-known meta-heuristic algorithms such as PSO, WOA, differential evolution (DE) are compared with proposed SCA in solving the ORPD problem. The results of this analysis clearly show that proposed algorithm is robust, effective, and computationally easy in solving the ORPD problem compared to the existing meta-heuristic algorithms.

An approximation to the inverse of left-sided truncated gaussian cumulative normal density function using Polya's model to generate random variates for simulation applications

The Gaussian or normal distribution is vital in most areas of industrial engineering, including simulation. For example, the inverse of the Gaussian cumulative density function is used in all simulation software (e.g., ARENA, ProModel) to generate a group of random numbers that fit Gaussian distribution. It is also used to estimate the life expectancy of new devices. However, the Gaussian distribution that is truncated from the left side is not defined in any simulation software. Estimation of the expected life of used devices needs left-sided truncated Gaussian distribution. Additionally, very few works examine generating random numbers from left-sided truncated Gaussian distribution. A high accuracy mathematical-based approximation to the left-sided truncated Gaussian cumulative density function is proposed in the current work. Our approximation is built based on Polya's approximation of the Gaussian cumulative density function. The current model is beneficial to approximate the inverse of the left-sided truncated Gaussian cumulative density function to generate random variates, which is necessary for simulation applications.

Aircraft total turnaround time estimation using fuzzy critical path method

Airport Collaborative Decision Making (ACDM) flight scheduling plans rely on accurate predictions of both optimal and reliable aircraft turnaround times, which is one of the most difficult things to complete. In order to account for the effects of randomness and fuzziness on the turnaround process duration, this paper deals with transforming the probability distribution of well-fitted sub-processes into a cumulative density function, which is equivalent to the fuzzy membership function (FMF), and then using goodness-of-fit to determine the fuzzy membership grade of each turnaround sub-process. The turnaround time is calculated using the fuzzy critical path method (FCPM), which is a combination of the Critical Path Method (CPM) and fuzzy set theory. In order to verify this estimate, we created the FMF using historical data and compared it to turnaround times based on FCPM. A linear regression model is used to examine the relationship between arrival delays and the FCPM-based turnaround process. We also use historical data to generate fuzzy sets of different arrival delays using Frankfurt airport data from the summer of 2017 and conclude that delays are positively correlated with the FCPM-based turnaround process.

Statistics of the Stokes Parameters for the Signal Scattered by Layered Structures With an Arbitrary Number of Slightly Rough Interfaces

Horizontally stratified structures are commonly used to represent naturally occurring structures, such as soils. The electromagnetic signature of such a medium illuminated by a radar and the polarization state of the scattered wave are fully determined by the knowledge of the four Stokes parameters. In this paper, we determine the statistics of the four Stokes parameters for the signal scattered by layered structures with an arbitrary number of slightly rough interfaces. The rough interfaces are realizations of second-order stationary centered Gaussian stochastic processes and the layered structure is illuminated by an elliptically polarized monochromatic wave. The zenithal and azimuthal components of the far scattered electric field are derived from the first-order small perturbation method. The derivation leads to a multivariate Gaussian model for the underlying complex scattered amplitudes and we establish the closed-form expressions of the probability density function, the cumulative density function and the first- and second-order moments for the four Stokes parameters. For an observation direction outside the incidence plane, we establish the condition on the incidence wave parameters for which the zenithal and azimuthal components are uncorrelated. For an air / snow cover / frozen soil / unfrozen soil structure, we analyze the marginal probabilities and validate the theory by comparison with Monte-Carlo simulations. More generally, when the two complex components of the field scattered by the illuminated zone are Gaussian random variables, these statistics offer possibilities for in-depth investigating the polarization of scattering processes from random media.

On the boundary properties of Bernstein estimators on the simplex

Abstract In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the d-dimensional simplex. Our results generalize those found by Leblanc (2012b), who treated the case d = 1, and complement the results from Ouimet (2021a) in the interior of the simplex. Since the “edges” of the d-dimensional simplex have dimensions going from 0 (vertices) up to d − 1 (facets) and our kernel function is multinomial, the asymptotic expressions for the bias, variance and mean squared error are not straightforward extensions of one-dimensional asymptotics as they would be for product-type estimators studied by almost all past authors in the context of Bernstein estimators or asymmetric kernel estimators. This point makes the mathematical analysis much more interesting.

On Statistical Properties of a New Bivariate Modified Lindley Distribution with an Application to Financial Data

There is an increasing interest in expanding the one‐parameter Lindley distribution to two‐parameter, three‐parameter, and five‐parameter. The univariate one‐parameter Lindley distribution is still one of the most applicable distributions in data analysis especially in lifetime data. Modeling dependent random quantities required bivariate parametric probability distributions. This study presents a new bivariate three‐parameter probability distribution called bivariate modified Lindley distribution. The one‐parameter modified Lindley distribution is used as a base line to construct the new model. Its statistical properties including cumulative function, density function, marginals, moments, conditional distributions, and copula are discussed. Simulation is constructed to declare theoretical properties, to show the flexibility of the new model and to investigate the goodness of fit. Two sets of real data, financial data and UEFA Champion’s League data, are used to show the applicability of the proposed model for different types of data.

Reconfigurable Intelligent Surface-Assisted HAPS Relaying Communication Networks for Multiusers Under AF Protocol: A Performance Analysis

This research paper studies the performance analysis of a reconfigurable intelligent surface (RIS)-assisted high altitude platform station (HAPS) relaying communication network for multiple ground users. Owing to transmission blockage between the source and the destination, the source transmits its information to the HAPS via the RIS where the HAPS acts as an amplify-and-forward (AF) relay node for sending the source information to ground users. To characterize the transmission links, the channels between the source-RIS-HAPS are assumed to follow approximate generalized-K fading distribution while the links between the HAPS and multiple ground users experience Shadowing-Rician fading distribution. Since HAPS employs AF relaying scheme, the closed-form expressions of the cumulative distribution function (CDF) and probability density function (PDF) of the system end-to-end signal-to-noise ratio (SNR) are derived by exploiting the Meijer’s G and Fox’s H functions. Based on these expressions, the analytical expressions of the system outage probability, average bit error rate (BER) and average capacity are obtained to examine the system performance. Moreover, the asymptotic expressions are obtained to provide more physical insight about the system. In addition, the results unveil the significant effect of the system and channel parameters on proposed system. The results show that the system with RIS offers better performance compared the system without RIS. The analytical results are verified by Monte-Carlo simulation to justify the accuracy of the derived expressions.

An Occupancy Mapping Method Based on K-Nearest Neighbours.

OctoMap is an efficient probabilistic mapping framework to build occupancy maps from point clouds, representing 3D environments with cubic nodes in the octree. However, the map update policy in OctoMap has limitations. All the nodes containing points will be assigned with the same probability regardless of the points being noise, and the probability of one such node can only be increased with a single measurement. In addition, potentially occupied nodes with points inside but traversed by rays cast from the sensor to endpoints will be marked as free. To overcome these limitations in OctoMap, the current work presents a mapping method using the context of neighbouring points to update nodes containing points, with occupancy information of a point represented by the average distance from a point to its k-Nearest Neighbours. A relationship between the distance and the change in probability is defined with the Cumulative Density Function of average distances, potentially decreasing the probability of a node despite points being present inside. Experiments are conducted on 20 data sets to compare the proposed method with OctoMap. Results show that our method can achieve up to 10% improvement over the optimal performance of OctoMap.

The pie sharing problem: Unbiased sampling of N+1 summative weights

A simple algorithm is provided for randomly sampling a set of N+1 weights such that their sum is constrained to be equal to one, analogous to randomly subdividing a pie into N+1 slices where the probability distribution of slice volumes are identically distributed. The cumulative density and probability density functions of the random weights are provided. The algorithmic implementation for the random number sampling are made available. This algorithm has potential applications in calibration, uncertainty analysis, and sensitivity analysis of environmental models. Three example applications are provided to demonstrate the efficiency and superiority of the proposed method compared to alternative sampling methods.

Assessing the performance of open-source, semi-automated pattern recognition software for harbour seal (P. v. vitulina) photo ID

Photographic identification (photo ID) is a well-established, non-invasive, and relatively cost-effective technique to collect longitudinal data from species that can be individually recognised based on natural markings. This method has been improved by computer-assisted pattern recognition software which speed up the processing of large numbers of images. Freely available algorithms exist for a wide range of species, but the choice of software can have significant effects on the accuracy of individual capture histories and derived demographic parameter estimates. We tested the performance of three open source, semi-automated pattern recognition software algorithms for harbour seal (Phoca vitulina vitulina) photo ID: ExtractCompare, I3S Pattern and Wild-ID. Performance was measured as the ability of the software to successfully score matching images higher than non-matching images using the cumulative density function (CDF). The CDF for the top ranked potential match was highest for Wild-ID (CDF1 = 0.34–0.58), followed by ExtractCompare (CDF1 = 0.24–0.36) and I3S pattern (CDF1 = 0.02–0.3). This trend emerged regardless of how many potential matches were inspected. The highest performing aspects in ExtractCompare were left heads, whereas in I3S Pattern and Wild-ID these were front heads. Within each aspect, images collected using a camera and lens performed higher than images taken by a camera and scope. Data processing within ExtractCompare took > 4 × longer than Wild-ID, and > 3 × longer than I3S Pattern. We found that overall, Wild-ID outperformed both ExtractCompare and I3S Pattern under tested scenarios, and we therefore recommend its assistance in harbour seal photo ID.

Kumaraswamy-Janardan Distribution: A Generalized Janardan Distribution with Application to Real Data

The quest to improve on flexibility of probability distributions motivated this research. Four-parameter Janardan generalized distribution known as Kumaraswamy-Janardan distribution is proposed through method of parameterization and studied. The probability density function, cumulative density function, survival rate function as well as hazard rate function of the distribution are established. Statistical properties such as moments, moment generating function as well as maximum likelihood of the model are discussed. The parameters are estimated using the simulated annealing optimization algorithm. Flexibility of the model in comparison with the baseline model as well as other competing sub-models is verified using Akaike Information Criteria (AIC). The model is tested with real data and is proven to be more flexible in fitting real data than any of its sub-models considered.

A Sensitivity Analysis Approach for Assessing the Effect of Design Parameters in Reducing Seismic Demand of Base-Isolated Storage Racks

The most used global sensitivity analysis (GSA) method is based on variance. This is performed using Monte Carlo Sampling (MCS) or Latin Hypercube Sampling (LHS). It requires a large sample to obtain accurate estimates. Density-based methods, such as the GSA PAWN, have been developed to reduce the sample size without compromising the result. PAWN is simpler than other methods because it uses cumulative density functions (CDF) instead of probability density. This method has been widely used in areas such as environmental engineering with very good results, reducing computation time. However, its use in structural engineering is incipient. The PAWN method was used to classify the design variables of the isolation system in relation to their sensitivity, and in relation to the seismic response of industrial storage racks. The above was analyzed in terms of the effectiveness of each variable to reduce the seismic demand using a novel base isolation kinematic device (BIKD). Racks with different combinations of their structural parameters such as the number of storage levels, the height between them, and isolation period, among others, were studied. The dimensions of the racks were chosen to match those that would later be experimentally tested on shaking table. An earthquake whose response spectrum matched the design spectrum of current Chilean regulations, was considered as seismic forcing. The maximum base shear load, the displacement of the top level of storage and the floor drift were considered as target responses to be studied. Fixed base racks (FBR), as reference, and base-isolated racks (BIR) were analyzed. The results showed the effectiveness of using the BIKD system in reducing all three-target responses up to one order of magnitude. Additionally, it was determined that the parameters that have the greatest influence on the response correspond to the number of storage levels and the height between them, both for FBR and BIR.