Showing papers on "Cumulative distribution function published in 2019"

Meta Distribution of SIR in Large-Scale Uplink and Downlink NOMA Networks

Abstract: We develop an analytical framework to derive the meta distribution and moments of the conditional success probability (CSP), which is defined as success probability for a given realization of the transmitters, in large-scale co-channel uplink and downlink non-orthogonal multiple access (NOMA) networks with one NOMA cluster per cell. The moments of CSP translate to various network performance metrics such as the standard success or signal-to-interference ratio (SIR) coverage probability (which is the 1-st moment), the mean local delay (which is the −1st moment in a static network setting), and the meta distribution (which is the complementary cumulative distribution function of the success or SIR coverage probability and can be approximated by using the 1st and 2nd moments). For the uplink NOMA network, to make the framework tractable, we propose two point process models for the spatial locations of the inter-cell interferers by utilizing the base station (BS)/user pair correlation function. We validate the proposed models by comparing the second moment measure of each model with that of the actual point process for the inter-cluster (or inter-cell) interferers obtained via simulations. For downlink NOMA, we derive closed-form solutions for the moments of the CSP, success (or coverage) probability, mean local delay, and meta distribution for the users. As an application of the developed analytical framework, we use the closed-form expressions to optimize the power allocations for downlink NOMA users in order to maximize the success probability of a given NOMA user with and without latency constraints. Closed-form optimal solutions for the transmit powers are obtained for two-user NOMA scenario. We note that maximizing the success probability with latency constraints can significantly impact the optimal power solutions for low SIR thresholds and favor orthogonal multiple access.

Optimal UAV Route Planning for Coverage Search of Stationary Target in River

Abstract: In this brief, we focus on the offline route planning of unmanned aerial vehicle (UAV) for the coverage search mission in a river region. Given the prior likelihood distribution of area importance, this brief aims to generate UAV feasible routes which maximize the cumulative probability of finding a single and stationary target within the required time. First, Gaussian mixture model is used to approximate the prior likelihood distribution, and several river segments with high detection probability corresponding to Gaussian components can be extracted. With the consideration of quantified factors, the river subregions are prioritized by the approximation insertion method and then allocated to UAVs. Moreover, to meet the terminal time constraint, the so-called positive/negative greedy method is proposed to expand or contract waypoints. Finally, the performance of our proposed algorithm is evaluated by simulations on a real river map, and the results verify its good performance in various scenarios.

Joint Beamforming and Power Allocation for UAV-Enabled Full-Duplex Relay

Abstract: This paper investigates an unmanned aerial vehicle (UAV) enabled full-duplex relaying system. By assuming that the UAV follows a circular trajectory and applies decode-and-forward relaying strategy, we study the joint design of beamforming and power allocation to maximize the instantaneous data rate, under both the individual and the sum power constraints over the source and relay nodes. As the problem is non-convex, we propose an efficient sub-optimal solution based on block-coordinate descent method by decomposing the problem into two sub-problems: a beamforming optimization sub-problem with given power allocation and a power allocation sub-problem with fixed beamforming. For the beamforming design sub-problem, the optimal solution is obtained based on the semi-definite relaxation technique. For the power allocation sub-problem, the optimal solution is obtained in closed form. Then, the closed-form cumulative distribution function and outage probability expressions for sub-optimal beamforming with both uniform power allocation and optimal power allocation are derived. In addition, simple and informative high signal-to-noise ratio (SNR) approximations for outage probability expressions are presented to gain insights. Finally, the optimal flying altitude that minimizes the average outage probability is obtained via one-dimensional search.

CA-CFAR Detection Performance in Homogeneous Weibull Clutter

Abstract: This letter presents a novel and exact formulation for the probability of detection of a cell-averaging, constant false-alarm rate (CFAR) radar system operating in a homogeneous Weibull clutter environment. We consider a realistic scenario with both target returns and clutter residues within the cell under test by the radar processing. In passing, we derive novel closed-form expressions for the probability density function and the cumulative distribution function of the sum of an exponentially fluctuating target embedded in Weibull clutter. The derived exact expressions are given in terms of both: 1) bivariate Fox H-function, for which we provide a portable and efficient MATHEMATICA code and 2) easily computable series representations. The validity of all expressions is confirmed via Monte Carlo simulation. The derived results are compared with the idealized Neyman–Pearson detector so as to quantify the CFAR losses, and they indicate that even a small change in the shape parameter of the clutter distribution can significantly affect the radar detection performance.

Unified Analysis of HetNets Using Poisson Cluster Processes Under Max-Power Association

Abstract: Owing to its flexibility in modeling real-world spatial configurations of users and base stations (BSs), the Poisson cluster process (PCP) has recently emerged as an appealing way to model and analyze heterogeneous cellular networks (HetNets). Despite its undisputed relevance to HetNets—corroborated by the models used in the industry—the PCP’s use in performance analysis has been limited. This is primarily because of the lack of analytical tools to characterize the performance metrics, such as the coverage probability of a user connected to the strongest BS. In this paper, we develop an analytical framework for the evaluation of the coverage probability, or equivalently the complementary cumulative density function (CCDF) of signal-to-interference-and-noise ratio ( SINR ), of a typical user in a $K$ -tier HetNet under a $\max $ power-based association strategy, where the BS locations of each tier follow either a Poisson point process (PPP) or a PCP. The key enabling step involves conditioning on the parent PPPs of all the PCPs, which allows us to express the coverage probability as a product of sum-product and probability generating functionals (PGFLs) of the parent PPPs. In addition to several useful insights, our analysis provides a rigorous way to study the impact of the cluster size on the ${\it SINR}$ distribution, which was not possible using the existing PPP-based models.

On Performance of Hexagonal, Cross, and Rectangular QAM for Multi-Relay Systems

Abstract: Error performance is considered as one of the most important performance measures, and deriving the closed-form expressions for efficient modulation techniques over generalized fading channels is important for future cellular systems. In this paper, the performance of a dual-hop amplify-and-forward multi-relay system with best relay selection is analyzed over independent and non-identically distributed (i.n.i.d.) Nakagami-m fading links with both integer and non-integer fading parameters. The impact of practical constraints of imperfect channel state information (CSI) and non-linear power amplifier (NLPA) at each of the relays are considered. Closed-form expressions for the outage probability are derived for both integer and non-integer fading parameters, and asymptotic analysis on the outage probability is performed to obtain the diversity order of the considered multi-relay system. Based on the cumulative distribution function approach, average symbol error rate (ASER) expressions for general order hexagonal QAM, general order rectangular QAM, and 32-cross QAM schemes are also derived. The comparative analysis of ASER for various QAM schemes with different constellations is also illustrated. Furthermore, the impact of the number of relays, fading parameter, channel estimation error, and non-linear distortion on the system performance is also highlighted. Finally, the derived analytical results are validated through Monte-Carlo simulations.

On the Sum of Fisher-Snedecor F Variates and its Application to Maximal-Ratio Combining

Abstract: Capitalizing on the recently proposed Fisher-Snedecor F composite fading model, in this letter, we investigate the sum of independent but not identically distributed (i.n.i.d.) Fisher-Snedecor F variates. First, a novel closed-form expression is derived for the moment generating function of the instantaneous signal-to-noise ratio. Based on this, the corresponding probability density function and cumulative distribution function of the sum of i.n.i.d. Fisher- Snedecor F variates are derived, which are subsequently employed in the analysis of multiple branch maximal-ratio combining (MRC). Specifically, we investigate the impact of multipath and shadowed fading on the outage probability and outage capacity of MRC based receivers. In addition, we derive exact closed-form expressions for the average bit error rate of coherent binary modulation schemes followed by an asymptotic analysis which provides further insights into the effect of the system parameters on the overall performance. Importantly, it is shown that the effect of multipath fading on the system performance is more pronounced than that of shadowing.

A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested

Abstract: One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling distribution) provides rich information for any estimation to be made. Distributions are split into two main groups: continuous and discrete. The present study applies to continuous distributions. One of the challenges of sampling is its accuracy, or, in other words, how representative the sample is of the population from which it was drawn. To answer this question, a series of statistics have been developed to measure the agreement between the theoretical (the population) and observed (the sample) distributions. Another challenge, connected to this, is the presence of outliers - regarded here as observations wrongly collected, that is, not belonging to the population subjected to study. To detect outliers, a series of tests have been proposed, but mainly for normal (Gauss) distributions—the most frequently encountered distribution. The present study proposes a statistic (and a test) intended to be used for any continuous distribution to detect outliers by constructing the confidence interval for the extreme value in the sample, at a certain (preselected) risk of being in error, and depending on the sample size. The proposed statistic is operational for known distributions (with a known probability density function) and is also dependent on the statistical parameters of the population—here it is discussed in connection with estimating those parameters by the maximum likelihood estimation method operating on a uniform U(0,1) continuous symmetrical distribution.

Sum of Squared Fluctuating Two-Ray Random Variables With Wireless Applications

Abstract: The fluctuating two-ray (FTR) channel fading offers good fit for small-scale experimental data in millimeter wave communications. We first investigate the statistical characterizations of the sum of independent nonidentically distributed squared FTR random variables. More specifically, the exact and approximate analytical expressions for moment generating function, probability density function, and cumulative distribution function are derived. Then, the performance of the maximal-ratio combining system over FTR fading channels is assessed in terms of the outage probability (OP) and the average bit error probability (ABEP). Moreover, asymptotic OP and ABEP expressions are obtained to obtain physical insights into the impact of the system and channel parameters on the overall performance. Finally, the correctness of our results is verified by simulation results.

Determining Distribution for the Product of Random Variables by Using Copulas

Abstract: Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers.

Cascaded Fluctuating Two-Ray Fading Channels

Abstract: In this letter, we derive the exact probability density function (PDF), cumulative distribution function (CDF), and generalized moment generating function of a product of $N$ independent fluctuating two-ray (FTR) random variables. Capitalizing on the derived expressions, new expressions for the PDF and CDF of the FTR fading model, i.e., $N=1$ , are obtained in terms of elementary functions, where the existing ones are expressed in terms of special functions. The derived distributions are then used to derive exact analytical expressions for the outage probability and average bit error rate. Monte Carlo simulations have been carried out to validate the correctness of our results.

Machine learning aided stochastic structural free vibration analysis for functionally graded bar-type structures

Abstract: This paper presents a machine learning aided stochastic free vibration analysis for functionally graded (FG) bar-type structures through finite element method (FEM). The considered system uncertainties including the constituent material properties, the dimensions of structural members, and the degree of the gradation of the FGM are incorporated. A novel kernel-based machine learning technique, namely the extended support vector regression (X-SVR), is presented to estimate the governing relationship between the uncertain system parameters and the structural natural frequencies. Subsequently, by applying the Monte-Carlo Simulation (MCS) through the established regression model, various types of statistical characteristics (i.e., mean, standard deviation, probability density function or PDF, and cumulative distribution function or CDF) of structural natural frequencies can be effectively established. Four numerical examples including test functions and practically stimulated engineering structures are thoroughly investigated herein to demonstrate the accuracy, applicability, and computational efficiency of the proposed approach.

Optimal Optical Omnidirectional Angle-of-Arrival Estimator With Complementary Photodiodes

Abstract: Angle-of-arrival (AOA) estimator is the core device in visible light positioning (VLP) systems with AOA algorithms. However, existing AOA estimators suffer from high computational complexity, narrow field-of-view (FOV), low accuracy, or high power consumption. In this work, we propose a novel AOA estimator based on an array of tilted complementary photodiodes (CPDs), where the estimator's FOV can be $2\pi$ rad, and the AOA estimation only requires the solution of a linear equation set. The orientations of the CPDs in the AOA estimator are optimized with respect to the average error power, resulting in closed-form optimal orientation expressions for an arbitrary number of CPDs. We also derive closed-form expressions for the probability density function and the cumulative distribution function (CDF) of the AOA estimation error. On the basis of CDF expression, we derive closed-form asymptotic bounds for the positioning outage probability of a fundamental VLP system. Analytical, simulation, and experimental results verify that the optimal AOA estimator can minimize the estimation error, and it can be employed in AOA positioning systems perusing high accuracy, low complexity, large FOV, low cost, low power consumption, and high response speed.

Higher Order Moments Using the Survival Function: The Alternative Expectation Formula

Abstract: Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a non-negative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a non-negative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both continuous and discrete non-negative integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed r...

A New Power Topp–Leone Generated Family of Distributions with Applications

Abstract: In this paper, we introduce a new general family of distributions obtained by a subtle combination of two well-established families of distributions: the so-called power Topp–Leone-G and inverse exponential-G families. Its definition is centered around an original cumulative distribution function involving exponential and polynomial functions. Some desirable theoretical properties of the new family are discussed in full generality, with comprehensive results on stochastic ordering, quantile function and related measures, general moments and related measures, and the Shannon entropy. Then, a statistical parametric model is constructed from a special member of the family, defined with the use of the inverse Lomax distribution as the baseline distribution. The maximum likelihood method was applied to estimate the unknown model parameters. From the general theory of this method, the asymptotic confidence intervals of these parameters were deduced. A simulation study was conducted to evaluate the numerical behavior of the estimates we obtained. Finally, in order to highlight the practical perspectives of the new family, two real-life data sets were analyzed. All the measures considered are favorable to the new model in comparison to four serious competitors.

On the Distribution of SINR for MMSE MIMO Systems

Abstract: While the performance analysis of linear zero-forcing (ZF) and minimum mean-square error (MMSE) receivers has received much attention, there has been no exact closed-form distribution of signal-to-interference-plus-noise ratio (SINR) for MMSE multiple-input multiple-output (MIMO) systems in the arbitrary fading environments for over the past 20 years. In this paper, an exact and general distribution of SINR for MMSE MIMO systems is presented under uncorrelated Rayleigh fading environments. We start our analysis by finding regular patterns from the probability density function (PDF) of SINR for small dimensions, and using the regular patterns, we extend the PDF to arbitrary dimensions. From the exact and general PDF of SINR for MMSE MIMO systems, the cumulative distribution function and the moment-generating function are derived in terms of the incomplete gamma function and the confluent hypergeometric function of the second kind, respectively. As applications, the error probability, outage probability, and achievable sum rate are investigated by using the distribution of SINR for MMSE MIMO systems. Most notably, the achievable sum rates for ZF and MMSE MIMO systems are defined by using the Meijer $G$ -function, which are the first closed-form expressions in the literature.

Performance of Decode-and-Forward Relaying in Mixed Beaulieu-Xie and $\mathcal{M}$ Dual-Hop Transmission Systems With Digital Coherent Detection

Abstract: Performance analysis is presented for a decode-and-forward protocol based mixed radio-frequency (RF) and free-space optical (FSO) dual-hop transmission system with digital coherent detection. The RF path is modeled by Beaulieu-Xie fading, while the FSO hop is characterised by the Malaga ( $\mathcal {M}$ ) distributed turbulence with pointing errors. We first derive novel and exact analytical expressions for the cumulative distribution function (CDF), the probability density function and the moment generating function (MGF) of the overall signal-to-noise ratio by means of Meijer’s G function, followed by the accurate infinite series expressions of the performance criterions, such as the outage probability, the average bit-error rate (BER) and the ergodic capacity (average channel capacity). Asymptotic analysis for the CDF, the MGF, the outage probability, the average BER, and the ergodic capacity is also provided. Monte Carlo simulations are performed to verify these derived expressions.

Localized Debiased Machine Learning: Efficient Inference on Quantile Treatment Effects and Beyond

Abstract: We consider the efficient estimation of a low-dimensional parameter in an estimating equation involving high-dimensional nuisances that depend on the parameter of interest. An important example is the (local) quantile treatment effect ((L)QTE) in causal inference, for which the efficient estimating equation involves as a nuisance the covariate-conditional cumulative distribution function evaluated at the quantile to be estimated. Debiased machine learning (DML) is a data-splitting approach to address the need to estimate nuisances using flexible machine learning methods that may not satisfy strong metric entropy conditions, but applying it to problems with parameter-dependent nuisances is impractical. For (L)QTE estimation, DML requires we learn the whole conditional cumulative distribution function, conditioned on potentially high-dimensional covariates, which is far more challenging than the standard supervised regression task in machine learning. We instead propose localized debiased machine learning (LDML), a new data-splitting approach that avoids this burdensome step and needs only estimate the nuisances at a single initial rough guess for the parameter. For (L)QTE estimation, this involves just learning two binary regression (i.e., classification) models, for which many standard, time-tested machine learning methods exist, and the initial rough guess may be given by inverse propensity weighting. We prove that under lax rate conditions on nuisances, our estimator has the same favorable asymptotic behavior as the infeasible oracle estimator that solves the estimating equation with the unknown true nuisance functions. Thus, our proposed approach uniquely enables practically-feasible and theoretically-grounded efficient estimation of important quantities in causal inference such as (L)QTEs and in other coarsened data settings.

Diffusive Mobile MC With Absorbing Receivers: Stochastic Analysis and Applications

Abstract: This paper presents a stochastic analysis of the time-variant channel impulse response (CIR) of a three dimensional diffusive mobile molecular communication (MC) system where the transmitter, the absorbing receiver, and the molecules can freely diffuse. In our analysis, we derive the mean, variance, probability density function (PDF), and cumulative distribution function (CDF) of the CIR. We also derive the PDF and CDF of the probability ${p}$ that a released molecule is absorbed at the receiver during a given time period. The obtained analytical results are employed for the design of drug delivery and MC systems with imperfect channel state information. For the first application, we exploit the mean and variance of the CIR to optimize a controlled-release drug delivery system employing a mobile drug carrier. We evaluate the performance of the proposed release design based on the PDF and CDF of the CIR. We demonstrate significant savings in the amount of released drugs compared to a constant-release scheme and reveal the necessity of accounting for the drug-carrier’s mobility to ensure reliable drug delivery. For the second application, we exploit the PDF of the distance between the mobile transceivers and the CDF of ${p}$ to optimize three design parameters of an MC system employing on-off keying modulation and threshold detection. Specifically, we optimize the detection threshold at the receiver, the release profile at the transmitter, and the time duration of a bit frame. We show that the proposed optimal designs can significantly improve the system performance in terms of the bit error rate and the efficiency of molecule usage.

Reliability measure approach for confidence-based design optimization under insufficient input data

Abstract: In most of the reliability-based design optimization (RBDO) researches, accurate input statistical model has been assumed to concentrate on the variability of random variables; however, only a limited number of data are available to quantify the input statistical model in many practical engineering applications. In other words, irreducible variability and uncertainty due to lack of knowledge exist simultaneously in random design variables, which may result in uncertainty of reliability. Therefore, the uncertainty induced by insufficient data has to be accounted for RBDO to guarantee the confidence of reliability. Using the Bayesian approach, the uncertainty of input distributions is successfully propagated to a cumulative distribution function (CDF) of reliability under reasonable assumptions, but it requires a number of function evaluations in double-loop Monte Carlo simulation (MCS). To tackle this challenge, the reliability measure approach (RMA) in confidence-based design optimization (CBDO) is proposed to handle the uncertainty of reliability following the idea of performance measure approach (PMA) in RBDO. Input distribution parameters are transformed to random variables following the standard normal distribution for the most probable point (MPP) search based on the proposed stochastic sensitivity analysis of reliability. Therefore, the reliability is approximated at MPP with respect to input distribution parameters. The proposed CBDO can treat confidence constraints employing the reliability value at the target confidence level that is approximated by MPP in standard normal space. In conclusion, CBDO can be performed in a probabilistic space of input distribution parameters corresponding to the conventional U-space in RBDO to yield the probability (confidence) that reliability is larger than the target reliability. The proposed method can significantly reduce the number of function evaluations by eliminating outer-loop MCS while maintaining acceptable accuracy. Numerical examples are used to demonstrate the effectiveness of the developed sensitivity analysis and RMA to estimate the confidence of reliability in CBDO.

Model for Non-Gaussian Sea Clutter Amplitudes Using Generalized Inverse Gaussian Texture

Abstract: In this letter, we focus on the statistical modeling of sea clutter amplitudes. Due to its non-Gaussian nature, the existing statistical models are sometimes difficult to represent well the heavy-tailed portion of amplitude distribution. To address this problem, we propose a compound Gaussian (CG) model with a generalized inverse Gaussian (GIG) texture to describe sea clutter amplitudes. In this regard, the probability density function and the cumulative distribution function of the clutter amplitudes for the proposed model are derived. Moreover, we provide an approach to estimate the unknown parameters of the proposed CG-GIG distribution. The experimental results indicate that the CG-GIG distribution is more suitable to describe the amplitudes of non-Gaussian sea clutter than its competitors.

Accurate Analysis of Weighted Centroid Localization

Abstract: Source localization of primary users (PUs) is a spectrum awareness feature that can be very useful in enhancing the functionality of cognitive radios (CRs). When the cooperating CRs have limited information about the PU, weighted centroid localization (WCL) based on received signal strength measurements represents an attractive low-complexity solution. This paper proposes a new analytical framework to accurately calculate the performance of WCL based on the statistical distribution of the ratio of two quadratic forms in normal variables. In particular, we derive an analytical expression for the root mean square error and an exact expression for the cumulative distribution function of the two-dimensional location estimate. The proposed framework accounts for the presence of independent and identically distributed shadowing as well as correlated shadowing with distance-dependent intensity. The methodology is general enough to include the analysis of the one-dimensional error, which leads also to the evaluation of the bias of the position estimate. Numerical results confirm that the analytical framework is able to predict the performance of WCL capturing all the essential aspects of propagation as well as CR network spatial topology.