Showing papers on "Cumulative distribution function published in 2017"

Modeling continuous response variables using ordinal regression

Abstract: We study the application of a widely used ordinal regression model, the cumulative probability model (CPM), for continuous outcomes. Such models are attractive for the analysis of continuous response variables because they are invariant to any monotonic transformation of the outcome and because they directly model the cumulative distribution function from which summaries such as expectations and quantiles can easily be derived. Such models can also readily handle mixed type distributions. We describe the motivation, estimation, inference, model assumptions, and diagnostics. We demonstrate that CPMs applied to continuous outcomes are semiparametric transformation models. Extensive simulations are performed to investigate the finite sample performance of these models. We find that properly specified CPMs generally have good finite sample performance with moderate sample sizes, but that bias may occur when the sample size is small. Cumulative probability models are fairly robust to minor or moderate link function misspecification in our simulations. For certain purposes, the CPMs are more efficient than other models. We illustrate their application, with model diagnostics, in a study of the treatment of HIV. CD4 cell count and viral load 6 months after the initiation of antiretroviral therapy are modeled using CPMs; both variables typically require transformations, and viral load has a large proportion of measurements below a detection limit.

Dual-Hop FSO Transmission Systems Over Gamma–Gamma Turbulence With Pointing Errors

Abstract: In this paper, we analyze the end-to-end performance of dual-hop free-space optical (FSO) fixed gain relaying systems under heterodyne detection and intensity modulation with direct detection techniques in the presence of atmospheric turbulence as well as pointing errors. In particular, we derive the cumulative distribution function (cdf) of the end-to-end signal-to-noise ratio (SNR) in exact closed form in terms of the bivariate Fox’s H function. Capitalizing on this cdf expression, novel closed-form expressions for the outage probability, the average bit-error rate (BER) for different modulation schemes, and the ergodic capacity of dual-hop FSO transmission systems are presented. Moreover, we present very tight asymptotic results for the outage probability and the average BER at high SNR regime in terms of simple elementary functions, and we derive the diversity order of the considered system. By using dual-hop FSO relaying, we demonstrate a better system performance as compared with the single FSO link. Numerical and Monte Carlo simulation results are provided to verify the accuracy of the newly proposed results, and a perfect agreement is observed.

Mixed mmWave RF/FSO Relaying Systems Over Generalized Fading Channels With Pointing Errors

Abstract: This paper studies the performance of mixed millimeter-wave radio-frequency (mmWave RF), free-space optics (FSO) systems in a highly scalable and cost-effective solution for fifth-generation (5G) mobile backhaul networks. The mmWave RF and FSO fading channels are, respectively, modeled by the Rician and the generalized Malaga ( $\mathcal {M}$ ) distributions. The effect of pointing errors due to the misalignment between the transmitter and the receiver in the FSO link is also included. Novel accurate closed-form expressions for the cumulative distribution function, the probability density function, and the moment generating function (MGF) in terms of Meijer's G functions are derived. Capitalizing on these new results, we analytically derive precise closed-form expressions for various performance metrics of the proposed system, including the outage probability, the average bit error rate (ABER), and the average capacity. Additionally, new asymptotic results are provided for the outage probability, the MGF, and the ABER in terms of simple elementary functions by applying the asymptotic expansion of the Meijer's G function at high signal-to-noise ratios (SNRs). Numerical results further validate the mathematical analysis by Monte-Carlo simulations.

Understanding characteristics in multivariate traffic flow time series from complex network structure

Abstract: Discovering dynamic characteristics in traffic flow is the significant step to design effective traffic managing and controlling strategy for relieving traffic congestion in urban cities. A new method based on complex network theory is proposed to study multivariate traffic flow time series. The data were collected from loop detectors on freeway during a year. In order to construct complex network from original traffic flow, a weighted Froenius norm is adopt to estimate similarity between multivariate time series, and Principal Component Analysis is implemented to determine the weights. We discuss how to select optimal critical threshold for networks at different hour in term of cumulative probability distribution of degree. Furthermore, two statistical properties of networks: normalized network structure entropy and cumulative probability of degree, are utilized to explore hourly variation in traffic flow. The results demonstrate these two statistical quantities express similar pattern to traffic flow parameters with morning and evening peak hours. Accordingly, we detect three traffic states: trough, peak and transitional hours, according to the correlation between two aforementioned properties. The classifying results of states can actually represent hourly fluctuation in traffic flow by analyzing annual average hourly values of traffic volume, occupancy and speed in corresponding hours.

Local Intrinsic Dimensionality I: An Extreme-Value-Theoretic Foundation for Similarity Applications

Abstract: Researchers have long considered the analysis of similarity applications in terms of the intrinsic dimensionality (ID) of the data. This theory paper is concerned with a generalization of a discrete measure of ID, the expansion dimension, to the case of smooth functions in general, and distance distributions in particular. A local model of the ID of smooth functions is first proposed and then explained within the well-established statistical framework of extreme value theory (EVT). Moreover, it is shown that under appropriate smoothness conditions, the cumulative distribution function of a distance distribution can be completely characterized by an equivalent notion of data discriminability. As the local ID model makes no assumptions on the nature of the function (or distribution) other than continuous differentiability, its extreme generality makes it ideally suited for the non-parametric or unsupervised learning tasks that often arise in similarity applications. An extension of the local ID model is also provided that allows the local assessment of the rate of change of function growth, which is then shown to have potential implications for the detection of inliers and outliers.

Quantile Selection Models With an Application to Understanding Changes in Wage Inequality

Abstract: We propose a method to correct for sample selection in quantile regression models. Selection is modelled via the cumulative distribution function, or copula, of the percentile error in the outcome equation and the error in the participation decision. Copula parameters are estimated by minimizing a method-of-moments criterion. Given these parameter estimates, the percentile levels of the outcome are re-adjusted to correct for selection, and quantile parameters are estimated by minimizing a rotated "check" function. We apply the method to correct wage percentiles for selection into employment, using data for the UK for the period 1978-2000. We also extend the method to account for the presence of equilibrium effects when performing counterfactual exercises.

Nearest-Neighbor and Contact Distance Distributions for Thomas Cluster Process

Abstract: We characterize the statistics of nearest-neighbor and contact distance distributions for Thomas cluster process (TCP), which is a special case of Poisson cluster process. In particular, we derive the cumulative distribution function of the distance to the nearest point of TCP from a reference point for three different cases: 1) reference point is not a part of the point process; 2) it is chosen uniformly at random from the TCP; and 3) it is a randomly chosen point from a cluster chosen uniformly at random from the TCP. While the first corresponds to the contact distance distribution, the other two provide two different viewpoints for the nearest-neighbor distance distribution. Closed-form bounds are also provided for the first two cases.

Cascaded FSO-VLC Communication System

Abstract: The proposed cascaded free space optics (FSO)-visible light communication (VLC) system consists of multiple VLC access points which caters the end users connected via a decode and forward relay to the FSO backhaul link. The FSO link is assumed to be affected by path-loss, pointing error, and atmospheric turbulence while the end-to-end signal-to-noise ratio (SNR) of VLC downlinks are statistically characterized considering the randomness of users position. In this letter, the novel closed form expressions of the statistics like probability density function and cumulative distribution function of the equivalent SNR are derived. Capitalizing on these, the closed form expressions for various performance metrics such as outage probability and error probability are provided. The simulation results are provided to verify the functional curves of mathematical analysis.

New results on the sum of Gamma random variates with application to the performance of wireless communication systems over Nakagami‐m fading channels

Abstract: The probability density function (PDF) and cumulative distribution function of the sum of L‐independent but not necessarily identically distributed Gamma variates, applicable to the output statistics of maximal ratio combining receiver operating over Nakagami‐m fading channels or in other words to the statistical analysis of the scenario where the sum of squared Nakagami‐m distributions is user‐of‐interest, are presented in closed form in terms of well‐known Meijer's G function and easily computable Fox's H function for integer‐valued and non‐integer‐valued m fading parameters. Further analysis, particularly on bit error rate via a PDF‐based approach, is also offered in closed form in terms of Meijer's G function and Fox's H function for integer‐valued fading parameters and extended Fox's H function ( urn:x-wiley:ett:media:ett2912:ett2912-math-0001) for non‐integer‐valued fading parameters. Our proposed results complement previous known results that are either expressed in terms of infinite sums, nested sums or higher‐order derivatives of the fading parameter m.

Local Intrinsic Dimensionality II: Multivariate Analysis and Distributional Support

Abstract: Distance-based expansion models of intrinsic dimensionality have had recent application in the analysis of complexity of similarity applications, and in the design of efficient heuristics. This theory paper extends one such model, the local intrinsic dimension (LID), to a multivariate form that can account for the contributions of different distributional components towards the intrinsic dimensionality of the entire feature set, or equivalently towards the discriminability of distance measures defined in terms of these feature combinations. Formulas are established for the effect on LID under summation, product, composition, and convolution operations on smooth functions in general, and cumulative distribution functions in particular. For some of these operations, the dimensional or discriminability characteristics of the result are also shown to depend on a form of distributional support. As an example, an analysis is provided that quantifies the impact of introduced random Gaussian noise on the intrinsic dimension of data. Finally, a theoretical relationship is established between the LID model and the classical correlation dimension.

New Results on the Fluctuating Two-Ray Model with Arbitrary Fading Parameters and Its Applications

Abstract: The fluctuating two-ray (FTR) fading model provides a much better fit than other fading models for small-scale fading measurements in millimeter wave communications. In this paper, using a mixture of gamma distributions, new exact analytical expressions for the probability density and cumulative distribution functions of the FTR distribution with arbitrary fading parameters are presented. Moreover, the performance of digital communication systems over the FTR fading channel is evaluated in terms of the channel capacity and the bit error rate. The interaction between channel fading parameters and system performance is further investigated. Our newly derived results extend and complement previous knowledge of the FTR fading model.

Fosgerau's travel time reliability ratio and the Burr distribution

Abstract: Recent international research has seen the development of methods for the inclusion of travel time reliability as a separate factor in economic analysis of transportation projects, including the valuation of travel time variability. Fosgerau's valuation method includes the consideration of travel time reliability in cost-benefit analysis by adding travel time variability to the set of generalised travel costs. This requires: (1) a defined unit of measurement for travel time variability, (2) estimates of the quantity of travel time variability, and (3) identification of the cost to travellers per unit of travel time variability. The chosen unit of measurement is the standard deviation of the travel time distribution, and the value of this unit of measurement can be defined relative to the average value of travel time by a reliability ratio that depends on user preference parameters (related to risk aversion) and the shape of the upper tail of the cumulative distribution function (cdf) of the travel time distribution. This shape is represented by a definite integral of the inverse of the cdf. Determining the shape of the cdf and its inverse function is facilitated if the distribution can be defined by an explicit algebraic function. The Burr (type XII) distribution is one distribution with this feature, and has been used to successfully represent observed travel time data. This paper describes the Burr distribution, demonstrates that it can provide a good representation of observed travel time data, and explains how it can be used to develop an exact expression for the reliability ratio and thus can aid the use of the method for the valuation of travel time reliability.

Outage Performance Analysis of Full-Duplex Relay-Assisted Device-to-Device Systems in Uplink Cellular Networks

Abstract: This paper proposes a full-duplex cooperative device-to-device (D2D) communication system, where the relay employed can receive and transmit signals simultaneously. We adopt such a system to assist with D2D transmission. We first derive the conditional cumulative distribution function and the probability density function (pdf) of a series of channel parameters when the interference to the base station is taken into consideration and power control is applied at the D2D transmitter and the relay node. Then, we obtain an exact expression for the outage probability as an integral and as a closed-form expression for a special case, which can be used as a good approximation to the general case when residual self-interference is small. Additionally, we also investigate the power allocation problem between the source and the relay and formulate a suboptimal allocation problem, which we prove to be quasi-concave. Our analysis is verified by the Monte Carlo simulations, and a number of important features of full-duplex cooperative D2D communications can, thereby, be revealed.

Interactive Q-learning for Quantiles.

Abstract: A dynamic treatment regime is a sequence of decision rules, each of which recommends treatment based on features of patient medical history such as past treatments and outcomes. Existing methods for estimating optimal dynamic treatment regimes from data optimize the mean of a response variable. However, the mean may not always be the most appropriate summary of performance. We derive estimators of decision rules for optimizing probabilities and quantiles computed with respect to the response distribution for two-stage, binary treatment settings. This enables estimation of dynamic treatment regimes that optimize the cumulative distribution function of the response at a prespecified point or a prespecified quantile of the response distribution such as the median. The proposed methods perform favorably in simulation experiments. We illustrate our approach with data from a sequentially randomized trial where the primary outcome is remission of depression symptoms. Supplementary materials for this arti...

Cover time for random walks on arbitrary complex networks.

Abstract: We present an analytical method for computing the mean cover time of a discrete-time random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This quantity is particularly important for random search processes and target localization on network structures. Based on the global mean first-passage time of target nodes, we derive a method for computing the cumulative distribution function of the cover time based on first-passage time statistics. Our method is viable for networks on which random walks equilibrate quickly. We show that it can be applied successfully to various model and real-world networks. Our results reveal an intimate link between first-passage and cover time statistics and offer a computationally efficient way for estimating cover times in network-related applications.

Blockage and Coverage Analysis with MmWave Cross Street BSs Near Urban Intersections

Abstract: Millimeter wave (mmWave) communication offers Gbps data transmission, which can support massive data sharing in vehicle-to-infrastructure (V2I) networks. In this paper, we analyze the blockage effects among different vehicles and coverage probability of a typical receiver, considering cross street BSs near urban intersections in a multi-lane mmWave vehicular network. First, a three-dimensional model of blockage among vehicles on different lanes is considered. Second, we compute the coverage probability considering the interference of cross street base stations. Incorporating the blockage effects, we derive an exact and semi closed-form expression of the cumulative distribution density (CDF) of the association link path gain. Then, a tight approximation of the coverage probability is computed. We provide numerical results to verify the accuracy of the analytic results. We demonstrate the effects of blockage and the cross street interference. Also, we compare coverage probability with different BSs intensities under various street settings. It is shown that in multi-lane V2I networks, blockage among vehicles is not significant. Also, deploying more BSs does not increase coverage probability efficiently in ultra-dense streets.

Relay selection in mixed RF/FSO system over generalized channel fading

Abstract: In this paper, a relay selection-based amplify-and-forward (AF) with mixed radio frequency/free space optical (RF/FSO) system is proposed. We have considered two relay selection schemes namely max-select, distributed switch and stay (DSS) and compared their performance when all relays are active. The AF-based relay employs (1) full-channel state information relaying (amplification gain dependent on instantaneous signal-to-noise ratio of source-relay link) (2) semi-blind relaying (fixed amplification gain) for the amplification purpose. The selected AF relay converts the received RF signal into an optical signal using the subcarrier intensity modulation scheme. The RF link is subject to a generalised η − μ distribution, while the channel for free space optical link is modelled as gamma–gamma distributed along with pointing errors. Novel analytical expressions for the probability density function, cumulative distribution function and the moment generating function of the equivalent signal-to-noise ratio is derived. Capitalising on these channel statistics, we provide the closed form expressions of the outage probability for different relay selection schemes. Copyright © 2016 John Wiley & Sons, Ltd.

Model-based clustering of Gaussian copulas for mixed data

Abstract: Clustering task of mixed data is a challenging problem. In a probabilistic framework, the main difficulty is due to a shortage of conventional distributions for such data. In this paper, we propose to achieve the mixed data clustering with a Gaussian copula mixture model, since copulas, and in particular the Gaussian ones, are powerful tools for easily modelling the distribution of multivariate variables. Indeed, considering a mixing of continuous, integer and ordinal variables (thus all having a cumulative distribution function), this copula mixture model defines intra-component dependencies similar to a Gaussian mixture, so with classical correlation meaning. Simultaneously, it preserves standard margins associated to continuous, integer and ordered features, namely the Gaussian, the Poisson and the ordered multinomial distributions. As an interesting by-product, the proposed mixture model generalizes many well-known ones and also provides tools of visualization based on the parameters. At a practical level, the Bayesian inference is retained and it is achieved with a Metropolis-within-Gibbs sampler. Experiments on simulated and real data sets finally illustrate the expected advantages of the proposed model for mixed data: flexible and meaningful parametrization combined with visualization features.

Random Access Delay Distribution of Multichannel Slotted ALOHA With Its Applications for Machine Type Communications

Abstract: An innovative iterative process is proposed to acquire the dynamic process of multichannel slotted ALOHA (S-ALOHA). It reveals the direct relation between the number of contending devices that perform their ${j}$ th random access (RA) attempt at the ${i}$ th RA slot and the newly arrived devices before the ${i}$ th RA slot. These results allow engineers to analytically derive the probability density function of RA delay of multichannel S-ALOHA, as well as its cumulative density function and average value. Under stable RA attempts assumption, simplified form of the above analysis is given, with which we prove the number of preamble transmissions follows truncated geometric distribution. Taking the two traffic models proposed for machine type communications as examples, numerical results are presented to verify the effectiveness of the proposed iterative process and the accuracy of its simplified form, and illustrate the delay characteristics of simplified long term evolution RA channel.

Analytical Expressions for Joint Distributions in Probabilistic Load Flow

Abstract: Adopting the Gaussian mixture model, we derive an analytical expression for the joint probability density function of active power on multiple transmission lines, in the presence of non-Gaussian stochastic power injections. The corresponding joint cumulative distribution function is further obtained by multiple integrals. Results of the proposed method coincide with that of Monte Carlo simulations.

Channel Modeling and Performance Analysis of Modulating Retroreflector FSO Systems Under Weak Turbulence Conditions

Abstract: This paper presents the fading channel of double-pass modulating retroreflector free-space optical systems under weak turbulence conditions that can be modeled by the distribution of the weighted product of two correlated Lognormal random variables. We first show the proposed channel model through using wave-optics simulation. Then, the probability density function (PDF) and cumulative distribution function (CDF) of the channel model are derived in closed form. The system bit error rate (BER) and outage probability are calculated by averaging the conditional BER of on–off keying (OOK) modulation over the PDF and substituting the signal-to-noise ratio threshold into the CDF, respectively. The obtained analytical formulas for the system performance are validated by Monte Carlo simulations. Moreover, the effect of the fading correlation and the aperture averaging on the performance is investigated.

A Unified Moment-Based Approach for the Evaluation of the Outage Probability With Noise and Interference

Abstract: In this paper, we develop a novel moment-based approach for the evaluation of the outage probability (OP) in a generalized fading environment with interference and noise. Our method is based on the derivation of a power series expansion of OP of the signal-to-interference-plus-noise ratio. It does not necessitate stringent requirements, the only major ones being the existence of a power series expansion of the cumulative distribution function of the desired user power and the knowledge of the cross moments of the interferers’ powers. The newly derived formula is shown to be applicable for most of the well-practical fading models of the desired user under some assumptions on the parameters of the powers’ distributions. A further advantage of our method is that it is applicable irrespective of the nature of the fading models of the interfering powers, the only requirement being the perfect knowledge of their cross moments. In order to illustrate the wide scope of applicability of our technique, we present a convergence study of the provided formula for the Generalized Gamma and the Rice fading models. Moreover, we show that our analysis has direct bearing on recent multi-channel applications using selection diversity techniques. Finally, we assess by simulations the accuracy of the proposed formula for various fading environments.

A Versatile Mixture Distribution and Its Application in Economic Dispatch with Multiple Wind Farms

Abstract: An improved versatile distribution for wind power is proposed, with higher representation accuracy and more suitable applications in economic dispatch (ED) compared with Versatile distribution. Further, versatile mixture distribution (VMD) is proposed to represent all possible wind power distribution with customized arbitrary errors. Compared with Gaussian Mixture distribution (GMD), it has more flexible forms and can represent wind power more accurately. Its cumulative distribution function has analytical forms with higher computational efficiency. Then, real-time dynamic ED model and algorithm based on VMD with multiple wind farms (WFs) is proposed. Based on the VMD model, the probability distributions of wind power of each WF and the summation of all WFs are modeled. The ED model and algorithm solve the problem of the correlation of multiple renewable energy random variables with high computation efficiency for real-time ED. The results show that compared with GMD or unimodal distributions, VMD can represent wind power more accurately and can greatly simplify and speed up the ED model and algorithm. Compared with other methods for multiple renewable energy random variables, the proposed model and algorithm can improve economy significantly.