Showing papers on "Poisson distribution published in 2017"

The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded

Abstract: The coefficient of determination R2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R2 for generalized linear mixed models (GLMMs) remains challenging. We have previously introduced a version of R2 that we called R2GLMM for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients ICC using Poisson and binomial GLMMs. In this article, we expand our methods to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling biological data. While expanding our approach, we highlight two useful concepts for biologists, Jensen9s inequality and the delta method, both of which help us in understanding the properties of GLMMs. Jensen9s inequality has important implications for biologically meaningful interpretation of GLMMs, while the delta method allows a general derivation of variance associated with non-Gaussian distributions. We also discuss some special considerations for binomial GLMMs with binary or proportion data. We illustrate the implementation of our extension by worked examples from the field of ecology and evolution in the R environment. However, our method can be used across disciplines and regardless of statistical environments.

Multi-rate Poisson tree processes for single-locus species delimitation under maximum likelihood and Markov chain Monte Carlo

Abstract: Motivation: In recent years, molecular species delimitation has become a routine approach for quantifying and classifying biodiversity. Barcoding methods are of particular importance in large-scale surveys as they promote fast species discovery and biodiversity estimates. Among those, distance-based methods are the most common choice as they scale well with large datasets; however, they are sensitive to similarity threshold parameters and they ignore evolutionary relationships. The recently introduced "Poisson Tree Processes" (PTP) method is a phylogeny-aware approach that does not rely on such thresholds. Yet, two weaknesses of PTP impact its accuracy and practicality when applied to large datasets; it does not account for divergent intraspecific variation and is slow for a large number of sequences. Results: We introduce the multi-rate PTP (mPTP), an improved method that alleviates the theoretical and technical shortcomings of PTP. It incorporates different levels of intraspecific genetic diversity deriving from differences in either the evolutionary history or sampling of each species. Results on empirical data suggest that mPTP is superior to PTP and popular distance-based methods as it, consistently yields more accurate delimitations with respect to the taxonomy (i.e., identifies more taxonomic species, infers species numbers closer to the taxonomy). Moreover, mPTP does not require any similarity threshold as input. The novel dynamic programming algorithm attains a speedup of at least five orders of magnitude compared to PTP, allowing it to delimit species in large (meta-) barcoding data. In addition, Markov Chain Monte Carlo sampling provides a comprehensive evaluation of the inferred delimitation in just a few seconds for millions of steps, independently of tree size. Availability and Implementation: mPTP is implemented in C and is available for download at http://github.com/Pas-Kapli/mptp under the GNU Affero 3 license. A web-service is available at http://mptp.h-its.org . Contact: : paschalia.kapli@h-its.org or alexandros.stamatakis@h-its.org or tomas.flouri@h-its.org. Supplementary information: Supplementary data are available at Bioinformatics online.

Negative-Poisson's-Ratio Materials: Auxetic Solids

Abstract: Poisson's ratio had long been considered to be an intrinsic material property, confined within a narrow domain and governed solely by the geometry of interatomic bonds. Materials with designed heterogeneity allow for control over the Poisson's ratio. Poisson's ratios of any value within the thermodynamically admissible domain may be attained, including negative Poisson's ratio (termed auxetic). In this article, we discuss the role of Poisson's ratio in elasticity, two-dimensional and three-dimensional materials, phase transformations, underlying causes in the microstructure, and other negative physical properties.

R and STATA code for: "Conditional Poisson models: a flexible alternative to conditional logistic case cross-over analysis"

Abstract: Illustration of conditional Poisson models as an alternative method in analyses of environmental data. In particular, this represents a computationally convenient alternative to both conditional logistic case-crossover models (when data are aggregated in time series form) and to standard Poisson regression for long time series (when control for time is achieved with computationally expensive spline functions). The code follows the examples included in the associated article that illustrates the methodology and some applications. The material includes: [1] londondataset2002_2006.csv stores the dataset used in the illustrative examples; [2] funccmake.R generates the R function to convert data from time series to case-crossover formats; [3] Rcode.R is the R code to reproduce the examples; and [4] Statacode.do is the Stata code to reproduce the examples.

Modeling and Analysis of Uplink Non-Orthogonal Multiple Access in Large-Scale Cellular Networks Using Poisson Cluster Processes

Abstract: Using the theory of Poisson cluster process (PCP), this paper provides a framework to analyze multi-cell uplink non-orthogonal multiple access (NOMA) systems Specifically, we characterize the rate coverage probability of an NOMA user who is at rank $m$ (in terms of the distance from its serving base station) among all users in a cell and the mean rate coverage probability of all users in a cell Since the signal-to-interference-plus-noise ratio of the $m$ th user relies on efficient successive interference cancellation (SIC), we consider three scenarios, ie, perfect SIC (in which the signals of $m-1$ interferers who are stronger than the $m$ th user are decoded successfully), imperfect SIC (in which the signals of $m-1$ interferers who are stronger than the $m$ th user may or may not be decoded successfully), and imperfect worst case SIC (in which the decoding of the signal of the $m$ th user is always unsuccessful whenever the decoding of its relative $m-1$ stronger users is unsuccessful) To derive the rate coverage expressions, we first characterize the Laplace transforms of the intra-cluster interferences in closed-form considering various SIC scenarios The Laplace transform of the inter-cluster interference is then characterized by exploiting distance distributions from geometric probability The derived expressions are customized for an equivalent OMA system Finally, numerical results are presented to validate the derived expressions The worst case SIC assumption provides remarkable simplifications in the mathematical analysis and is found to be highly accurate for higher user target rate requirements A comparison of Poisson point process-based and PCP-based modeling is also conducted

Tests Based on EDF Statistics

Abstract: This chapter considers tests of fit based on the empirical distribution function (EDF). The EDF is a step function, calculated from the sample, which estimates the population distribution function. From the basic definitions of the supremum statistics and the quadratic statistics given above, suitable computing formulas must be found. A second group of statistics for censored samples is of the general Cramer-von Mises type. The modifications for all the statistics were calculated from points for finite n obtained by Monte Carlo methods. Green and Hegazy have shown that slight modifications of the basic EDF statistics can improve power in tests for normality against selected alternatives. Data may appear to be discrete either because the sample genuinely arises from a discrete distribution like the Binomial or Poisson, for example, in measurements of counts, or alternatively because originally continuous data may have been grouped.

Analysis of D2D Underlaid Cellular Networks: SIR Meta Distribution and Mean Local Delay

Abstract: We study the performance of device-to-device (D2D) communication underlaying cellular wireless network in terms of the meta distribution of the signal-to-interference ratio (SIR), which is the distribution of the conditional SIR distribution given the locations of the wireless nodes. Modeling D2D transmitters and base stations as Poisson point processes (PPPs), moments of the conditional SIR distribution are derived in order to calculate analytical expressions for the meta distribution and the mean local delay of the typical D2D receiver and cellular downlink user. It turns out that for D2D users, the total interference from the D2D interferers and base stations is equal in distribution to that of a single PPP, while for downlink users, the effect of the interference from the D2D network is more complicated. We also derive the region of transmit probabilities for the D2D users and base stations that result in a finite mean local delay and give a simple inner bound on that region. Finally, the impact of increasing the base station density on the mean local delay, the meta distribution, and the density of users reliably served is investigated with numerical results.

The Poisson distribution, abstract fractional difference equations, and stability

Abstract: We study the initial value problem (∗) { C∆ u(n) = Au(n+ 1), n ∈ N0; u(0) = u0 ∈ X. when A is a closed linear operator with domain D(A) defined on a Banach space X. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem (*), using operator-theoretical conditions on A. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator A in case of Hilbert spaces.

Casimir Meets Poisson: Improved Quark/Gluon Discrimination with Counting Observables

Abstract: Charged track multiplicity is among the most powerful observables for discriminating quark- from gluon-initiated jets. Despite its utility, it is not infrared and collinear (IRC) safe, so perturbative calculations are limited to studying the energy evolution of multiplicity moments. While IRC-safe observables, like jet mass, are perturbatively calculable, their distributions often exhibit Casimir scaling, such that their quark/gluon discrimination power is limited by the ratio of quark to gluon color factors. In this paper, we introduce new IRC-safe counting observables whose discrimination performance exceeds that of jet mass and approaches that of track multiplicity. The key observation is that track multiplicity is approximately Poisson distributed, with more suppressed tails than the Sudakov peak structure from jet mass. By using an iterated version of the soft drop jet grooming algorithm, we can define a “soft drop multiplicity” which is Poisson distributed at leading-logarithmic accuracy. In addition, we calculate the next-to-leading-logarithmic corrections to this Poisson structure. If we allow the soft drop groomer to proceed to the end of the jet branching history, we can define a collinear-unsafe (but still infrared-safe) counting observable. Exploiting the universality of the collinear limit, we define generalized fragmentation functions to study the perturbative energy evolution of collinear-unsafe multiplicity.

A review of multivariate distributions for count data derived from the Poisson distribution

Abstract: The Poisson distribution has been widely studied and used for modeling univariate count-valued data. However, multivariate generalizations of the Poisson distribution that permit dependencies have been far less popular. Yet, real-world, high-dimensional, count-valued data found in word counts, genomics, and crime statistics, for example, exhibit rich dependencies and motivate the need for multivariate distributions that can appropriately model this data. We review multivariate distributions derived from the univariate Poisson, categorizing these models into three main classes: 1 where the marginal distributions are Poisson, 2 where the joint distribution is a mixture of independent multivariate Poisson distributions, and 3 where the node-conditional distributions are derived from the Poisson. We discuss the development of multiple instances of these classes and compare the models in terms of interpretability and theory. Then, we empirically compare multiple models from each class on three real-world datasets that have varying data characteristics from different domains, namely traffic accident data, biological next generation sequencing data, and text data. These empirical experiments develop intuition about the comparative advantages and disadvantages of each class of multivariate distribution that was derived from the Poisson. Finally, we suggest new research directions as explored in the subsequent Discussion section. WIREs Comput Stat 2017, 9:e1398. doi: 10.1002/wics.1398

Modeling and Analysis of D2D Millimeter-Wave Networks With Poisson Cluster Processes

Abstract: This paper investigates the performance of millimeter wave (mmWave) communications in clustered device-to-device (D2D) networks. The locations of D2D transceivers are modeled as a Poisson Cluster Process. In each cluster, devices are equipped with multiple antennas, and the active D2D transmitter (D2D-Tx) utilizes mmWave to serve one of the proximate D2D receivers. Specifically, we introduce three user association strategies: 1) uniformly distributed D2D-Tx model; 2) nearest D2D-Tx model; and 3) closest line-of-site (LOS) D2D-Tx model. To characterize the performance of the considered scenarios, we derive new analytical expressions for the coverage probability and area spectral efficiency (ASE). Additionally, in order to efficiently illustrating the general trends of our system, a closed-form lower bound for the special case interfered by intra-cluster LOS links is derived. We provide Monte Carlo simulations to corroborate the theoretical results and show that: 1) the coverage probability is mainly affected by the intra-cluster interference with LOS links; 2) there exists an optimum number of simultaneously active D2D-Txs in each cluster for maximizing ASE; and 3) the closest LOS model outperforms the other two scenarios but at the cost of extra system overhead.

Differential expression analysis for RNAseq using Poisson mixed models.

Abstract: Identifying differentially expressed (DE) genes from RNA sequencing (RNAseq) studies is among the most common analyses in genomics. However, RNAseq DE analysis presents several statistical and computational challenges, including over-dispersed read counts and, in some settings, sample non-independence. Previous count-based methods rely on simple hierarchical Poisson models (e.g. negative binomial) to model independent over-dispersion, but do not account for sample non-independence due to relatedness, population structure and/or hidden confounders. Here, we present a Poisson mixed model with two random effects terms that account for both independent over-dispersion and sample non-independence. We also develop a scalable sampling-based inference algorithm using a latent variable representation of the Poisson distribution. With simulations, we show that our method properly controls for type I error and is generally more powerful than other widely used approaches, except in small samples (n <15) with other unfavorable properties (e.g. small effect sizes). We also apply our method to three real datasets that contain related individuals, population stratification or hidden confounders. Our results show that our method increases power in all three data compared to other approaches, though the power gain is smallest in the smallest sample (n = 6). Our method is implemented in MACAU, freely available at www.xzlab.org/software.html.

Mean-parametrized Conway-Maxwell-Poisson regression models for dispersed counts

Abstract: Conway–Maxwell–Poisson (CMP) distributions are flexible generalizations of the Poisson distribution for modelling overdispersed or underdispersed counts. The main hindrance to their wider use in pr...

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.

The Exponentiated Generalized-G Poisson Family of Distributions

Abstract: In this article we propose and study a new family of distributions which is defined by using the genesis of the truncated Poisson distribution and the exponentiated generalized-G distribution Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics and their moments, reliability and Shannon entropy are derived Estimation of the parameters using the method of maximum likelihood is discussed Although this generalization technique can be used to generalize many other distributions, in this study we present only two special models The importance and flexibility of the new family is exemplified using real world data

Study on a Poisson's Equation Solver Based On Deep Learning Technique

Abstract: In this work, we investigated the feasibility of applying deep learning techniques to solve Poisson's equation. A deep convolutional neural network is set up to predict the distribution of electric potential in 2D or 3D cases. With proper training data generated from a finite difference solver, the strong approximation capability of the deep convolutional neural network allows it to make correct prediction given information of the source and distribution of permittivity. With applications of L2 regularization, numerical experiments show that the predication error of 2D cases can reach below 1.5\% and the predication of 3D cases can reach below 3\%, with a significant reduction in CPU time compared with the traditional solver based on finite difference methods.

Poisson Multi-Bernoulli Mapping Using Gibbs Sampling

Abstract: This paper addresses the mapping problem. Using a conjugate prior form, we derive the exact theoretical batch multiobject posterior density of the map given a set of measurements. The landmarks in the map are modeled as extended objects, and the measurements are described as a Poisson process, conditioned on the map. We use a Poisson process prior on the map and prove that the posterior distribution is a hybrid Poisson, multi-Bernoulli mixture distribution. We devise a Gibbs sampling algorithm to sample from the batch multiobject posterior. The proposed method can handle uncertainties in the data associations and the cardinality of the set of landmarks, and is parallelizable, making it suitable for large-scale problems. The performance of the proposed method is evaluated on synthetic data and is shown to outperform a state-of-the-art method.

Unimodal Probability Distributions for Deep Ordinal Classification

Abstract: Probability distributions produced by the cross-entropy loss for ordinal classification problems can possess undesired properties. We propose a straightforward technique to constrain discrete ordinal probability distributions to be unimodal via the use of the Poisson and binomial probability distributions. We evaluate this approach in the context of deep learning on two large ordinal image datasets, obtaining promising results.

Bayesian Inference and Model Assessment for Spatial Point Patterns Using Posterior Predictive Samples

Abstract: Spatial point pattern data describes locations of events observed over a given domain, with the number of and locations of these events being random. Historically, data analysis for spatial point patterns has focused on rejecting complete spatial randomness and then on fitting a richer model specification. From a Bayesian standpoint, the literature is growing but primarily considers versions of Poisson processes, focusing on specifications for the intensity. However, the Bayesian literature on, e.g., clustering or inhibition processes is limited, primarily attending to model fitting. There is little attention given to full inference and scant with regard to model adequacy or model comparison. The contribution here is full Bayesian analysis, implemented through generation of posterior point patterns using composition. Model features, hence broad inference, can be explored through functions of these samples. The approach is general, applicable to any generative model for spatial point patterns. The approach is also useful in considering model criticism and model selection both in-sample and, when possible, out-of-sample. Here, we adapt or extend familiar tools. In particular, for model criticism, we consider Bayesian residuals, realized and predictive, along with empirical coverage and prior predictive checks through Monte Carlo tests. For model choice, we propose strategies using predictive mean square error, empirical coverage, and ranked probability scores. For simplicity, we illustrate these methods with standard models such as Poisson processes, log-Gaussian Cox processes, and Gibbs processes. The utility of our approach is demonstrated using a simulation study and two real datasets.

Poisson Random Fields for Dynamic Feature Models

Abstract: We present the Wright-Fisher Indian buffet process (WF-IBP), a probabilistic model for time-dependent data assumed to have been generated by an unknown number of latent features. This model is suitable as a prior in Bayesian nonparametric feature allocation models in which the features underlying the observed data exhibit a dependency structure over time. More specifically, we establish a new framework for generating dependent Indian buffet processes, where the Poisson random field model from population genetics is used as a way of constructing dependent beta processes. Inference in the model is complex, and we describe a sophisticated Markov Chain Monte Carlo algorithm for exact posterior simulation. We apply our construction to develop a nonparametric focused topic model for collections of time-stamped text documents and test it on the full corpus of NIPS papers published from 1987 to 2015.

Approaches for dealing with various sources of overdispersion in modeling count data: Scale adjustment versus modeling.

Abstract: Overdispersion is a common problem in count data. It can occur due to extra population-heterogeneity, omission of key predictors, and outliers. Unless properly handled, this can lead to invalid inference. Our goal is to assess the differential performance of methods for dealing with overdispersion from several sources. We considered six different approaches: unadjusted Poisson regression (Poisson), deviance-scale-adjusted Poisson regression (DS-Poisson), Pearson-scale-adjusted Poisson regression (PS-Poisson), negative-binomial regression (NB), and two generalized linear mixed models (GLMM) with random intercept, log-link and Poisson (Poisson-GLMM) and negative-binomial (NB-GLMM) distributions. To rank order the preference of the models, we used Akaike's information criteria/Bayesian information criteria values, standard error, and 95% confidence-interval coverage of the parameter values. To compare these methods, we used simulated count data with overdispersion of different magnitude from three different sources. Mean of the count response was associated with three predictors. Data from two real-case studies are also analyzed. The simulation results showed that NB and NB-GLMM were preferred for dealing with overdispersion resulting from any of the sources we considered. Poisson and DS-Poisson often produced smaller standard-error estimates than expected, while PS-Poisson conversely produced larger standard-error estimates. Thus, it is good practice to compare several model options to determine the best method of modeling count data.

Bayesian causality test for integer‐valued time series models with applications to climate and crime data

Abstract: Summary We investigate the causal relationship between climate and criminal behaviour. Considering the characteristics of integer-valued time series of criminal incidents, we propose a modified Granger causality test based on the generalized auto-regressive conditional heteroscedasticity type of integer-valued time series models to analyse the relationship between the number of crimes and the temperature as an environmental factor. More precisely, we employ the Poisson, negative binomial and log-linear Poisson integer-valued generalized auto-regressive conditional heteroscedasticity models and particularly adopt a Bayesian method for our analysis. The Bayes factors and posterior probability of the null hypothesis help to determine the causality between the variables considered. Moreover, employing an adaptive Markov chain Monte Carlo sampling scheme, we estimate model parameters and initial values. As an illustration, we evaluate our test through a simulation study and, to examine whether or not temperature affects crime activities, we apply our method to data sets categorized as sexual offences, drug offences, theft of motor vehicles, and domestic-violence-related assault in Ballina, New South Wales, Australia. The result reveals that more sexual offences, drug offences and domestic-violence-related assaults occur during the summer than in other seasons of the year. This evidence strongly advocates a causal relationship between crime and temperature.

Poisson algebras via model theory and differential-algebraic geometry

Abstract: Brown and Gordon asked whether the Poisson Dixmier–Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier–Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier–Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

Reliability Analysis of V2V Communications on Orthogonal Street Systems

Abstract: The analysis of vehicle-to-vehicle communications is generally limited to vehicles on street segments, which are modeled using 1-D point processes. However, it is essential to model the intersections which are crucial for vehicle safety. In this paper, we focus on orthogonal street systems involving intersections with Poisson distributed vehicles on each street. We derive analytical expressions for the success probabilities of two types of users-- the typical general user and the typical intersection user. We show that the the orthogonal street system shares some properties of both 1-D and 2-D Poisson networks. Specifically, the vehicles on the street system behave like 1-D and 2-D Poisson point processes of vehicles in the high-reliability and low-reliability regimes, respectively. Also, we deduce that the success probability of the typical general/intersection user is upper bounded by the minimum of the success probabilities of the 1-D and 2-D Poisson networks.

Connection Establishment in LTE-A Networks: Justification of Poisson Process Modeling

Abstract: The connection establishment in Long-Term Evolution Advanced (LTE-A) is often executed for distributed user equipment (UE) nodes with frequent small data sets for transmission to the central enhanced Node B. LTE-A connection establishment consists mainly of an access barring check (ABC) followed by preamble transmission (contention). Previous studies of connection establishment have often assumed Poisson characteristics (without verifying the Poisson assumption). In this paper, we introduce a simple equilibrium analysis framework for comprehensively evaluating the LTE-A connection establishment, including both access barring and preamble contention. We conduct a detailed analysis of the backlog arising from the uniform backoff over up to $T_{o}^{\max}$ slots by UE requests that failed the barring check or collided in the preamble contention. We verify that the process representing the numbers of backlogged UE requests rejoining the connection establishment tends to Poisson process characteristics for high barring probability and long maximum timeout $T_{o}^{\max}$ . We present numerical comparisons of our equilibrium model with simulations for practical parameter settings. The comparisons illustrate the effects of the parameter settings on the convergence of the LTE-A connection establishment dynamics to Poisson characteristics for nonsynchronized and synchronized request arrivals.

Error Performance Analysis of Diffusive Molecular Communication Systems With On-Off Keying Modulation

Abstract: In this paper, we analyze the bit error rate (BER) of the diffusive molecular communication (DMC) systems employing on-off keying (OOK) modulation. We also analyze the BER of the OOK-modulated DMC systems with inter-symbol interference cancellation (ISIC). Our main motivation is to introduce alternative tools for analyzing and efficiently computing the BER of the DMC systems without or with ISIC. Specifically, for the OOK-modulated DMC systems without ISIC, we first derive an exact BER expression based on the Poisson modeling of DMC systems. Then, the Gaussian- and Gamma-approximation approaches are introduced to approximate the discrete Poisson distribution, and based on the approximation approaches, the corresponding BER expressions are derived. Furthermore, in order to reduce the computation complexity imposed by long inter-symbol interference, we propose the Monte-Carlo, simplified Poisson, simplified Gaussian, and the simplified Gamma approaches for BER computation. In the context of the OOK-modulated DMC systems with ISIC, we consider both the Poisson and Gaussian-approximation approaches for BER analysis. Again, exact and approximate BER expressions are derived under the Poisson, Gaussian-approximation, simplified Poisson, and simplified Gaussian approaches. Finally, the considered approaches are compared and validated by a range of performance results obtained from evaluation of the derived expressions or by simulations. Our studies show that the alternative approaches are in general effective for providing near-accurate BER estimation.