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The trust of Mobile Nodes inside the network is considered on the grounds that the potential factor that needs to be explored for encouraging prevalent network execution. This network execution impacting trust is corrupted by the presence of deliberately carrying on selfish Nodes that doesn't advance information parcels to their local agreeable Mobile Nodes. These narrow minded Nodes decrease the parcel forward action depending on its own vitality state for moderating its vitality in order to gauge dynamic inside the network. Here, a Modified Skellam Distribution - Based Selfish Node Detection Technique (MSD-SNDT) is proposed for authorizing powerful and productive location and segregation of selfishly carrying on portable Nodes from the network. The reproduction tests is directed for evaluating the greatness of the proposed MSD-SNDT approach regarding parcel conveyance, throughput, complete overhead and vitality utilizations under differing quantities of portable Nodes and narrow minded Nodes inside the network.

The trust of nodes in Mobile Ad hoc NETwork (MANET) is the potential factor that needs to be investigated for facilitating predominant network performance. The network performance influencing trust is degraded due to the presence of intentionally behaving selfish nodes that do not forward data packets to their adjoining cooperative mobile nodes. The selfish nodes lessen packet forwarding based on the energy level for conserving their energy and remain active in the network. In this paper, Skellam Distribution Inspired Trust Factor-based Selfish Node Detection Technique (SDITF-SNDT) is propounded for enforcing effective and efficient detection, and isolation of selfishly behaving mobile nodes from the network. The proposed SDITF-SNDT initiates the process of selfish node detection by determining the mean packet deviation through which the Standard Deviation (SD) and Variance are estimated for computing the Skellam Distribution Inspired Trust Factor (SDITF). This computation aids in reliability estimation of nodes in order to categorize them into selfish and cooperative nodes. From the investigations conducted for the proposed SDITF-SNDT approach, excellent improvement in Packet Delivery Ratio (PDR) and phenomenal minimization in energy consumption are confirmed for increased number of mobile nodes in the network.

Using European soccer data sets, which contain data related to common European soccer leagues, players basic information, and teams’ goals, etc., this paper analyzes the characteristics of European soccer and players, explores data visualization regarding European soccer, and makes predictions of results of matches. Based on Python 3 and some of the packages inside, such as numpy, the author improves the data set to make it clear and user-friendly. Visualizations of data and basic statistics, including Poisson Distribution, are then utilized to determine the results. Finally, this paper analyzes the attacking and defending abilities of different leagues and teams in Europe, ascertains distributions of players’ attributes, and predicts match results by using Poisson distribution and Skellam Distribution. Generally, this paper analyzes data from leagues to matches to players. All these analyses are meaningful for the public to understand the characteristics of European soccer and the world behind the numbers.

The degree that the environment determines what genes contribute towards adaptation is a fundamental question in microbial evolution. Microbial populations are often experimentally passaged in different environments and sequenced in order to identify candidates for adaptation in a particular environment. However, there remains the need to develop an appropriate statistical framework to identify genes that acquired more mutations in one environment over the other (i.e., divergent evolution). Here we demonstrate how the evolutionary outcomes among replicate populations in the same environment, known as parallel evolution, can be leveraged to construct an intuitive statistical test for identifying the genes that contribute towards divergent evolution. To accomplish this task, we examined publicly available evolve-and-resequence experiment datasets and found that the distribution of mutation counts among genes can be predicted using an ensemble of independent Poisson random variables. Building on this result, we propose that the degree of divergent evolution at a given gene between populations from two different environments can be modeled as the difference between two Poisson random variables, known as the Skellam distribution. We then propose and apply a statistical test to identify specific genes that contribute towards divergent evolution. IMPORTANCE: There is currently no existing framework that can be leveraged to identify genes that contribute towards divergent evolution in microbial evolution experiments. To correct for this absence, we investigated the distribution of mutation counts among genes in order to identify an appropriate null model. Our observations suggest that divergent evolution within a given gene can be modeled as the difference in the total number of mutations observed between two environments. This quantity is described by a probability distribution known as the Skellam distribution, providing an appropriate statistical test for researchers seeking to identify the set of genes that contribute towards divergent evolution in evolution experiments.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

Photon-limited Poisson image denoising is urgent demand in many application fields, but is particularly challenging because the image structures are often damaged seriously. The effective using of the image prior is very important for improving the quality of the denoised image. In this paper, we exploit the prior of inner statistical relationships among pixels in local spatial neighborhood of the Poisson noisy image and introduce a new Skellam distribution based inner interaction potential function as a distance measurement between the pixels. In the framework of conditional random field (CRF) modeling, we propose a novel Poisson-Skellam distribution based regularization CRF model for photon-limited Poisson noise removal. Using the alternating direction method of multipliers technique, we extend our method to a flexible plug-and-play scheme in which we can combine powerful Gaussian denoising method to improve the denoising performance better.

We investigate the network selection problem over heterogeneous networks. We formulate the decision making problem as a Markov decision process (MDP). Two Poisson processes are introduced to model user arrival and departure, and the change of the number of user in a specific network is described by Skellam Distribution. We apply the Total order preference by similarity to the ideal solution (TOPSIS) method to address the issue of state space explosion in MDP model. To acquire the optimal network selection scheme, the value iteration algorithm is employed to solve the MDP model. We take simulations in Matlab to prove the effectiveness of the proposed scheme. From the experiment results, it is found that our proposed scheme can reduce handover probability by 7.5% and 40.4% at the early stage of connection, compared with the Infinite MDP-based (IMDP) strategy and TOPSISonly strategy. It can also reduce blocking probability by 6.1% compared with IMDP, while maintaining desirable throughput for the target user.

The Inter-Symbol Interference (ISI) in Diffusive Molecular Communication (DMC) is unavoidable due to the stochastic behavior of molecular Brownian motion in a biological environment. In this paper, we analyze the channel capacity of DMC model by considering the ISI from all the previous time slots and the channel transmission probability in each time slot. We first use the Poisson distribution to approximate the Binomial distribution in order to obtain the total number of molecules received in the current time slot. Then we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. On this basis, we formulate the mutual information and bit error probability. Our numerical results are presented to show the channel capacity can be maximized with optimal lower bit error probability by controlling the channel transmission probability in each time slot and using the Skellam distribution. More importantly, under the same physical parameters, a higher capacity is achieved with using fewer molecules in our work.

The aim of this paper is to study and develop Bayesian models for the analysis of volleyball match outcomes as recorded by the set-difference. Due to the peculiarity of the outcome variable (set-difference) which takes discrete values from $-3$ to $3$, we cannot consider standard models based on the usual Poisson or binomial assumptions used for other sports such as football/soccer. Hence, the first and foremost challenge was to build models appropriate for the set-differences of each volleyball match. Here we consider two major approaches: a) an ordered multinomial logistic regression model and b) a model based on a truncated version of the Skellam distribution. For the first model, we consider the set-difference as an ordinal response variable within the framework of multinomial logistic regression models. Concerning the second model, we adjust the Skellam distribution in order to account for the volleyball rules. We fit and compare both models with the same covariate structure as in Karlis & Ntzoufras (2003). Both models are fitted, illustrated and compared within Bayesian framework using data from both the regular season and the play-offs of the season 2016/17 of the Greek national men's volleyball league A1., IMA Journal of Management Mathematics, 2021

We determine the net-baryon multiplicity distribution which reproduces all cumulants measured so far by lattice QCD. We present the dependence on the volume and temperature of this distribution. We find that for temperatures and volumes encountered in heavy ion reactions, the multiplicity distribution is very close to the Skellam distribution, making the experimental determination of it rather challenging. We further provide estimates for the statistics required to measure cumulants of the net-baryon and net-proton distributions., Comment: 13 pages, 6 figures; Extended version. Now include statistics estimate for RHIC and LHC based on delta method

Two bivariate extensions of the Skellam distribution were introduced in 2014 by a subset of the present authors, who also proposed moment estimators for the dependence parameters in these models. The limiting distribution of these estimators is established here, and their asymptotic efficiency is compared with that of the corresponding maximum likelihood estimators. © 2017 The Authors. Stat Published by John Wiley & Sons Ltd.

We derive a set of recurrence relations for the calculation of the probabilities of a large class of integer-valued random variables. We show that the probability mass function can be recursively computed for random variables with a probability generating function satisfying certain functional form.

The tick structure of the financial markets entails that price changes observed at very high frequency are discrete. Departing from this empirical evidence we develop a new model to describe the dynamic properties of multivariate time-series of high frequency price changes, including the high probability of observing no variations (price staleness). We assume the existence of two independent latent/hidden Markov processes determining the dynamic properties of the price changes and the excess probability of the occurrence of zeros. We study the probabilistic properties of the model that generates a zero-inflated mixture of Skellam distributions and we develop an EM estimation procedure with closed-form M step. In the empirical application, we study the joint distribution of the price changes of four assets traded on NYSE. Particular focus is dedicated to the precision of the univariate and multivariate density forecasts, to the quality of the predictions of quantities like the volatility and correlations across assets, and to the possibility of disentangling the different sources of zero price variation as generated by absence of news, microstructural frictions or by the offsetting positions taken by the traders.

In this paper, an integer-valued autoregressive model of order one (INAR(1)) with time-varying parameters and driven by a periodic sequence of innovations is introduced. The proposed INAR(1) model is based on the signed thinning operator defined by Kachour and Truquet (2011) and conveniently adapted to the periodic case. Basic notations and definitions concerning the periodic signed thinning operator are provided. Based on this thinning operator, Chesneau and Kachour (2012) established a signed INAR(1) model. Motivated by the work of Chesneau and Kachour (2012), we introduce a periodic model, denoted by S-PINAR(1), with period s. In contrast to conventional INAR(1) models, these models are defined in \(\mathbb {Z}\) allowing for negative values both for the series and its autocorrelation function. For a proper \(\mathbb {Z}\)-valued time series, a distribution for the innovation term defined on \(\mathbb {Z}\) is required. The S-PINAR(1) model assumes a specific innovation distribution, the Skellam distribution. Regarding parameter estimation, two methods are considered: conditional least squares and conditional maximum likelihood. The performance of the S-PINAR(1) model is assessed through a simulation study.

This paper considers the modeling of nonstationary integer valued time series with conditional heteroskedasticity using Skellam distribution. Two approaches of estimation of the model’s parameters are treated and discussed. The obtained results are verified through some numerical simulation. In addition, the proposed model is applied to real time series.

The probability of informed trading (PIN) is a commonly used market microstructure measure for detecting the level of information asymmetry. Estimating PIN can be problematic due to corner solutions, local maxima and floating point exceptions (FPE). Yan and Zhang [J. Bank. Finance, 2012, 36, 454–467] show that whilst factorization can solve FPE, boundary solutions appear frequently in maximum likelihood estimation for PIN. A grid search initial value algorithm is suggested to overcome this problem. We present a faster method for reducing the likelihood of boundary solutions and local maxima based on hierarchical agglomerative clustering (HAC). We show that HAC can be used to determine an accurate and fast starting value approximation for PIN. This assists the maximum likelihood estimation process in both speed and accuracy.

The pixel values of images taken by an image sensor are said to be corrupted by Poisson noise. To date, multiscale Poisson image denoising techniques have processed Haar frame and wavelet coefficients—the modeling of coefficients is enabled by the Skellam distribution analysis. We extend these results by solving for shrinkage operators for Skellam that minimizes the risk functional in the multiscale Poisson image denoising setting. The minimum risk shrinkage operator of this kind effectively produces denoised wavelet coefficients with minimum attainable $L^{2}$ error.

In this work we investigate the problem of private statistical analysis of time-series data in the distributed and semi-honest setting. In particular, we study some properties of Private Stream Aggregation (PSA), first introduced by Shi et al. 2011. This is a computationally secure protocol for the collection and aggregation of data in a distributed network and has a very small communication cost. In the non-adaptive query model, a secure PSA scheme can be built upon any key-homomorphic weak pseudo-random function as shown by Valovich 2017, yielding security guarantees in the standard model which is in contrast to Shi et al. We show that every mechanism which preserves \((\epsilon ,\delta )\)-differential privacy in effect preserves computational \((\epsilon ,\delta )\)-differential privacy when it is executed through a secure PSA scheme. Furthermore, we introduce a novel perturbation mechanism based on the symmetric Skellam distribution that is suited for preserving differential privacy in the distributed setting, and find that its performances in terms of privacy and accuracy are comparable to those of previous solutions. On the other hand, we leverage its specific properties to construct a computationally efficient prospective post-quantum protocol for differentially private time-series data analysis in the distributed model. The security of this protocol is based on the hardness of a new variant of the Decisional Learning with Errors (DLWE) problem. In this variant the errors are taken from the symmetric Skellam distribution. We show that this new variant is hard based on the hardness of the standard Learning with Errors (LWE) problem where the errors are taken from the discrete Gaussian distribution. Thus, we provide a variant of the LWE problem that is hard based on conjecturally hard lattice problems and uses a discrete error distribution that is similar to the continuous Gaussian distribution in that it is closed under convolution. A consequent feature of the constructed prospective post-quantum protocol is the use of the same noise for security and for differential privacy.

Two-stage non-local methods represented by the Poisson non-local means (PNLM) method [1] and the non-local principal component analysis (NLPCA) [2] method perform well for photon-limited Poisson image reconstruction, where patch-similarity computation in the non-local reconstruction stage is guided and affected by a pre-reconstructed image obtained at the first stage. In this paper, we propose a new method to provide a better pre-reconstructed image with low computational cost. Firstly, we propose an adaptive method for fitting the linear relationship between image intensity and Poisson parameter; Secondly, we obtain an initial estimated image according to this relationship. Lastly, we obtain new pre-reconstructed image by adjusting the initial estimation according to Skellam distribution. Numerical experiments show that our method can provide a better pre-reconstructed image, and therefore can improve the performance of the PNLM method and reduce the computational cost of the NLPCA method efficiently.

What are the most surprising, most impressive and most exciting matches in the history of the highest division of English football? Fans, clubs, critics and broadcasters often attempt to interpret and compare the outcomes of football matches. This can be useful as a means of building a reputation for and interest in a particular club, match or competition. However, such analysis is often subjective with fan bias, personal preference and the prize on offer influencing expectation and interpretation of the results. In this paper statistical measures are developed to identify the most surprising, most impressive and most exciting outcomes from an objective ex post point of view. To achieve this it is assumed that goal scoring in soccer can approximately be considered as a Poisson process. First, the most surprising outcomes are estimated as those with the lowest Poisson probability to occur. Second, the most impressive outcomes are identified by their so-called Skellam percentile position. This measure varie...

While models for integer valued time series are now abundant, there is a shortage of similar models when the time series refer to data defined on Z, i.e., in both the positive and negative integers. Such data occur in certain disciplines and the need for such models also appear when taking differences of a positive integer count time series. In addition one would often like to include covariates to explain variations in the variable of interest. In this article we construct a model doing all these assuming a specific innovation distribution and provide fully parametric inference, including prediction. Real data applications on accidents and financial returns are given. Finally we also discuss alternative models and extensions.

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.

In recent years, there has been a growing interest in modelling integred-valued time series. In this article, we propose a modified and generalized version of the first order rounded integer-valued autoregressive RINAR(1) model, originally introduced by Kachour and Yao (2009). Indeed, this class can be considered as an alternative of classical models based on the thinning operators. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. Least squares estimator of the model parameters is considered and its consistence is established. Finally, we describe the price change data using a model of the new class.

Financial prices are usually modelled as continuous, often involving geometric Brownian motion with drift, leverage and possibly jump components. An alternative modelling approach allows financial observations to take integer values that are multiples of a fixed quantity, the ticksize - the monetary value associated with a single change during the price evolution. In the case of high-frequency data, the sample exhibits diverse trading operations in a few seconds. In this context, the observables are assumed to be conditionally independent and identically distributed from either of two flexible likelihoods: the Skellam distribution - defined as the difference between two independent Poisson distributions - or a mixture of Geometric distributions. Posterior inference is obtained via adaptive Gibbs sampling algorithms. Comparisons of the models applied to high-frequency financial data is provided.

Ratios of cumulants of net proton-number fluctuations measured by the STAR Collaboration show strong deviations from a skellam distribution, which should describe thermal properties of cumulant ratios, if proton-number fluctuations are generated in equilibrium and a hadron resonance gas (HRG) model would provide a suitable description of thermodynamics at the freeze-out temperature. We present some results on sixth order cumulants entering the calculation of the QCD equation of state at non-zero values of the baryon chemical potential (mu_B) and discuss limitations on the applicability of HRG thermodynamics deduced from a comparison between QCD and HRG model calculations of cumulants of conserved charge fluctuations. We show that basic features of the $\mu_B$-dependence of skewness and kurtosis ratios of net proton-number fluctuations measured by the STAR Collaboration resemble those expected from a O(mu_B^2) QCD calculation of the corresponding net baryon-number cumulant ratios., Comment: 6 pages, 10 figures, plenary talk given at Strange Quark Matter 2016, Berkeley, USA, 27 June 2016 - 1 July 2016

Rankings based on counts are often presented to identify popular items, such as baby names, English words, or Web sites. This article shows that, in some examples, the number of correctly identified items can be very small. We introduce a standard error versus rank plot to diagnose possible misrankings. Then to explain the slowly growing number of correct ranks, we model the entire set of count data via a Zipf–Poisson ensemble with independent Xi ∼ Poi(Ni − α) for α > 1 and N > 0 and integers i ⩾ 1. We show that as N → ∞, the first n′(N) random variables have their proper order relative to each other, with probability tending to 1 for n′ up to (AN/log (N))1/(α + 2) for A = α2(α + 2)/4. We also show that the rate N 1/(α + 2) cannot be achieved. The ordering of the first n′(N) entities does not preclude for some interloping m > n′. However, we show that the first n″ random variables are correctly ordered exclusive of any interlopers, with probability tending to 1 if n″ ⩽ (BN/log (N))1/(α + 2) for any B < A....

By the laws of quantum physics, pixel intensity does not have a true value, but should be a random variable. Contrary to the conventional assumptions, the distribution of intensity may not be an additive Gaussian. We propose to directly model the intensity difference and show its validity by an experimental comparison to the conventional additive model. As a model of the intensity difference, we present a Skellam distribution derived from the Poisson photon noise model. This modeling induces a linear relationship between intensity and Skellam parameters, while conventional variance computation methods do not yield any significant relationship between these parameters under natural illumination. The intensity-Skellam line is invariant to scene, illumination, and even most of camera parameters. We also propose practical methods to obtain the line using a color pattern and an arbitrary image under natural illumination. Because the Skellam parameters that can be obtained from this linearity determine a noise distribution for each intensity value, we can statistically determine whether any intensity difference is caused by an underlying signal difference or by noise. We demonstrate the effectiveness of this new noise model by applying it to practical applications of background subtraction and edge detection.

Motivated by features of low latency data in financial econometrics we study in detail integer-valued Levy processes as the basis of price processes for high-frequency econometrics. We propose using models built out of the difference of two subordinators. We apply these models in practice to low latency data for a variety of different types of futures contracts.

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer substantial performance improvements over existing alternatives., 27 pages, 8 figures, slight formatting changes; submitted for publication

We propose a method that compares Poisson distributed outcomes. Our method uses the exact distribution of the difference between two Poisson variables to calculate the sample size required to detect a given difference with prespecified power. When the true difference between the two Poisson rates is more than 1.2 units, the number of subjects and events needed at the desired power and Type I error rate is 5–10% less than that computed by simulation based on the normal approximation method. The normal approximation method is more comparable to the exact sample size method when the difference between the rates is less than 1.2 units. The proposed method is more intuitive, efficient, and less subjective than the normal approximation method. A simple code is developed in R to estimate the sample size and critical values.