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Proportional data arise frequently in a wide variety of fields of study. Such data often exhibit extra variation such as over/under dispersion, sparseness and zero inflation. For example, the hepatitis data present both sparseness and zero inflation with 19 contributing non-zero denominators of 5 or less and with 36 having zero seropositive out of 83 annual age groups. The whitefly data consists of 640 observations with 339 zeros (53%), which demonstrates extra zero inflation. The catheter management data involve excessive zeros with over 60% zeros averagely for outcomes of 193 urinary tract infections, 194 outcomes of catheter blockages and 193 outcomes of catheter displacements. However, the existing models cannot always address such features appropriately. In this paper, a new two-parameter probability distribution called Lindley–binomial (LB) distribution is proposed to analyze the proportional data with such features. The probabilistic properties of the distribution such as moment, moment generating function are derived. The Fisher scoring algorithm and EM algorithm are presented for the computation of estimates of parameters in the proposed LB regression model. The issues on goodness of fit for the LB model are discussed. A limited simulation study is also performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates. The proposed model is illustrated through three aforementioned proportional datasets. [ABSTRACT FROM AUTHOR]

In unsupervised learning, clustering is a common starting point for data processing. The convex or concave fusion clustering method is a novel approach that is more stable and accurate than traditional methods such as k-means and hierarchical clustering. However, the optimization algorithm used with this method can be slowed down significantly by the complexity of the fusion penalty, which increases the computational burden. This paper introduces a random projection ADMM algorithm based on the Bernoulli distribution and develops a double random projection ADMM method for high-dimensional fusion clustering. These new approaches significantly outperform the classical ADMM algorithm due to their ability to significantly increase computational speed by reducing complexity and improving clustering accuracy by using multiple random projections under a new evaluation criterion. We also demonstrate the convergence of our new algorithm and test its performance on both simulated and real data examples. [ABSTRACT FROM AUTHOR]

Compositional data (CoDa) often appear in various fields such as biology, medicine, geology, chemistry, economics, ecology and sociology. Although existing Dirichlet and related models are frequently employed in CoDa analysis, sometimes they may provide unsatisfactory performances in modelling CoDa as shown in our first real data example. First, this paper develops a multivariate compositional inverse Gaussian (CIG) model as a new tool for analysing CoDa. By incorporating the stochastic representation (SR), the expectation–maximization (EM) algorithm (aided by a one-step gradient descent algorithm) can be established to solve the parameter estimation for the proposed distribution (model). Next, zero observations may be often encountered in the real CoDa analysis. Therefore, the second aim of this paper is to propose a new model (called as ZCIG model) through a novel mixture SR based on both the CIG random vector and a so-called zero-truncated product Bernoulli random vector to model CoDa with zeros. Corresponding statistical inference methods are also developed for both cases without/with covariates. Two real data sets are analysed to illustrate the proposed statistical methods by comparing the proposed CIG and ZCIG models with existing Dirichlet and logistic-normal models. [ABSTRACT FROM AUTHOR]

This paper propose a binomial distribution model based on Bayesian statistics in the field of monitoring artificial radionuclides. A multi time-interval analysis method is adopted to extract the 214Bi–214Po correlated events counts in the 222Rn decay series. Then the model evaluates natural background using the ratio of correlated events counts to total α counts. The Beta distribution is used as the prior distribution of this model. We detailed discussed the process of posterior estimation, and compared the new approach with the conventional approaches. The results show the new approach has a more sensitive response to artificial radionuclides. [ABSTRACT FROM AUTHOR]

In this paper, a new threshold INAR(1) model based on modified negative binomial operator with random coefficient is proposed. Basic probabilistic and statistical properties of this process are established. Then the conditional least squares (CLS) and the conditional maximum likelihood (CML) methods are applied to estimate the model parameters when the threshold value is known or not. The asymptotic properties of the CLS-estimator and the CML-estimator are also been discussed. A method to test the constancy of the autoregressive parameters is provided. As an illustration, a simulation study is conducted to illustrate the performances of these estimators and present an empirical analysis of monthly counts of break and enter non-dwelling in Bellingen. [ABSTRACT FROM AUTHOR]

Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work. [ABSTRACT FROM AUTHOR]

Summary: The binomial proportion is a classic parameter with many applications and has also been extensively studied in the literature. By contrast, the reciprocal of the binomial proportion, or the inverse proportion, is often overlooked, even though it also plays an important role in various fields. To estimate the inverse proportion, the maximum likelihood method fails to yield a valid estimate when there is no successful event in the Bernoulli trials. To overcome this zero‐event problem, several methods have been introduced in the previous literature. Yet to the best of our knowledge, there is little work on a theoretical comparison of the existing estimators. In this paper, we first review some commonly used estimators for the inverse proportion, study their asymptotic properties, and then develop a new estimator that aims to eliminate the estimation bias. We further conduct Monte Carlo simulations to compare the finite sample performance of the existing and new estimators, and also apply them to handle the zero‐event problem in a meta‐analysis of COVID‐19 data for assessing the relative risks of physical distancing on the infection of coronavirus. [ABSTRACT FROM AUTHOR]

In this paper, event-triggered control for uncertain delayed neural networks (DNNs) with actuator saturation against deception attack is discussed. We propose a novel framework into which an event-triggered mechanism (ETM), actuator saturation, system uncertainty, and deception attack are combined. In the framework, the discrete ETM is employed to determine whether the sampled signal should be transmitted to controller so as to save network bandwidth, a random occurrence deception attack model satisfying Bernoulli distribution is introduced to construct the system robust, and actuator saturation is considered due to the actual complex network environment. Based on the framework, we discussed the stochastic stability of a novel delayed neural network model in a closed-loop system. By resorting to the appropriate Lyapunov–Krasovskii functional (LKF), we derive some new sufficient conditions for the stochastically stable of the system and obtain the gain of the system controller using efficient linear matrix inequality (LMI) method. Finally, a numerical example, in the end, demonstrates the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The shared frailty models are now popular for modelling heterogeneity in survival analysis. It assumes that the same frailty is shared by all individual members within the families. Also, it is believed that all the individuals in the population are susceptible to the event of interest and will eventually experience the event. This may not always be the situation in reality. There may be a certain fraction of the population which is non-susceptible for an event and hence may not experience the event under study. Non-susceptibility is modelled by frailty models with compound frailty distribution. Further, susceptibility may be different for different families. This can be attained by randomizing the parameter of frailty distribution. This paper incorporates both the things, non-susceptibility and different susceptibility for different families by considering compound negative binomial distribution with random probability of susceptibility as frailty distribution. The inferential problem is solved in a Bayesian framework using Markov Chain Monte Carlo methods. The proposed model is then applied to a real-life data set. [ABSTRACT FROM AUTHOR]

The object of research is the method and algorithm of arithmetic addition of binomial numbers generated by binary binomial counting systems. The lack of binomial arithmetic, in particular the operation of adding binary binomial numbers, in a certain way prevents their introduction into information systems and the construction of information and communication technologies based on them for combinatorial optimization, generation of combinatorial objects, data compression and encryption. In the framework of the proposed approach, instead of operating with binomial coefficients, only operations with their upper and lower parameters are carried out. At the same time, the weighting coefficients of binary binomial numbers, which are added to each other, are represented in the form of two-component tuples. Taking this into account, this paper presents an algorithm for binomial arithmetic addition using dynamic arrays. The main idea, which is included in the structure of the algorithm of binomial arithmetic addition based on dynamic arrays, is that the transition from a two-dimensional model of summation to a one-dimensional one is carried out. At the same time, only available, existing binomial coefficients are placed in the dynamic array. Accordingly, the search for binomial coefficients equal to or greater than the quantitative equivalent takes place in much smaller areas. In comparison with the algorithm based on matrix models, this quite significantly reduces the amount of time spent when performing the summation operation, and also reduces the requirements for the amount of memory required for placing two-component tuples of the assembly array. In the course of the research, a several-fold decrease in the number of machine cycles required to search for the necessary elements in the dynamic array was practically confirmed. This leads to an increase in the performance of the presented algorithm of binomial arithmetic addition based on dynamic arrays. In turn, this leads to the acceleration of solving information tasks of combinatorial optimization, generation of combinatorial objects, data compression and encryption, for the solution of which the operation of adding binary binomial numbers is used. [ABSTRACT FROM AUTHOR]

For state estimation of multi-source asynchronous measurement systems with measurement missing phenomena, this paper proposes a distributed sequential inverse covariance intersection (DSICI) fusion algorithm based on conditional Kalman filtering method. It is mainly divided into synchronized state space module, local filtering module and fusion estimation module. The missing measurements occurring in the system are modelled and described by a set of random variables obeying a Bernoulli distribution. The synchronized state space module uses a state iteration method to synchronize the asynchronous measurement system at the moment of measurement update and it ensures the integrity of the measurement information. The local filtering module uses a conditional Kalman filtering algorithm for filter estimation. The reliability of the local filtering results is guaranteed because the local estimator designs a method to interact information with the domain sensors. The fusion estimation module designs a DSICI fusion algorithm with higher accuracy and satisfying consistency, which fuses the filtering results provided by each sensor when the relevant information between multiple sensors is unknown. Simulation examples demonstrate the excellent performance of the proposed algorithm, with a 33% improvement in accuracy over existing algorithms and an iteration time of less than 3 ms. [ABSTRACT FROM AUTHOR]

This paper considers the problem of randomly varying trial lengths of a class of two-dimensional (2D) linear systems represented by Roesser model, and two new iterative learning control (ILC) schemes are proposed. One is a conventional P-type control rule with a modified tracking error; the other is an ILC law with an iteration-average operator. Both learning control schemes are developed by a 2D stochastic variable to describe varying trial lengths. Under the Bernoulli distribution assumption and the initial state conditions, the convergence analysis is performed rigorously in the probability sense. Finally, illustrative examples have been provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Sensitive data, such as financial, personal, or classified governmental information, must be protected throughout its cycle. This paper studies the problem of safeguarding transmitted data based on data categorization techniques. This research aims to use a novel routine as a new meta-heuristic to enhance a novel data categorization based-traffic classification technique where private data is classified into multiple confidential levels. As a result, two packets belonging to the same confidentiality level cannot be transmitted through two routers simultaneously, ensuring a high data protection level. Such a problem is determined by a non-deterministic polynomial-time hardness (NP-hard) problem; therefore, a scheduling algorithm is applied to minimize the total transmission time over the two considered routers. To measure the proposed scheme's performance, two types of distribution, uniform and binomial distributions used to generate packets transmission time datasets. The experimental result shows that the most efficient algorithm is the Best-Random Algorithm (BR˜), recording 0.028 s with an average gap of less than 0.001 in 95.1% of cases compared to all proposed algorithms. In addition, BR˜ is compared to the best-proposed algorithm in the literature which is the Modified decreasing Estimated-Transmission Time algorithm (MDETA). The results show that BR˜ is the best one in 100% of cases where MDETA reaches the best results in only 48%. [ABSTRACT FROM AUTHOR]

This paper establishes the strong consistency and asymptotic normality of the negative binomial quasi-maximum likelihood estimator (NBQMLE) for a general class of integer-valued time series models whose parameters are periodic time-varying. This class of models is specified via its conditional mean, which is expressed as a periodic parametric function of (infinite) past observations. An illustration on specific models, namely the Poisson periodic INGARCH model, the negative binomial periodic INGARCH model, and the periodic INAR(1) model is given. In addition, the performance of the NBQMLE is assessed through a simulation study, and an application of the proposed methodology to a real dataset is provided. [ABSTRACT FROM AUTHOR]

The standard binomial distribution is based on a sequence of Bernoulli trials. The probability of success is the same for each trial. In the Generalized Binomial Distribution (GBD), the probability of success changes by the history of successes up to that point. For example, if a student correctly answers a large proportion of the first few questions, the probability of correctly answering the next question increases because the history of successes may indicate skill. The Bernoulli events are correlated. Alternatively, the standard binomial distribution does not consider the history of successes, and assumes that successes are not correlated. In this paper we prove some new properties of the GBD and apply the concept to five new areas in sport competition, finance, biology, meteorology, and testing pseudo-random number generators. We prove that the GBD predicts the results in the collected data significantly better than the standard binomial distribution thus proving that the events are indeed correlated. We provide an R code, and a supplementary Excel file that provide users with an easy way to analyze any collected data set based on Bernoulli processes. [ABSTRACT FROM AUTHOR]

We propose an approach to construct a new family of generalized Farlie–Gumbel–Morgenstern (GFGM) copulas that naturally scales to high dimensions. A GFGM copula can model moderate positive and negative dependence, cover different types of asymmetries, and admits exact expressions for many quantities of interest such as measures of association or risk measures in actuarial science or quantitative risk management. More importantly, this paper presents a new method to construct high-dimensional copulas based on mixtures of power functions and may be adapted to more general contexts to construct broader families of copulas. We construct a family of copulas through a stochastic representation based on multivariate Bernoulli distributions and Coxian-2 distributions. This paper will cover the construction of a GFGM copula and study its measures of multivariate association and dependence properties. We explain how to sample random vectors from the new family of copulas in high dimensions. Then, we study the bivariate case in detail and find that our construction leads to an asymmetric modified Huang–Kotz FGM copula. Finally, we study the exchangeable case and provide insights into the most negative dependence structure within this new class of high-dimensional copulas. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of asymmetric beta-binomial generalized autoregressive conditional heteroscedastic (GARCH) models for bounded integer-valued time series, which can capture the asymmetric impact of positive and negative observations. We study the stationarity conditions of the process and derive the moment and covariance functions. Furthermore, we estimate the unknown parameters using the conditional maximum likelihood (CML) method. The asymptotic properties of the estimators are discussed, as well as their finite-sample performance. Finally, we illustrate the model to real time series data in the field of meteorology. • A new asymmetric beta-binomial GARCH model for bounded Z-valued time series. • Asymmetric impact of positive and negative observations. • CML estimation with asymptotic properties. • A real time series data in the field of meteorology. [ABSTRACT FROM AUTHOR]

In this article, the hybrid cyber-attacks for singular Markov jump systems (MJSs) are mainly solved by designing the asynchronous mixed H ∞ and passive controller. A kind of dynamic event-triggered (ETM) based strategy is adopted to maximize the savings of network bandwidth resources. And the hybrid cyber-attacks including two common attacks, the spoofing attack and the denial of service (DoS) attack are considered, which are assumed to be mutually independent and meet Bernoulli distribution. The singular MJSs under cyber-attacks are converted into the singular Markov jump time-delay systems. Then, the criteria of stochastically stable for the closed-loop singular MJSs meeting the mixed H ∞ and passive performance are obtained by linear matrix inequalities (LMIs). And by optimizing the obtained LMIs, the gains of asynchronous resilient controller are further derived, and the weighting matrices of dynamic event-triggered based strategy are also gained. Eventually, the feasibility of the offered way is verified by an industrial example of DC motor control system. • In this paper, the resilient controller for the singular Markov jump systems is considered. • A dynamic event-triggered based strategy is introduced to further save network bandwidth. • Occurrence of hybrid cyber-attacks are described by two mutually unrelated random variables satisfying Bernoulli distribution. • Criteria of stochastically admissible for singular Markov jump CPSs under hybrid cyber-attacks are obtained by a set of LMIs. • The codesign of resilient controller and the dynamic event-triggered based strategy is presented. [ABSTRACT FROM AUTHOR]

Benchmarking and comparative testing of three approaches for direct counting of stability failures are described. These approaches are based on the estimation of the failure rate from sample data using exponential distribution of time to failure, statistical frequency of failures and binomial distribution. All three approaches were included in the Explanatory Notes for the Second-Generation IMO Intact Stability Criteria. A brief description of the respective step-by-step procedures is included in the paper for convenience. The benchmarking is carried out using data synthesized from a Poisson process. A practical issue of decision making, based on direct counting, is discussed and the influence of the sample size is analyzed. • Benchmarking and comparative testing of three approaches for direct counting of stability failures are described. • The benchmarking is carried out using data synthesized from a Poisson distribution. • The three tested approaches were included in the draft Explanatory Notes for the Second-Generation IMO Intact Stability Criteria. • The rate estimators, their bias, variance, and the confidence intervals for failure rate are analyzed. • A practical issue of decision making, based on direct counting, is discussed and the influence of the sample size is analyzed. [ABSTRACT FROM AUTHOR]

This paper proposes an approach for anomaly/damage detection via structural health monitoring (SHM) systems based on a Binomial Distribution Classifier (BDC). The approach consists of two monitoring levels, labeled as alert and alarm states, respectively, where a damage index (DI) is computed and tracked over time. The alert state is reached when the DI exceeds a given threshold. At this stage, the BDC algorithm starts counting the number of DI values above the threshold within an observation window and computing the related probability of occurrence by using the binomial probability distribution. If the probability falls below a desired limit, the alarm state is triggered. Conversely, the SHM system returns in the non-alert condition. The proposed approach is discussed and evaluated through case studies involving both simulated and experimental data. In the examples, the DI is computed using the Mahalanobis distance of the monitored modal frequencies. The results demonstrate the capability of the BDC to reduce false alarms while preserving the probability of detection. [ABSTRACT FROM AUTHOR]

In this paper we introduce the mixed censored δ -shock model which combines the censored δ -shock model and the classical extreme shock model. Under the mixed censored δ -shock model, the system fails whenever no shock occurs within a δ -length time period from the last shock, or the magnitude of the shock is larger than another critical threshold γ > 0. For the discrete-time case of occurrence of shocks, by assuming the dependence between intershock times and the corresponding magnitudes of shocks we derive the probability generating function (pgf) of the lifetime of the system, and a matrix-based expression is obtained for the exact distribution of the system's lifetime when the distribution of intershock times and the magnitudes of shocks have a discrete bivariate phase-type distribution. Similar results are obtained by assuming the independence between intershock times and the corresponding magnitudes of shocks, and by proving that the distribution of the shifted lifetime at δ , is the convolution of a discrete compound geometric distribution and a discrete compound Bernoulli distribution, we get several results concerning the distribution of system's lifetime, like as, simple and efficient recursions for evaluating the survival function and the probability mass function (pmf), the mean and the variance of system's lifetime, as well as discrete Lundberg-type upper bounds for the reliability function. For the continuous-time case of occurrence of shocks we obtain an exact formula for the reliability function and the Laplace–Stieltjes transform of system's lifetime by assuming the dependence between intershock times and the corresponding magnitudes of shocks, and under the independence setup, we obtain the reliability function when the intershock times have the uniform distribution, and we give an asymptotic result under the Poisson process for the arrival of shocks. Similar to the discrete-time case, it is shown that the distribution of the shifted lifetime at δ , is the convolution of a compound geometric distribution and a compound Bernoulli distribution and using this we obtain two-sided Lundberg-type bounds for the survival function. Finally, some numerical examples to illustrate our results, are also given. [ABSTRACT FROM AUTHOR]