Modeling Time Series Of Counts Research Articles

Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series

Many studies have used extensions of ARMA models for the analysis of non-Gaussian time series. One of them is the Generalized Autoregressive Moving Average, GARMA, enabling the modeling of count time series with distributions such as Poisson. The GARMA class is being expanded to accommodate other distributions, aiming to capture the typical characteristics of count data, including under or overdispersion and excess zeros. This study aims to propose an approach based on the GARMA class in order to analyze count time series with excess zeros, assuming a time-varying zero-adjusted Poisson distribution. This approach allows for capturing serial correlation, forecasting the future values, and estimating the future probability of zeros. For inference, a Bayesian analysis was adopted using the Hamiltonian Monte Carlo (HMC) algorithm for sampling from the joint posterior distribution. We conducted a simulation study and presented an application to influenza mortality reported in Brazil. Our findings demonstrated the usefulness of the model in estimating the probability of non-occurrence and the number of counts in future periods.

Modeling COVID-19 cases using NB-INGARCH and ARIMA models: A case study in Iligan City, Philippines

Modeling COVID-19 cases using count data approach has been scarce in the Philippine setting. This study compares the NB-INGARCH with the traditional ARIMA in modeling daily COVID-19 cases in Iligan City (August 14, 2020 – October 31, 2021). We employ the maximum likelihood estimation method and compare the models using the Akaike's information criterion (AIC). Empirical results reveal that the NB-INGARCH(7,0) outperforms ARIMA(2,1,3) in terms of forecast evaluation measures. However, the results show that rainfall and air pressure have no significant effects on the cases. We conclude that the NB-INGARCH model is a viable alternative approach to modeling count time series.

Investigation of femicide in Turkey: modeling time series of counts.

This study aims to reveal the relationship between the number of femicide in Turkey, the female unemployment rate, the male unemployment rate, and inflation. The contribution of the study to the literature is that it estimates the relationship between femicides and macroeconomic variables with a method that incorporates the count data. The dataset was analyzed by using the INGARCH model, one of the most popular approaches for count time series data. This model is particularly attractive for dealing with serial dependence and over-dispersion. The findings revealed that an increase in the female unemployment rate and a decrease in the male unemployment rate increases the number of femicide. In addition, it was observed that the number of femicide in the previous month had a negative effect on the current month's number. Increasing the employment rate of women, and women's economic freedom generally, are essential factors in reducing femicide. To prevent femicides, policymakers should aim to increase women's employment.

Robust Estimation for Poisson Integer-Valued GARCH Models Using a New Hybrid Loss

The Poisson integer-valued GARCH model is a popular tool in modeling time series of counts. The commonly used maximum likelihood estimator is strongly influenced by outliers, so there is a need to develop a robust M-estimator for this model. This paper has three aims. First, the authors propose a new loss function, which is a hybrid of the tri-weight loss for relatively small errors and the exponential squared loss for relatively large ones. Second, Mallows’ quasi-likelihood estimator (MQLE) is proposed as an M-estimator and its existence, uniqueness, consistency and asymptotic normality are established. In addition, a data-adaptive algorithm for computing MQLE is given based on a data-driven selection of tuning parameters in the loss function. Third, simulation studies and analysis of a real example are conducted to illustrate the performance of the new estimator, and a comparison with existing estimators is made.

Time Series Regression Models for COVID-19 Deaths

This article develops nonlinear functional forms for modeling count time series of daily deaths due to the COVID-19 virus. Our models explain the mean levels of the time series while accounting for the time-varying variances. A Bayesian approach using Markov chain Monte Carlo (MCMC) is adopted for analysis, inference and forecasting of the time series under the proposed models. Applications are shown for time series of death counts from several countries affected by the pandemic.

Poisson GSTAR model: spatial temporal modeling count data follow generalized linear model and count time series models

This paper discusses the formation of one temporal spatial model, the Generalized Space Time Autoregressive (GSTAR) model, if the data of model is count data. The development of the GSTAR model is an update or problem completion of the count data which tends to be stationary and non-normal / Gaussian data, because GSTAR model is assumed normal distributed and stationarity. GSTAR modelling for count data refers to the Time series model for count data, which are the Generalized Autoregressive Moving Average (GARMA) model and modeling Count Time Series which has the Generalized Linear Model (GLM) concept. The model formed is called the Poisson GSTAR model

Likelihood Inference for Generalized Integer Autoregressive Time Series Models

For modeling count time series data, one class of models is generalized integer autoregressive of order p based on thinning operators. It is shown how numerical maximum likelihood estimation is possible by inverting the probability generating function of the conditional distribution of an observation given the past p observations. Two data examples are included and show that thinning operators based on compounding can substantially improve the model fit compared with the commonly used binomial thinning operator.

Count Time-Series Analysis: A Signal Processing Perspective

Signal processing techniques are constantly expanding to accommodate a wider range of data structures and applications. A time series is a sequence of observations taken sequentially in time. Time-series analysis is concerned with techniques for the analysis of serial dependence and their use in practical applications, including 1) forecasting of future values from current and past values and 2) outlier detection and intervention analysis. Traditionally, time-series analysis has been applied to continuously varying data. However, in many areas of science and engineering we encounter count variables, i.e., variables that take on nonnegative integer values. Time series of counts are obtained in various disciplines whenever many events are counted during certain time periods. Examples include the monthly number of car accidents in a region, the weekly number of new cases in epidemiology, the number of transactions at a stock market per minute in finance, or the number of photon arrivals per microsecond in a focal-plane array. In some cases, the counts are large numbers and it makes sense to approximate them by continuous variables. However, there are many applications where the counts tend to be small and include many zeros. In this case, the observations cannot be adequately modeled with a continuous distribution. During the last three decades, there has been significant progress in the area of count time-series analysis [1], [2]. The main objective of this article is to present the state-of-the-art developments for modeling count time series in a signal processing framework by emphasizing the key theoretical, methodological, and practical application issues.

A negative binomial mixture integer-valued GARCH model

This paper generalizes the negative binomial integer-valued GARCH model (NBINGARCH) to a negative binomial mixture integer-valued GARCH (NBMINGARCH) for modeling time series of counts with presence of overdispersion. This class of models consists of a mixture of K stationary or non-stationary negative binomial integer-valued GARCH components. The advantage of these models over the NBINGARCH models includes the ability to handle multimodality and nonstationary components. Compared to the MINGARCH models, this class of models is more flexible to describe the greater degrees of overdispersion. The necessary and sufficient first and second order stationarity conditions are investigated. The estimation of parameters is done through an EM algorithm and the model is selected by some information criterions. Some simulation results and real data application are provided. Keywords: Time series of counts, Mixture model, Negative binomial, Overdispersion. AMS 2010 Mathematics Subject Classification : 62F10, 62M10

Mean targeting estimator for the integer-valued GARCH(1, 1) model

The integer-valued GARCH model is commonly used in modeling time series of counts. Maximum likelihood estimation (MLE) is used to estimate unknown parameters, but numerical results for MLE are sensitive to the choice of initial values, which also occurs in estimating the GARCH model. To alleviate this numerical difficulty, we propose an alternative to MLE and name it as mean targeting estimation (MTE), which is an analogue to variance targeting estimation used in the GARCH model. Consistency and asymptotic normality for MTE are established. Comparisons with the standard MLE are provided and the merits of the mean targeting method are discussed. In particular, it is shown that MTE can be superior to MLE for estimating parameters or prediction when the model is well specified and misspecified. We conduct numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposals.

An INAR(1) process for modeling count time series with equidispersion, underdispersion and overdispersion

We present a novel first-order nonnegative integer-valued autoregressive model for stationary count data processes with Bernoulli-geometric marginals based on a new type of generalized thinning operator. It can be used for modeling time series of counts with equidispersion, underdispersion and overdispersion. The main properties of the model are derived, such as probability generating function, moments, transition probabilities and zero probability. The maximum likelihood method is used for estimating the model parameters. The proposed model is fitted to time series of counts of iceberg orders and of cases of family violence illustrating its capabilities in challenging cases of overdispersed and equidispersed count data.

Poisson–geometric INAR(1) process for modeling count time series with overdispersion

In this paper, we propose a new first‐order non‐negative integer‐valued autoregressive [INAR(1)] process with Poisson–geometric marginals based on binomial thinning for modeling integer‐valued time series with overdispersion. Also, the new process has, as a particular case, the Poisson INAR(1) and geometric INAR(1) processes. The main properties of the model are derived, such as probability generating function, moments, conditional distribution, higher‐order moments, and jumps. Estimators for the parameters of process are proposed, and their asymptotic properties are established. Some numerical results of the estimators are presented with a discussion of the obtained results. Applications to two real data sets are given to show the potentiality of the new process.

A Binomial Integer-Valued ARCH Model.

We present an integer-valued ARCH model which can be used for modeling time series of counts with under-, equi-, or overdispersion. The introduced model has a conditional binomial distribution, and it is shown to be strictly stationary and ergodic. The unknown parameters are estimated by three methods: conditional maximum likelihood, conditional least squares and maximum likelihood type penalty function estimation. The asymptotic distributions of the estimators are derived. A real application of the novel model to epidemic surveillance is briefly discussed. Finally, a generalization of the introduced model is considered by introducing an integer-valued GARCH model.

Empirical likelihood for linear and log-linear INGARCH models

The integer-valued GARCH model is a popular tool for modeling time series of counts. This paper develops empirical likelihood methods for the linear and log-linear integer-valued GARCH models, respectively. We consider three aspects of the empirical likelihood method: estimates, confidence regions and test. For both models, an empirical log-likelihood ratio statistic is derived and its asymptotic distribution is shown to be a chi-squared distribution. Simulation studies lead to superior performance of the empirical likelihood method compared with the conventional treatments in the literature. An application to a real data example is also provided.

Modeling time series of counts with COM-Poisson INGARCH models

Frequently count time series exhibit overdispersion, but the opposite phenomenon of underdispersion is well documented in some situations, thus may be encountered in real applications. The INGARCH model is a popular tool for modeling time series of counts. The Poisson and negative binomial models can only deal with overdispersion, and the double Poisson and generalized Poisson models can treat both of them, but the latter two models have some shortcomings or limitations. The revived COM-Poisson distribution is flexible in modeling a wide range of overdispersion and underdispersion with only two parameters, while possessing properties that make it methodologically appealing and useful in practice, thus we introduce a new INGARCH model based on this distribution. We give an approximate condition for stationarity and expressions for mean, variance and autocorrelation function and test the accuracy of these expressions via simulations and real examples. We discuss the maximum likelihood estimation procedure for the parameters of interest. In addition, we also try to find influences of different distributions on INGARCH models using three real data sets. The results show that the generalized Poisson and COM-Poisson models perform well in most cases and the COM-Poisson model is a powerful competitor compared with the generalized Poisson model.

Thinning operations for modeling time series of counts—a survey

The analysis of time series of counts is an emerging field of science. To obtain an ARMA-like autocorrelation structure, many models make use of thinning operations to adapt the ARMA recursion to the integer-valued case. Most popular among these probabilistic operations is the concept of binomial thinning, leading to the class of INARMA models. These models are proved to be useful, especially for processes of Poisson counts, but may lead to difficulties in the case of different count distributions. Therefore, several alternative thinning concepts have been developed. This article reviews such thinning operations and shows how they are successfully applied to define integer-valued ARMA models.

INAR(1) modeling of overdispersed count series with an environmental application

AbstractThis paper is concerned with a novel version of the INAR(1) model, a non‐linear auto‐regressive Markov chain on ℕ, with innovations following a finite mixture distribution of $m \geq 1$ Poisson laws. For $m > 1$, the stationary marginal probability distribution of the chain is overdispersed relative to a Poisson, thus making INAR(1) suitable for modeling time series of counts with arbitrary overdispersion. The one‐step transition probability function of the chain is also a finite mixture, of m Poisson‐Binomial laws, facilitating likelihood‐based inference for model parameters. An explicit EM‐algorithm is devised for inference by maximization of a conditional likelihood. Alternative options for inference are discussed along with criteria for selecting m. Integer‐valued prediction (IP) is developed by a parametric bootstrap approach to ‘coherent’ forecasting, and a certain test statistic based on predictions is introduced for assessing performance of the fitted model. The proposed model is fitted to time series of counts of pixels where spatially averaged rain rate exceeds a given threshold level, illustrating its capabilities in challenging cases of highly overdispersed count data. Copyright © 2007 John Wiley & Sons, Ltd.

A New Type of Discrete Self-Decomposability and Its Application to Continuous-Time Markov Processes for Modeling Count Data Time Series

We propose a family of extended thinning operators, indexed by a parameter γ in [0, 1), with the boundary case of γ=0 corresponding to the well-known binomial thinning operator. The extended thinning operators can be used to construct a class of continuous-time Markov processes for modeling count time series data. The class of stationary distributions of these processes is called generalized discrete self-decomposable, denoted by DSD (γ). We obtain characterization results for the DSD (γ) class and investigate relationships among the classes for different γ's.