Integer-valued Time Series Research Articles

Locally Adaptive Shrinkage Priors for Trends and Breaks in Count Time Series

Non-stationary count time series characterized by features such as abrupt changes and fluctuations about the trend arise in many scientific domains including biophysics, ecology, energy, epidemiology, and social science domains. Current approaches for integer-valued time series lack the flexibility to capture local transient features while more flexible models for continuous data types are inadequate for universal applications to integer-valued responses such as settings with small counts. We present a modeling framework, the negative binomial Bayesian trend filter (NB-BTF), that offers an adaptive model-based solution to capturing multiscale features with valid integer-valued inference for trend filtering. The framework is a hierarchical Bayesian model with a dynamic global-local shrinkage process. The flexibility of the global-local process allows for the necessary local regularization while the temporal dependence induces a locally smooth trend. In simulation, the NB-BTF outperforms a number of alternative trend filtering methods. Then, we demonstrate the method on weekly power outage frequency in Massachusetts townships. Power outage frequency is characterized by a persistent low level trend with occasional spikes. These illustrations show the estimation of a smooth, non-stationary trend with adequate uncertainty quantification.

A zero-inflated Poisson integer-valued autoregressive model with time-varying coefficients covariates

For integer-valued time series, we may be interested in how a covariate affects the series. In the existing models, it is commonly assumed that the coefficients of the covariates are constant, and the innovation process follows a Poisson distribution. However, in practice, the influence of covariates may vary with time, and there may be a large number of zeros. To address this situation, this paper proposes an integer-valued autoregressive model with time-varying coefficients on covariates, incorporating a zero-inflated Poisson innovation process. The binomial thinning parameters and unknown functions of the proposed model are estimated based on approximate conditional least squares and the cubic spline method. The consistency and asymptotic properties of the binomial thinning parameters are investigated. Simulation studies and real data analysis support the flexibility and good performance of the proposed model.

A bilinear modeling in counts time series with applications

This paper introduces a modified bilinear model for integer-valued time series in which thinning operators are applied in bilinear terms involving the product of the input and state process separately. The proposed model is able to consider overdispersion. Furthermore, it connects a feature of the integer-valued autoregressive conditional heteroskedasticity and integer-valued autoregressive processes. Important properties of the proposed model as well as the sufficient condition for stationarity are derived. After considering some estimation methods based on time domain and frequency domain approaches, simulation studies are conducted to check the performance of the estimates. The analysis of practical cases in social science is accomplished to highlight the usefulness of the proposed model in applications, and the model’s adequacy is provided. Further, the suggested model discusses the problem of data forecasting.

A mixed BAR(1) model driven by serially dependent innovation with application

To better analyze dependent count data with bounded support, this paper proposes a first-order mixture binomial autoregressive (BAR(1)) model, which is constructed via making use of the Pegram operator, where the innovation processes are assumed to be serially dependent to the observation rather than independent. The new model not only inherits the advantages of existing mixture BAR(1) models while also overcoming their shortcomings, but also does well in explaining binomial overdispersion and zero inflation. We infer various statistical properties of the process, including conditional moments, moments and autocovariance functions. The EM algorithm is employed to calculate the conditional maximum likelihood estimate. To evaluate the performance of obtained estimates, some simulation studies are conducted. Furthermore, the forecasting problem for the proposed model is also discussed. To show the applicability of our model, an application to proposition data set is conducted and we compare our model to some competitive integer-valued time series models with bounded support.

Negative binomial community network vector autoregression for multivariate integer-valued time series

Modeling multivariate integer-valued time series with appropriate methods is currently a popular research topic. In this paper, we propose a multivariate integer-valued autoregressive time series model based on a fixed network community structure. We use the negative binomial distribution as the conditional marginal distribution and a copula to construct the conditional joint distribution. The newly proposed model introduces the heterogeneity of nodes. Stability conditions are provided for both fixed and increasing dimensions. We estimate the parameters of the proposed model by maximizing the quasi-likelihood function with known and unknown community membership matrices, respectively. Corresponding asymptotic properties of parameter estimates are also provided. A simulation study is conducted to demonstrate the asymptotic behavior of the proposed model, and two real datasets are employed to compare the proposed model with other competitive models.

First-order multivariate integer-valued autoregressive model with multivariate mixture distributions

The univariate integer-valued time series has been extensively studied, but literature on multivariate integer-valued time series models is quite limited and the complex correlation structure among the multivariate integer-valued time series is barely discussed. In this study, we proposed a first-order multivariate integer-valued autoregressive model to characterize the correlation among multivariate integer-valued time series with higher flexibility. Under the general conditions, we established the stationarity and ergodicity of the proposed model. With the proposed method, we discussed the models with multivariate Poisson-lognormal distribution and multivariate geometric-logitnormal distribution and the corresponding properties. The estimation method based on EM algorithm was developed for the model parameters and extensive simulation studies were performed to evaluate the effectiveness of proposed estimation method. Finally, a real crime dataset was analysed to demonstrate the advantage of the proposed model with comparison to the other models.

Randomness Test of Thinning Parameters for the NBRCINAR(1) Process

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications.

Self-exciting threshold -valued autoregressive processes for non-stationary time series of counts

To handle non-stationary integer-valued time series of counts with piecewise characteristics and linear trends, this paper introduced a class of self-exciting threshold Z -valued autoregressive processes. The process is defined with a free type distribution for the innovation, which enhances the flexibility of the model. The basic probabilistic and statistical properties of the proposed model are discussed. Conditional least squares (CLS) estimator and modified quasi-likelihood (MQL) estimator, as well as their asymptotic properties are obtained. A searching algorithm for estimating the threshold parameter is also provided. Some simulation studies are conducted to show the performances of the proposed methods. Finally, an application to a real data example is provided.

Asymmetric beta-binomial GARCH models for time series with bounded support

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.

Diagnostic checks in time series models based on a new correlation coefficient of residuals

For checking time series models, the Ljung–Box, Li–Mak and Zhu–Wang statistics play an important role, which use the Pearson's correlation coefficient to implement (squared) residual (partial) autocorrelation tests. In this paper, we replace the Pearson's correlation coefficient with a new rank correlation coefficient and propose a new test statistic to conduct diagnostic checks for residuals in autoregressive moving average models, autoregressive conditional heteroscedasticity models and integer-valued time series models, respectively. We conduct simulations to assess the performance of the new test statistic, and compare it with existing ones, and the results show the superiority of the proposed one. We use three real examples to exhibit its usefulness.

Some developments on seasonal INAR processes with application to influenza data

Influenza epidemic data are seasonal in nature. Zero-inflation, zero-deflation, overdispersion, and underdispersion are frequently seen in such number of cases of disease (count) data. To explain these counts’ features, this paper introduces a flexible model for nonnegative integer-valued time series with a seasonal autoregressive structure. Some probabilistic properties of the model are discussed for general seasonal INAR(p) model and three estimation methods are used to estimate the model parameters for its special case seasonal INAR(1) model. The performance of the estimation procedures has been studied using simulation. The proposed model is applied to analyze weekly influenza data from the Breisgau- Hochschwarzwald county of Baden–Württemberg state, Germany. The empirical findings show that the suggested model performs better than existing models.

Robust estimation for general integer-valued autoregressive models based on the exponential-polynomial divergence

In this study, we develop a robust estimator for integer-valued one-parameter exponential family autoregressive models, named general integer-valued autoregressive models. This model accommodates a broad class of integer-valued time series models. In particular, we propose a robust estimation method that minimizes the exponential-polynomial divergence (EPD) belonging to the Brègman divergence family. EPD subsumes the density power divergence (DPD), which has been extensively studied by many authors for the past decades. Under regularity conditions, the minimum EPD estimator (MEPDE) is shown to be consistent and asymptotically normal. Comparing the performance of MEPDE with the minimum DPD estimator, we substantiate the validity of MEPDE through a simulation study and real data analysis.

Density Power Divergence Estimator for General Integer-Valued Time Series with Exogenous Covariates

Density Power Divergence Estimator for General Integer-Valued Time Series with Exogenous Covariates

Consistent Model Selection Procedure for Random Coefficient INAR Models.

In the realm of time series data analysis, information criteria constructed on the basis of likelihood functions serve as crucial instruments for determining the appropriate lag order. However, the intricate structure of random coefficient integer-valued time series models, which are founded on thinning operators, complicates the establishment of likelihood functions. Consequently, employing information criteria such as AIC and BIC for model selection becomes problematic. This study introduces an innovative methodology that formulates a penalized criterion by utilizing the estimation equation within conditional least squares estimation, effectively addressing the aforementioned challenge. Initially, the asymptotic properties of the penalized criterion are derived, followed by a numerical simulation study and a comparative analysis. The findings from both theoretical examinations and simulation investigations reveal that this novel approach consistently selects variables under relatively relaxed conditions. Lastly, the applications of this method to infectious disease data and seismic frequency data produce satisfactory outcomes.

Bayesian modeling of spatial integer-valued time series

Many infectious diseases spread through person-to-person contact, either directly or indirectly. The proposal incorporates spatial-temporal patterns into multivariate integer-valued GARCH models with either a generalized Poisson distribution or zero-inflated generalized Poisson distribution in order to describe these features of the data. By considering the neighboring locations of the target series, the set-up incorporates a flexible and continuous conceptualization of distance to present the spatial components, thereby highlighting for the non-separability of space and time. Such an approach eliminates the need to pre-assign a spatial weight matrix. Newly designed models are utilized to investigate time-series counts of infectious diseases, enabling inference, prediction, and model selection within a Bayesian framework through Markov chain Monte Carlo (MCMC) algorithms. As an illustration, design simulation studies and multivariate weekly dengue cases are scrutinized for the performance of the Bayesian methods. The proposed models successfully capture the characteristics of spatial dependency, over-dispersion, and a large portion of zeros, providing a comprehensive model for the observed phenomena in the data.

Bayesian log-linear beta-negative binomial integer-valued Garch model

When dealing with time series with outlying and atypical data, a commonly used approach is to develop models based on heavy-tailed distributions. The literature coping with continuous-valued time series with extreme observations is well explored. However, current literature on modelling integer-valued time series data with heavy-tailedness is less considered. The state of the art research on this topic is presented by Gorgi (J R Stat Soc Ser B (Stat Methodol) 82:1325–1347, 2020) very recently, which introduced a linear Beta-negative binomial integer-valued generalized autoregressive conditional heteroscedastic (BNB-INGARCH) model. However, such proposed process allows for positive correlation only. This paper develops a log-linear version of the BNB-INGARCH model, which accommodates both negative and positive serial correlations. Moreover, we adopt Bayesian inference for better quantifying the uncertainty of unknown parameters. Due to the high computational demand, we resort to adaptive Markov chain Monte Carlo sampling schemes for parameter estimations and inferences. The performance of the proposed method is evaluated via a simulation study and empirical applications.

Inference and forecasting for continuous-time integer-valued trawl processes

This paper develops likelihood-based methods for estimation, inference, model selection, and forecasting of continuous-time integer-valued trawl processes. The full likelihood of integer-valued trawl processes is, in general, highly intractable, motivating the use of composite likelihood methods, where we consider the pairwise likelihood in lieu of the full likelihood. Maximizing the pairwise likelihood of the data yields an estimator of the parameter vector of the model, and we prove consistency and, in the short memory case, asymptotic normality of this estimator. When the underlying trawl process has long memory, the asymptotic behaviour of the estimator is more involved; we present some partial results for this case. The pairwise approach further allows us to develop probabilistic forecasting methods, which can be used to construct the predictive distribution of integer-valued time series. In a simulation study, we document the good finite sample performance of the likelihood-based estimator and the associated model selection procedure. Lastly, the methods are illustrated in an application to modelling and forecasting financial bid–ask spread data, where we find that it is beneficial to carefully model both the marginal distribution and the autocorrelation structure of the data.

A statistical study for some classes of first-order mixed generalized binomial autoregressive models

In order to solve the modeling problems of integer-valued time series with complex and dependent structures, this article proposes some classes of first-order mixed generalized binomial autoregressive models. Basic probabilistic and statistical properties of the models are discussed. The parameters of models are estimated by the conditional maximum likelihood method. Also, the asymptotic properties and the numerical results of these estimators are obtained. Finally, the good performances of these models are illustrated, among other competitive models in the literature, by an application to the data of weekly rainy days.

On Periodic Generalized Poisson INAR(1) Model

In this paper, we introduce a first-order Periodic Generalized Poisson Integer-Valued Autoregressive model which has been shown to be useful to describe overdispersion, equidispersion and underdispersion feature encountered in periodically correlated Integer-Valued time series. Some probabilistic and statistical properties are established, such as the periodically correlated stationarity conditions, in the first and the second moments are provided and the closed-forms of these moments are, under these conditions, derived. Moreover, the structure of the periodic autocovariance is obtained. The estimation problem is addressed through the Yule-Walker the Two-Stage Conditional Least Squares and the Conditional Maximum Likelihood methods. The performance of these methods is done through an intensive simulation study and an application on real data set is accomplished. Keywords and phrases: Periodic Generalized Poisson, Integer-Valued Autoregressive, Periodically correlated process, periodically stationary condition.

Epidemic change-point detection in general integer-valued time series

ABSTRACT In this paper, we consider the structural change in a class of discrete valued time series, where the true conditional distribution of the observations is assumed to be unknown. The conditional mean of the process depends on a parameter θ ∗ which may change over time. We provide sufficient conditions for the consistency and the asymptotic normality of the Poisson quasi-maximum likelihood estimator (QMLE) of the model. We consider an epidemic change-point detection and propose a test statistic based on the QMLE of the parameter. Under the null hypothesis of a constant parameter (no change), the test statistic converges to a distribution obtained from increments of a Browninan bridge. The test statistic diverges to infinity under the epidemic alternative, which establishes that the proposed procedure is consistent in power. The effectiveness of the proposed procedure is illustrated by simulated and real data examples.