Time Series Count Data Research Articles

A Copula Discretization of Time Series-Type Model for Examining Climate Data

The study presents a comparative analysis of climate data under two scenarios: a Gaussian copula marginal regression model for count time series data and a copula-based bivariate count time series model. These models, built after comprehensive simulations, offer adaptable autocorrelation structures considering the daily average temperature and humidity data observed at a regional airport in Mobile, AL.

Bivariate Random Coefficient Integer-Valued Autoregressive Model Based on a ρ-Thinning Operator

While overdispersion is a common phenomenon in univariate count time series data, its exploration within bivariate contexts remains limited. To fill this gap, we propose a bivariate integer-valued autoregressive model. The model leverages a modified binomial thinning operator with a dispersion parameter ρ and integrates random coefficients. This approach combines characteristics from both binomial and negative binomial thinning operators, thereby offering a flexible framework capable of generating counting series exhibiting equidispersion, overdispersion, or underdispersion. Notably, our model includes two distinct classes of first-order bivariate geometric integer-valued autoregressive models: one class employs binomial thinning (BVGINAR(1)), and the other adopts negative binomial thinning (BVNGINAR(1)). We establish the stationarity and ergodicity of the model and estimate its parameters using a combination of the Yule–Walker (YW) and conditional maximum likelihood (CML) methods. Furthermore, Monte Carlo simulation experiments are conducted to evaluate the finite sample performances of the proposed estimators across various parameter configurations, and the Anderson-Darling (AD) test is employed to assess the asymptotic normality of the estimators under large sample sizes. Ultimately, we highlight the practical applicability of the examined model by analyzing two real-world datasets on crime counts in New South Wales (NSW) and comparing its performance with other popular overdispersed BINAR(1) models.

EXPLORING THE INAR MODEL ON HEAVY TAILED TIME SERIES DATA WITH OUTLIERS

Count data are intrinsically measures of event frequency; it is clear that there is an intrinsic relationship with recurring time to event. Events are typically tallied within time intervals for practical and convenient reasons. The existence of outliers is one issue that prevents count data from being stationary in time series analysis; this has an impact on the effectiveness of fitting several common stationary models to the count data collected over time. Thus, the purpose of this study was to examine how well the Integer Valued Autoregressive (INAR) model performed while modeling count data that included outliers. While this model has been studied for count time series data, it has not been studied for varying degrees of outliers. A monte-carlo simulation was carried out to select the best INAR(p), where p=1,2,3 and 4 on data with 10%, 20% and 30% outliers at different sample sizes. The INAR (4) has the best fit across the sample sizes at the larger percentages of outliers while INAR (3) at the lowest percentage with smallest information criteria of assessment and they are therefore recommended for such modeling.

Inferring stochastic group interactions within structured populations via coupled autoregression

The internal behaviour of a population is an important feature to take account of when modelling its dynamics. In line with kin selection theory, many social species tend to cluster into distinct groups in order to enhance their overall population fitness. Temporal interactions between populations are often modelled using classical mathematical models, but these sometimes fail to delve deeper into the, often uncertain, relationships within populations. Here, we introduce a stochastic framework that aims to capture the interactions of animal groups and an auxiliary population over time. We demonstrate the model’s capabilities, from a Bayesian perspective, through simulation studies and by fitting it to predator–prey count time series data. We then derive an approximation to the group correlation structure within such a population, while also taking account of the effect of the auxiliary population. We finally discuss how this approximation can lead to ecologically realistic interpretations in a predator–prey context. This approximation also serves as verification to whether the population in question satisfies our various assumptions. Our modelling approach will be useful for empiricists for monitoring groups within a conservation framework and also theoreticians wanting to quantify interactions, to study cooperation and other phenomena within social populations.

Poisson-Lindley minification INAR process with application to financial data

<p>This paper introduces the Poisson-Lindley minification integer-valued autoregressive (PL-MINAR) process, a novel statistical model for analyzing count time series data. The modified negative binomial thinning and the Poisson-Lindley (PL) marginal distribution served as the foundation for the model. The proposed model was examined in terms of its basic stochastic properties, especially related to conditional stochastic measures (e.g., transition probabilities, conditional mean and variance, autocorrelation function). Through comprehensive simulations, the effectiveness of various parameter estimation techniques was validated. The PL-MINAR model's practical utility was demonstrated in analyzing the number of Bitcoin transactions and stock trades, showing its superior or comparable performance to the established INAR model. By offering a robust tool for financial time series analysis, this research holds potential for significant improvements in forecasting and understanding market dynamics.</p>

A multivariate heavy-tailed integer-valued GARCH process with EM algorithm-based inference

A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic (GARCH) process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the expectation-maximization (EM) algorithm are proposed for estimating the parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed process. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-tailedness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. Two empirical applications of our approach are provided. The first application concerns modeling bivariate count time series data on cannabis possession-related offenses in Australia, while the second one involves modeling intraday high-frequency financial transactions data from multiple holdings in the U.S. financial market.

Bayesian tensor factorisations for time series of counts

We propose a flexible nonparametric Bayesian modelling framework for multivariate time series of count data based on tensor factorisations. Our models can be viewed as infinite state space Markov chains of known maximal order with non-linear serial dependence through the introduction of appropriate latent variables. Alternatively, our models can be viewed as Bayesian hierarchical models with conditionally independent Poisson distributed observations. Inference about the important lags and their complex interactions is achieved via MCMC. When the observed counts are large, we deal with the resulting computational complexity of Bayesian inference via a two-step inferential strategy based on an initial analysis of a training set of the data. Our methodology is illustrated using simulation experiments and analysis of real-world data.

Integer-Valued First Order Autoregressive (INAR(1)) Model With Negative Binomial (NB) Innovation For The Forecasting Of Time Series Count Data

This paper is about the theoretical investigation of integer-valued first order autoregressive (INAR(1)) model with negative binomial (NB) innovation for the forecasting of time series count data. The study makes use of the Conditional Least squares (CLS) estimator to estimate the parameter of INAR(1) model, and Maximum Likelihood Estimator (MLE) to estimate the mean (μ ) and the dispersion parameter (K) of the NB distribution. A simulation experiment based on theoretical generated data were addressed under different parameter values α =0.2, 0.6, 0.8, different sample sizes n=30, 90, 120, 600 for the class of INAR(1) model, and μ  =0.85, 1.5, 2,  K=1,2, 4 for the NB distribution. The Monte Carlo simulations were conducted with codes written in R, all results were based on 1000 runs. The estimation of parameter for the class of INAR(1) model gives a better result when the number of observations is small and the parameter value is high. The NB estimation gives a better result when the number of observations is small and with large K values. The forecasting accuracy of the model at different lead time period l =1, 3, 5, 7, 9, 15 were investigated with codes written in R. The results showed that the minimum mean square error (MMSE) produced when the number of lead times forecasts is between one and five were less than that produced when the numbers of lead times forecast were greater than five. The MMSE increased when the number of lead time periods increases. This result indicates that forecasting with this class of model is better with short time frame of predictions. The study was applied to the number of deaths arising from COVID-19 in Nigeria which consist of count time series data of 48 observations (weekly data), from January 2021 to December 2021.

Flexible Lévy-Based Models for Time Series of Count Data with Zero-Inflation, Overdispersion, and Heavy Tails

The explosion of time series count data with diverse characteristics and features in recent years has led to a proliferation of new analysis models and methods. Significant efforts have been devoted to achieving flexibility capable of handling complex dependence structures, capturing multiple distributional characteristics simultaneously, and addressing nonstationary patterns such as trends, seasonality, or change points. However, it remains a challenge when considering them in the context of long-range dependence. The Lévy-based modeling framework offers a promising tool to meet the requirements of modern data analysis. It enables the modeling of both short-range and long-range serial correlation structures by selecting the kernel set accordingly and accommodates various marginal distributions within the class of infinitely divisible laws. We propose an extension of the basic stationary framework to capture additional marginal properties, such as heavy-tailedness, in both short-term and long-term dependencies, as well as overdispersion and zero inflation in simultaneous modeling. Statistical inference is based on composite pairwise likelihood. The model’s flexibility is illustrated through applications to rainfall data in Guinea from 2008 to 2023, and the number of NSF funding awarded to academic institutions. The proposed model demonstrates remarkable flexibility and versatility, capable of simultaneously capturing overdispersion, zero inflation, and heavy-tailedness in count time series data.

Spatial asynchrony and cross-scale climate interactions in populations of a coldwater stream fish.

Climate change affects populations over broad geographic ranges due to spatially autocorrelated abiotic conditions known as the Moran effect. However, populations do not always respond to broad-scale environmental changes synchronously across a landscape. We combined multiple datasets for a retrospective analysis of time-series count data (5-28 annual samples per segment) at 144 stream segments dispersed over nearly 1,000 linear kilometers of range to characterize the population structure and scale of spatial synchrony across the southern native range of a coldwater stream fish (brook trout, Salvelinus fontinalis), which is sensitive to stream temperature and flow variations. Spatial synchrony differed by life stage and geographic region: it was stronger in the juvenile life stage than in the adult life stage and in the northern sub-region than in the southern sub-region. Spatial synchrony of trout populations extended to 100-200 km but was much weaker than that of climate variables such as temperature, precipitation, and stream flow. Early life stage abundance changed over time due to annual variation in summer temperature and winter and spring stream flow conditions. Climate effects on abundance differed between sub-regions and among local populations within sub-regions, indicating multiple cross-scale interactions where climate interacted with local habitat to generate only a modest pattern of population synchrony over space. Overall, our analysis showed higher degrees of response heterogeneity of local populations to climate variation and consequently population asynchrony than previously shown based on analysis of individual, geographically restricted datasets. This response heterogeneity indicates that certain local segments characterized by population asynchrony and resistance to climate variation could represent unique populations of this iconic native coldwater fish that warrant targeted conservation. Advancing the conservation of this species can include actions that identify such priority populations and incorporate them into landscape-level conservation planning. Our approach is applicable to other widespread aquatic species sensitive to climate change.

Nearly unstable integer‐valued ARCH process and unit root testing

AbstractThis paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom.

An integer GARCH model for a Poisson process with time-varying zero-inflation.

A serially dependent Poisson process with time-varying zero-inflation is proposed. Such formulations have the potential to model count data time series arising from phenomena such as infectious diseases that ebb and flow over time. The model assumes that the intensity of the Poisson process evolves according to a generalized autoregressive conditional heteroscedastic (GARCH) formulation and allows the zero-inflation parameter to vary over time and be governed by a deterministic function or by an exogenous variable. Both the expectation maximization (EM) and the maximum likelihood estimation (MLE) approaches are presented as possible estimation methods. A simulation study shows that both parameter estimation methods provide good estimates. Applications to two real-life data sets on infant deaths due to influenza show that the proposed integer-valued GARCH (INGARCH) model provides a better fit in general than existing zero-inflated INGARCH models. We also extended a non-linear INGARCH model to include zero-inflation and an exogenous input. This extended model performed as well as our proposed model with respect to some criteria, but not with respect to all.

A multiplicative thinning‐based integer‐valued GARCH model

In this article, we introduce a multiplicative integer‐valued time series model, which is defined as the product of a unit‐mean integer‐valued independent and identically distributed (i.i.d.) sequence, and an integer‐valued dependent process. The latter is defined as a binomial thinning operation of its own past and of the past of the observed process. Furthermore, it combines some features of the integer‐valued GARCH (INGARCH), the autoregressive conditional duration (ACD), and the integer autoregression (INAR) processes. The proposed model has an unspecified distribution and is able to parsimoniously generate very high overdispersion, persistence, and heavy‐tailedness. The dynamic probabilistic structure of the model is first analyzed. In addition, parameter estimation is considered by using a two‐stage weighted least squares estimate (2SWLSE), consistency and asymptotic normality (CAN) of which are established under mild conditions. Applications of the proposed formulation to simulated and actual count time series data are provided.

Non-linear INAR(1) processes under an alternative geometric thinning operator

We propose a novel class of first-order integer-valued AutoRegressive (INAR(1)) models based on a new operator, the so-called geometric thinning operator, which induces a certain non-linearity to the models. We show that this non-linearity can produce better results in terms of prediction when compared to the linear case commonly considered in the literature. The new models are named non-linear INAR(1) (in short NonLINAR(1)) processes. We explore both stationary and non-stationary versions of the NonLINAR processes. Inference on the model parameters is addressed and the finite-sample behavior of the estimators investigated through Monte Carlo simulations. Two real data sets are analyzed to illustrate the stationary and non-stationary cases and the gain of the non-linearity induced for our method over the existing linear methods. A generalization of the geometric thinning operator and an associated NonLINAR process are also proposed and motivated for dealing with zero-inflated or zero-deflated count time series data.

Identification of an influence network using ensemble-based filtering for Hawkes processes driven by count data

Many networks have event-driven dynamics (such as communication, social media and criminal networks), where the mean rate of the events occurring at a node in the network changes according to the occurrence of other events in the network. In particular, events associated with a node of the network could increase the rate of events at other nodes, depending on their influence relationship. Thus, it is of interest to use temporal data to uncover the directional, time-dependent, influence structure of a given network while also quantifying uncertainty even when knowledge of a physical network is lacking. Typically, methods for inferring the influence structure in networks require knowledge of a physical network or are only able to infer small network structures. In this paper, we model event-driven dynamics on a network by a multidimensional Hawkes process. We then develop a novel ensemble-based filtering approach for a time-series of count data (i.e., data that provides the number of events per unit time for each node in the network) that not only tracks the influence network structure over time but also approximates the uncertainty via ensemble spread. The method overcomes several deficiencies in existing methods such as existing methods for inferring multidimensional Hawkes processes are too slow to be practical for any network over ∼50 nodes, can only deal with timestamp data (i.e. data on just when events occur not the number of events at each node), and that we do not need a physical network to start with. Our method is massively parallelizable, allowing for its use to infer the influence structure of large networks (∼10,000 nodes). We demonstrate our method for large networks using both synthetic and real-world email communication data.

A Hybrid Deep Learning Model for Link Dynamic Vehicle Count Forecasting with Bayesian Optimization

The link dynamic vehicle count is a spatial variable that measures the traffic state of road sections, which reflects the actual traffic demand. This paper presents a hybrid deep learning method that combines the gated recurrent unit (GRU) neural network model with automatic hyperparameter tuning based on Bayesian optimization (BO) and the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) model. There are four steps in this hybrid approach. First, the ICEEMDAN is employed to decompose the link dynamic vehicle count time series data into several intrinsic components. Second, the components are predicted by the GRU model. At the same time, the Bayesian optimization method is utilized to automatically optimize the hyperparameters of the GRU model. Finally, the predicted subcomponents are reconstructed to obtain the final prediction results. The proposed hybrid deep learning method is tested on two roads of Hangzhou, China. Results show that, compared with the 12 benchmark models, the proposed hybrid deep learning model achieves the best performance in link dynamic vehicle count forecasting.

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.

Classical inference for time series of count data in parameter-driven models

We study estimation on parameter-driven models for time series of counts. This class of models follows the structure of a generalized linear model in which the serial dependency is included in the model by the link function through a time-dependent latent process. The likelihood function for this class of models commonly cannot be calculated explicitly and computationally intensive methods like importance sampling and Markov chain Monte Carlo are used to estimate the model parameters. Here, we propose a simple and fast estimation procedure in a wide class of models that accommodate both discrete and continuous data. The maximum likelihood methodology is used to obtain the parameter estimates for the models under study. The simplicity of the procedure allows for build bootstrap confidence intervals for the hyperparameters and latent states of parameter-driven models. We perform extensive simulation studies to verify the asymptotic behavior of the parameter estimates, as well as present application of the proposed procedure through set of real data.

Different Forecasting Model Comparison for Near Future Crash Prediction

A traffic crash is becoming one of the major factors that leads to unexpected death in the world. Short window traffic crash prediction in the near future is becoming more pragmatic with the advancements in the fields of artificial intelligence and traffic sensor technology. Short window traffic prediction can monitor traffic in real time, identify unsafe traffic dynamics, and implement suitable interventions for traffic conflicts. Crash prediction being an important component of intelligent traffic systems, it plays a crucial role in the development of proactive road safety management systems. Some near future crash prediction models were put forward in recent years; further improvements need to be implemented for actual applications. This paper utilizes traffic accident data from the study Freeway in China to build a time series-based count data model for daily crash prediction. Lane traffic flow, weather information, vehicle speed, and truck to car ratio were extracted from the deployment of non-intrusive detection systems with support of the Bridge Management Administration study and were input into the model as independent variables. Different types of prediction models in machine learning and time series forecasting methods such as boosting, ARIMA, time-series count data model, etc. are compared within the paper. Results show that integrating time series with a count data model can capture traffic accident features and account for the temporal structure for variable serial correlation. A prediction error of 0.7 was achieved according to Root Mean Squared Deviation.

Bayesian Inference for the Negative Binomial-Quasi Lindley Model for Time Series Count Data on the COVID-19 Pandemic

In statistical models, the generalized linear model (GLMs) plays a role in studying to describe a response variable as a function of 1 or more predictor variables. Computational methods and mixed distributions are frequently used to build predictive models to perform time-to-event data analysis. To develop a statistical model so that the model can make predictions appropriately and accurately, it starts with developing a suitable distribution for the nature of the actual data. This paper proposes a new mixed negative binomial distribution for count data with over-dispersion, the so-called negative binomial-quasi Lindley (NB-QL) distribution. A new GLMs framework for the NB-QL model to build the time series count data model is introduced, and its application is carried out based on the actual data sets of the COVID-19 epidemic in Thailand. The models are related to GLMs as they are linear relationships between outcome variables and covariates. Where the response variable was in the form of time series count data under the exponential family distribution function, with the random components and link functions. In this study, we study the factors that affect the number of COVID-19 death cases in Thailand and provide the predictive modeling of the number of the COVID-19 death cases from 1 January 2020 to 31 December 2020, for which this data set has the observed sample of 366 days. In contrast, a model with an NB-QL distribution and NB has approached the uniform. Based on the deviance, DIC, and the probability integral transform histogram, we can see that the proposed model is also suitable for forecasting the number of the COVID-19 death cases daily in Thailand, indicating that the NB-QL time series model was another efficient alternative to modeling count data that has an over-dispersion problem. According to the NB-QL time series model about the number of the COVID-19 death cases daily in Thailand, it is indicated that the average number of daily COVID-19 deaths is influenced by the number of the COVID-19 death cases in the previous 3 days. The average number of COVID-19 death cases in Thailand is also influenced by the previous 2 days. At the same time, the number of infected cases daily in Thailand is influenced by the number of the COVID-19 death cases daily. In addition, there are also the components interventions of internal covariate effects due to the data, as there was a surge in the number of the COVID-19 death cases daily in Thailand at the time. HIGHLIGHTS A new mixture NB distribution to be a flexible alternative to analyze count data with over-dispersion. The new distribution is a mix of the NB and QL distributions; a name is Negative Binomial-Quasi Lindley A new mixed negative binomial distribution for time series count data with over-dispersion, and the Bayesian approach is the method used to estimate the parameters of the proposed model. We will apply the GLMs framework to build the time series count data The new mixed NB distribution in this study is an extremely effective alternative for modeling count data in the context of over-dispersion GRAPHICAL ABSTRACT