Time Series Count Data Research Articles

Understanding near-miss count data on construction sites using greedy D-vine copula marginal regression

Near misses are an important type of accident precursor because they provide insights into failure-generating mechanisms, help understand safety risks, and guide necessary interventions before they develop into real accidents. A high occurrence frequency of near-miss incidents typically signals a warning of small safety margins. However, the occurrences of near-miss incidents are stochastic and serially dependent in nature. Ignoring these features will typically lead to a misunderstanding of a project's safety level. This paper presents a D-vine copula marginal regression model for count time series data. Incident counts are expressed as a function of predictors. A time-varying discrete marginal distribution is used to describe the uncertainty of incident occurrences at an arbitrary time point, while the dependence between consecutive observations at different time points is captured with copula functions. To allow for long-range and non-Gaussian dependences between incidents, the D-vine structure is used to build the multivariate copula function where the bivariate copula associated with each edge is not necessarily Gaussian. To avoid evaluating a large number of candidate models that use different marginals and bivariate copulas as building blocks, a greedy algorithm is proposed to estimate relevant parameters and select the best model. An information criterion is used to determine the tree level of the D-vine decomposition, which implies the Markov structure of the incident occurrences. The proposed method is applied to a set of near-miss count data collected from a construction project over 5 years. We show that the hidden dependence has a relatively large time-lag and is strongly non-Gaussian. Comparison with conventional methods shows that the proposed method is efficient and yields better predictive performance.

Statistical modeling of COVID-19 deaths with excess zero counts

Abstract Objectives Modeling and forecasting possible trajectories of COVID-19 infections and deaths using statistical methods is one of the most important topics in present time. However, statistical models use different assumptions and methods and thus yield different results. One issue in monitoring disease progression over time is how to handle excess zeros counts. In this research, we assess the statistical empirical performance of these models in terms of their fit and forecast accuracy of COVID-19 deaths. Methods Two types of models are suggested in the literature to study count time series data. The first type of models is based on Poisson and negative binomial conditional probability distributions to account for data over dispersion and using auto regression to account for dependence of the responses. The second type of models is based on zero-inflated mixed auto regression and also uses exponential family conditional distributions. We study the goodness of fit and forecast accuracy of these count time series models based on autoregressive conditional count distributions with and without zero inflation. Results We illustrate these methods using a recently published online COVID-19 data for Tunisia, which reports daily death counts from March 2020 to February 2021. We perform an empirical analysis and we compare the fit and the forecast performance of these models for death counts in presence of an intervention policy. Our statistical findings show that models that account for zero inflation produce better fit and have more accurate forecast of the pandemic deaths. Conclusions This paper shows that infectious disease data with excess zero counts are better modelled with zero-inflated models. These models yield more accurate predictions of deaths related to the pandemic than the generalized count data models. In addition, our statistical results find that the lift of travel restrictions has a significant impact on the surge of COVID-19 deaths. One plausible explanation of the outperformance of zero-inflated models is that the zero values are related to an intervention policy and therefore they are structural.

Generalized Linear Model Approach for Time Series Count Data on Number of Foreign Tourists Modeling in West Java

This study aimed to model the number of foreign tourist visits in West Java. The number of a tourist visit is part of the discrete data so that the normal distribution approach become less precise in common modeling. In this study, the number of tourist visits was carried out using the Generalized Linear Model approach, combining the Poisson distribution and negative binomial distribution with the identity and log link function. The effect of internal covariates due to a surge in the number of tourists in a given month was also added to the modeling. tscount: An R Package for Analysis of Count Time Series Following Generalized Linear Models. The results showed that the four models obtained were equally good based on the mean absolute percent error (MAPE) values, while the model obtained with the negative binomial distribution integral probability log link function is the best model based on the Akaike information criterion (AIC), Bayesian information criterion (BIC) and integral probability transform (PIT) histogram values. The negative binomial with the log link function approach was then used to model and to predict the number of foreign visits. Plot using negative binomial and log link function, has a value closer to the actual data plot, also strong with the smallest AIC and BIC values.

Modelling and monitoring of INAR(1) process with geometrically inflated Poisson innovations

ABSTRACT To analyse count time series data inflated at the r + 1 values , we propose a new first-order integer-valued autoregressive process with r-geometrically inflated Poisson innovations. Some statistical properties together with conditional maximum likelihood estimate are provided. For the purpose of statistical monitoring, we focus on the cumulative sum chart, exponentially weighted moving average chart and combined jumps chart towards the proposed process. Numerical simulations indicate that the conditional maximum likelihood estimator is unbiased. Moreover, the cumulative sum chart is the best choice to monitor our model in practice. Some applications about telephone complaints data are provided to illustrate the proposed methods.

On Abrupt Changes of the Approximate and Sample Entropy Values in Supercomputer Power Consumption

When calculating the Approximate Entropy (ApEn) and the Sample Entropy (SampEn) of a time series, representing the course of supercomputer power consumption, seemingly unexplained abrupt changes sometimes appear. These sudden changes occur when calculating the course of ApEn and SampEn values using a floating time window and can be manifested also in other discrete-valued time series, such as the development of prices on stock exchanges, time series from birth-death models, count-data time series, etc. These abrupt changes are not continuous and it is clear that they do not reflect real changes in the complexity degree of the analyzed time series. This can cause erroneous detection of a change in the degree of complexity of the time series, which can trigger a false alarm that something unusual is happening in the complex system being monitored (human brain, engine gearbox, financial market, supercomputer infrastructure, etc.). This work reveals in detail the mechanism of this phenomenon and also proposes measures to prevent its occurrence.

A first-order integer-valued autoregressive process with zero-modified Poisson-Lindley distributed innovations

In this paper, we introduce a first-order integer-valued autoregressive process with zero-modified Poisson-Lindley distributed innovations based on the binomial thinning operator. Some statistical and conditional properties of the model are presented, several estimation methods are applied to estimate the unknown parameters and, for the Yule-Walker and conditional least square estimators, the asymptotic distributions are obtained. Besides, a simulation study is conducted to compare the performance of the proposed estimators and, finally, two real count time series data are applied to demonstrate the usefulness of the proposed model.

Modeling Multivariate Count Time Series Data with a Vector Poisson Log-Normal Additive Model: Applications to Testing Treatment Effects in Single-Case Designs

In education and psychology, single-case designs (SCDs) have been used to detect treatment effects using time series data in the presence or absence of intervention. One popular design variant of SCDs is a multiple-baseline design for multiple outcomes, which often collects outcomes with some form of a count. A Poisson model is a natural choice for the count outcome. However, the assumption of the Poisson model that the outcome variable’s mean is equal to its variance is often violated in SCDs, as the variance is often larger than the mean (called overdispersion). In addition, when multiple outcomes are from the same participant, it is likely that they are correlated. In this paper, we present a vector Poisson log-normal additive (V-PLN-A) model to deal with (a) change processes (auto- and cross-correlations and data-driven trend) and (b) correlation and overdispersion in multivariate count time series. A multivariate normal distribution was adapted to account for correlation among multiple outcomes as well as possible overdispersion. The V-PLN-A model was applied to an educational intervention study to test treatment effects. Simulation study results showed that parameter recovery of the V-PLN-A model was satisfactory in a large number of timepoints using Bayesian analysis, and that ignoring change processes and overdispersion led to biased estimates of the treatment effects.

Mixtures of Nonlinear Poisson Autoregressions

We study nonlinear infinite order Markov switching integer‐valued ARCH models for count time series data. Markov switching models take into account complex dynamics and can deal with several stylistic facts of count data including proper modelling of nonlinearities, overdispersion and outliers. We study structural properties of those models. Under mild conditions, we prove consistency and asymptotic normality of the maximum likelihood estimator for the case of finite order autoregression. In addition, we give conditions which imply that the marginal likelihood ratio test, for testing the number of regimes, converges to a Gaussian process. This result enables us to prove that the BIC provides a consistent estimator for selecting the true number of regimes. A real data example illustrates the methodology and compares this approach with alternative methods.

Risk Interactions of Coronavirus Infection across Age Groups after the Peak of COVID-19 Epidemic.

Background: The COVID-19 pandemic has incurred significant disease burden worldwide, particularly on the elderly population. This study aims to explore how risks of coronavirus infection interact across age groups using data from South Korea. Methods: Daily new COVID-19 cases from 10 March to 30 April 2020 were scraped from online open sources. A multivariate vector autoregressive model for time series of count data was used to examine the risk interactions across age groups. Case counts from previous days were included as predictors to dynamically examine the change of risk patterns. Results: In South Korea, the risk of coronavirus infection among elderly people was significantly affected by other age groups. An increase in virus infection among people aged 20–39 was associated with a double risk of infection among elderly people. Meanwhile, an increase in virus infection among elderly people was also significantly associated with risks of infection among other age groups. The risks of infection among younger people were relatively unaffected by that of other age groups. Conclusions: Protecting elderly people from coronavirus infection could not only reduce the risk of infection among themselves but also ameliorate the risks of virus infection among other age groups. Such interventions should be effective and for the long term.

Modeling the effects of lake-effect snow related weather conditions on daily traffic crashes: A time series count data approach

Modeling the effects of lake-effect snow related weather conditions on daily traffic crashes: A time series count data approach

Random parameter models for estimating statewide daily bicycle counts using crowdsourced data

Persistent lack of non-motorized traffic counts can affect the evidence-based decisions of transportation planning and safety-concerned agencies in making reliable investments in bikeway and other non-motorized facilities. Researchers have used various approaches to estimate bicycles counts, such as scaling, direct-demand modeling, time series, and others. In recent years, an increasing number of studies have tried to use crowdsourced data for estimating the bicycle counts. Crowdsourced data only represents a small percentage of cyclists. This percentage, on the other hand, can change based on the location, facility type, meteorological, and other factors. Moreover, the autocorrelation observed in bicycle counts may be different from the autocorrelation structure observed among crowdsourced platform users, such as Strava. Strava users are more consistent; hence, the time series count data may be stationary, while bicycle demand may vary based on seasonal factors. In addition to seasonal variation, several time-invariant contributing factors (e.g., facility type, roadway characteristics, household income) affect bicycle demand, which needs to be accounted for when developing direct demand models. In this paper, we use a mixed-effects model with autocorrelated errors to predict daily bicycle counts from crowdsourced data across the state of Texas. Additionally, we supplement crowdsourced data with other spatial and temporal factors such as roadway facility, household income, population demographics, population density and weather conditions to predict bicycle counts. The results show that using a robust methodology, we can predict bicycle demand with a 29% margin of error, which is significantly lower than merely scaling the crowdsourced data (41%).

Time Series Regression for Zero-Inflated and Overdispersed Count Data: A Functional Response Model Approach

Count time series data feature prominently in epidemiology, business, and environmental sciences. Often, such data exhibit zero-inflation and overdispersion in addition to serial dependence. Parametric models such as the negative binomial distribution are employed to account for overdispersion. In practice, the conditional variance structure may be unknown or may not be negative binomial. In this paper, a distribution-free approach for estimation of regression parameters of conditionally overdispersed and zero-inflated time series models is developed. Parameter estimates are optimal in the Godambe-information sense. Simulation studies indicate that our method is robust to model misspecification with small relative bias and nearly the same efficiency as that of the MLE for some observation-driven count time series processes. A case study comparing our method with fully parametric methods using weekly syphilis counts from 2007–2010 in Virginia, USA, illustrates the benefit of our method.

Copula-based Markov zero-inflated count time series models with application

ABSTRACT Count time series data with excess zeros are observed in several applied disciplines. When these zero-inflated counts are sequentially recorded, they might result in serial dependence. Ignoring the zero-inflation and the serial dependence might produce inaccurate results. In this paper, Markov zero-inflated count time series models based on a joint distribution on consecutive observations are proposed. The joint distribution function of the consecutive observations is constructed through copula functions. First- and second-order Markov chains are considered with the univariate margins of zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), or zero-inflated Conway–Maxwell–Poisson (ZICMP) distributions. Under the Markov models, bivariate copula functions such as the bivariate Gaussian, Frank, and Gumbel are chosen to construct a bivariate distribution of two consecutive observations. Moreover, the trivariate Gaussian and max-infinitely divisible copula functions are considered to build the joint distribution of three consecutive observations. Likelihood-based inference is performed and asymptotic properties are studied. To evaluate the estimation method and the asymptotic results, simulated examples are studied. The proposed class of models are applied to sandstorm counts example. The results suggest that the proposed models have some advantages over some of the models in the literature for modeling zero-inflated count time series data.

Forecasting evaluation via parametric bootstrap for threshold-INARCH models

This article is concerned with the issue of forecasting and evaluation of threshold-asymmetric volatility models for time series of count data. In particular, threshold integer-valued models with conditional Poisson and conditional negative binomial distributions are highlighted. Based on the parametric bootstrap method, some evaluation measures are discussed in terms of one-step ahead forecasting. A parametric bootstrap procedure is explained from which directional measure, magnitude measure and expected cost of misclassification are discussed to evaluate competing models. The cholera data in Bangladesh from 1988 to 2016 is analyzed as a real application.

Multivariate count autoregression

We are studying linear and log-linear models for multivariate count time series data with Poisson marginals. For studying the properties of such processes we develop a novel conceptual framework which is based on copulas. Earlier contributions impose the copula on the joint distribution of the vector of counts by employing a continuous extension methodology. Instead we introduce a copula function on a vector of associated continuous random variables. This construction avoids conceptual difficulties related to the joint distribution of counts yet it keeps the properties of the Poisson process marginally. Furthermore, this construction can be employed for modeling multivariate count time series with other marginal count distributions. We employ Markov chain theory and the notion of weak dependence to study ergodicity and stationarity of the models we consider. Suitable estimating equations are suggested for estimating unknown model parameters. The large sample properties of the resulting estimators are studied in detail. The work concludes with some simulations and a real data example.

Generalized Poisson Hidden Markov Model for Overdispersed or Underdispersed Count Data

The most suitable statistical method for explaining serial dependency in time series count data is that based on Hidden Markov Models (HMMs). These models assume that the observations are generated from a finite mixture of distributions governed by the principle of Markov chain (MC). Poisson-Hidden Markov Model (P-HMM) may be the most widely used method for modelling the above said situations. However, in real life scenario, this model cannot be considered as the best choice. Taking this fact into account, we, in this paper, go for Generalised Poisson Distribution (GPD) for modelling count data. This method can rectify the overdispersion and underdispersion in the Poisson model. Here, we develop Generalised Poisson Hidden Markov model (GP-HMM) by combining GPD with HMM for modelling such data. The results of the study on simulated data and an application of real data, monthly cases of Leptospirosis in the state of Kerala in South India, show good convergence properties, proving that the GP-HMM is a better method compared to P-HMM.

Data Analytics Driven Controlling: Bridging Statistical Modeling and Managerial Intuition

Strategic planning in a corporate environment is often based on experience and intuition, although internal data is usually available and can be a valuable source of information. Predicting merger & acquisition (M&A) events is at the heart of strategic management, yet not sufficiently motivated by data analytics driven controlling. One of the main obstacles in using e.g. count data time series for M&A seems to be the fact that the intensity of M&A is time varying at least in certain business sectors, e.g. communications. We propose a new automatic procedure to bridge this obstacle using novel statistical methods. The proposed approach allows for a selection of adaptive windows in count data sets by detecting significant changes in the intensity of events. We test the efficacy of the proposed method on a simulated count data set and put it into action on various M&A data sets. It is robust to aberrant behavior and generates accurate forecasts for the evaluated business sectors. It also provides guidance for an a-priori selection of fixed windows for forecasting. Furthermore, it can be generalized to other business lines, e.g. for managing supply chains, sales forecasts, or call center arrivals, thus giving managers new ways for incorporating statistical modeling in strategic planning decisions.

A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Integer‐valued time series data have an ever‐increasing presence in various applications (e.g., the number of purchases made in response to a marketing strategy, or the number of employees at a business) and need to be analyzed properly. While a Poisson autoregressive (PAR) model would seem like a natural choice to model such data, it is constrained by the equi‐dispersion assumption (i.e., that the variance and the mean equal). Hence, data that are over‐ or under‐dispersed (i.e., have the variance greater or less than the mean respectively) are improperly modeled, resulting in biased estimates and inaccurate forecasts. This work instead develops a flexible integer‐valued autoregressive model for count data that contain over‐ or under‐dispersion. Using the Conway–Maxwell–Poisson (CMP) distribution and related distributions as motivation, we develop a first‐order sum‐of‐CMP's autoregressive (SCMPAR(1)) model that will instead offer a generalizable construct that captures the PAR, and versions of what we refer to as a negative binomial AR model, and binomial AR model respectively as special cases, and serve as an overarching representation connecting these three special cases through the dispersion parameter. We illustrate the SCMPAR model's flexibility and ability to effectively model count time series data containing data dispersion through simulated and real data examples.

Time series of count data: A review, empirical comparisons and data analysis

Observation and parameter driven models are commonly used in the literature to analyse time series of counts. In this paper, we study the characteristics of a variety of models and point out the main differences and similarities among these procedures, concerning parameter estimation, model fitting and forecasting. Alternatively to the literature, all inference was performed under the Bayesian paradigm. The models are fitted with a latent AR($p$) process in the mean, which accounts for autocorrelation in the data. An extensive simulation study shows that the estimates for the covariate parameters are remarkably similar across the different models. However, estimates for autoregressive coefficients and forecasts of future values depend heavily on the underlying process which generates the data. A real data set of bankruptcy in the United States is also analysed.

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.