Time Series Of Counts Research Articles

Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data

INAR models have the great advantage of being able to capture the conditional distribution of a count time series based on their past observations, thus allowing it to be tailored to meet the unique characteristics of count data. This paper reviews the two-parameter Poisson extended exponential (PEE) distribution and its corresponding INAR(1) process. Then the INAR of order p (INAR(p)) model that incorporates PEE innovations is proposed, its statistical properties are presented, and its parameters are estimated using conditional least squares and conditional maximum likelihood estimation methods. Two practical data sets are analyzed and compared with competing INAR models in an effort to gauge the performance of the proposed model. It is found that the proposed model performs better than the competitors.

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.

Modelling and diagnostic tests for Poisson and negative-binomial count time series

When modelling unbounded counts, their marginals are often assumed to follow either Poisson (Poi) or negative binomial (NB) distributions. To test such null hypotheses, we propose goodness-of-fit (GoF) tests based on statistics relying on certain moment properties. By contrast to most approaches proposed in the count-data literature so far, we do not restrict ourselves to specific low-order moments, but consider a flexible class of functions of generalized moments to construct model-diagnostic tests. These cover GoF-tests based on higher-order factorial moments, which are particularly suitable for the Poi- or NB-distribution where simple closed-form expressions for factorial moments of any order exist, but also GoF-tests relying on the respective Stein’s identity for the Poi- or NB-distribution. In the time-dependent case, under mild mixing conditions, we derive the asymptotic theory for GoF tests based on higher-order factorial moments for a wide family of stationary processes having Poi- or NB-marginals, respectively. This family also includes a type of NB-autoregressive model, where we provide clarification of some confusion caused in the literature. Additionally, for the case of independent and identically distributed counts, we prove asymptotic normality results for GoF-tests relying on a Stein identity, and we briefly discuss how its statistic might be used to define an omnibus GoF-test. The performance of the tests is investigated with simulations for both asymptotic and bootstrap implementations, also considering various alternative scenarios for power analyses. A data example of daily counts of downloads of a TeX editor is used to illustrate the application of the proposed GoF-tests.

Change-points analysis for generalized integer-valued autoregressive model via minimum description length principle

This article considers the problem of modeling a class of count time series with multiple change-points using segmented generalized integer-valued autoregressive (S-GINAR) processes. The minimum description length principle (MDL) is applied to study the statistical inference for the S-GINAR model, and the consistency results of the MDL model selection procedure are established respectively under the condition of known and unknown number of change-points. To find the “best” combination of the number of change-points, the locations of change-points, the order of each segment and its parameters, a genetic algorithm with simulated annealing is implemented to solve this difficult optimization problem. In particular, the simulated annealing process makes up for the precocious problem of the traditional genetic algorithm. Numerical results from simulation experiments and three examples of real data analysis show that the procedure has excellent empirical properties.

Artificial Neural Networks and Time Series of Counts: A Class of Nonlinear INGARCH Models

Abstract Time series of counts are frequently analyzed using generalized integer-valued autoregressive models with conditional heteroskedasticity (INGARCH). These models employ response functions to map a vector of past observations and past conditional expectations to the conditional expectation of the present observation. In this paper, it is shown how INGARCH models can be combined with artificial neural network (ANN) response functions to obtain a class of nonlinear INGARCH models. The ANN framework allows for the interpretation of many existing INGARCH models as a degenerate version of a corresponding neural model. Details on maximum likelihood estimation, marginal effects and confidence intervals are given. The empirical analysis of time series of bounded and unbounded counts reveals that the neural INGARCH models are able to outperform reasonable degenerate competitor models in terms of the information loss.

Hypergraph reconstruction from uncertain pairwise observations

The network reconstruction task aims to estimate a complex system’s structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities—the pairwise case. Here, using Bayesian inference, we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.

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.

Zero-modified count time series modeling with an application to influenza cases

Zero-modified count time series modeling with an application to influenza cases

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.

Doubly-Inflated Poisson INGARCH Models for Count Time Series

Doubly-Inflated Poisson INGARCH Models for Count Time Series

Conditional-mean multiplicative operator models for count time series

Multiplicative error models (MEMs) are commonly used for real-valued time series, but they cannot be applied to discrete-valued count time series as the involved multiplication would not preserve the integer nature of the data. Thus, the concept of a multiplicative operator for counts is proposed (as well as several specific instances thereof), which are then used to develop a kind of MEMs for count time series (CMEMs). If equipped with a linear conditional mean, the resulting CMEMs are closely related to the class of so-called integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models and might be used as a semi-parametric extension thereof. Important stochastic properties of different types of INGARCH-CMEM as well as relevant estimation approaches are derived, namely types of quasi-maximum likelihood and weighted least squares estimation. The performance and application are demonstrated with simulations as well as with two real-world data examples.

True and spurious anomalies in ambient dose rate monitoring.

Ambient dose rate surveys reveal zones of an elevated dose rate. As the observation results from two physical processes-the 'true' process in nature that one wants to assess and the observation process that consists of measuring under certain conditions-an observed anomaly can have its cause in either process. Anomalous effects rooted in the observation process are called spurious. Distinction between true and spurious anomalies may not be straightforward in many cases. As an example, dose rate surveying using a specific instrument is discussed, whose analysis shows the presence of both types of anomalies. Examples of both types are shown. The physical cause of a true anomaly can often be identified by inspecting the environment or by more sophisticated investigation, e.g. by gamma spectrometry. Identification of spurious anomalies is more difficult and requires investigation of count time series.

Efficient probabilistic reconciliation of forecasts for real-valued and count time series

Hierarchical time series are common in several applied fields. The forecasts for these time series are required to be coherent, that is, to satisfy the constraints given by the hierarchy. The most popular technique to enforce coherence is called reconciliation, which adjusts the base forecasts computed for each time series. However, recent works on probabilistic reconciliation present several limitations. In this paper, we propose a new approach based on conditioning to reconcile any type of forecast distribution. We then introduce a new algorithm, called Bottom-Up Importance Sampling, to efficiently sample from the reconciled distribution. It can be used for any base forecast distribution: discrete, continuous, or in the form of samples, providing a major speedup compared to the current methods. Experiments on several temporal hierarchies show a significant improvement over base probabilistic forecasts.

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.

Using state‐space models to estimate recreational angling effort and infer processes that regulate angler dynamics

AbstractObjectiveState‐space models are a flexible modeling approach and are often fit to ecological time series data exhibiting temporal autocorrelation. Traditionally, angler effort data collected using on‐site creel surveys are analyzed using design‐based methods. With some exceptions, state‐space models are rarely used to model creel survey data that are also generally a time series of temporally autocorrelated counts.MethodsIn this study, we demonstrated how to fit state‐space models to a time series of angler counts in 11 sections of three trout fisheries in Idaho. The basic model was extended to make inference about processes that may regulate angling dynamics, such as “population growth rate” of angling effort and recreational carrying capacity.ResultEstimated angling effort varied from 21,179 h in the lowermost section of the Henrys Fork Snake River to 199,457 h in the uppermost section of the South Fork Snake River. The finite population growth rate of angling effort was 1.82 when transitioning from a weekday to a weekend, suggesting that angling effort was 1.82 times greater, on average, on Saturdays than on Fridays, and the population growth rate was 0.38 (i.e., 0.38 times smaller) when transitioning from a Sunday to a Monday. Estimated carrying capacity among fishery sections varied from 129 daily hours of angling effort on the Big Lost River to 693 h on the middle section of the South Fork Snake River. Carrying capacity was 1.88 times higher on fishery sections that had ≥0.5 access points/km than on sections with <0.5 access points/km.ConclusionThe state‐space models used in this study can be modified or extended to fit a variety of data types or can be used to evaluate additional hypotheses regarding the angling process.

State Space Modeling of Event Count Time Series.

This paper proposes a class of algorithms for analyzing event count time series, based on state space modeling and Kalman filtering. While the dynamics of the state space model is kept Gaussian and linear, a nonlinear observation function is chosen. In order to estimate the states, an iterated extended Kalman filter is employed. Positive definiteness of covariance matrices is preserved by a square-root filtering approach, based on singular value decomposition. Non-negativity of the count data is ensured, either by an exponential observation function, or by a newly introduced "affinely distorted hyperbolic" observation function. The resulting algorithm is applied to time series of the daily number of seizures of drug-resistant epilepsy patients. This number may depend on dosages of simultaneously administered anti-epileptic drugs, their superposition effects, delay effects, and unknown factors, making the objective analysis of seizure counts time series arduous. For the purpose of validation, a simulation study is performed. The results of the time series analysis by state space modeling, using the dosages of the anti-epileptic drugs as external control inputs, provide a decision on the effect of the drugs in a particular patient, with respect to reducing or increasing the number of seizures.

Cyber Risk Contagion

Financial technologies (fintechs) are continuously expanding, across different markets and financial services. While financial technologies bring many opportunities, such as reduced costs and extended inclusion, they also bring risks, among which include cyber risks, that are difficult to measure. One of the difficulties that arise in the measurement of cyber risks is the interdependence among cyber losses, a problem that has not yet been solved. To fill the gap, this paper proposes a multivariate model for cyber risks, based on their observed time series of counts. The time-varying intensity parameter of the model determines the probability that a cyber attack occurs, and its specification takes not only time but also sectorial interdependence into account. The effectiveness of the proposed model is demonstrated by means of a real cyber loss dataset, in which there exists time and sectorial dependence among different events.

Analysis of Count Time Series: A Bayesian GARMA(p, q) Approach

Extensions of the Autoregressive Moving Average, ARMA(p, q), class for modeling non-Gaussian time series have been proposed in the literature in recent years, being applied in phenomena such as counts and rates. One of them is the Generalized Autoregressive Moving Average, GARMA(p, q), that is supported by the Generalized Linear Models theory and has been studied under the Bayesian perspective. This paper aimed to study models for time series of counts using the Poisson, Negative binomial and Poisson inverse Gaussian distributions, and adopting the Bayesian framework. To do so, we carried out a simulation study and, in addition, we showed a practical application and evaluation of these models by using a set of real data, corresponding to the number of vehicle thefts in Brazil.