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Discrete-Valued Time Series
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Discrete-Valued Time Series

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (31 May 2023) | Viewed by 23006

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Christian H. Weiss

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, Helmut-Schmidt-University, PO-Box 700822, D-22008 Hamburg, Germany
Interests: time series analysis; count time series; categorical time series; statistical process control; discrete data; computational statistics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The first methods for analyzing time series were developed about 100 years ago. Since then, the field of time series analysis has enjoyed growing interest among researchers and practitioners. For most of this period, however, the focus was on real-valued time series, i.e., time series having a continuous range consisting of real numbers or real vectors. Starting slowly in the 1980s, and then rapidly since the 2000s, also discrete-valued time series began to attract more and more attention. Here, discrete-valued time series might be of various types. Undoubtedly most popular are count time series, the range of which is quantitative and consists of non-negative integers (either the full set of non-negative integers or a finite subset thereof with some specified upper bound). However, also truly integer-valued time series (where the range also includes negative integers) are increasingly considered. By contrast, categorical time series, with the range being qualitative, are still somewhat disregarded until now. Here, one has to distinguish the case where the attainable categories exhibit a natural ordering, so-called ordinal time series, and those where not even an ordering exists (nominal time series). Finally, we also sometimes end up with discrete-valued time series although the raw time series have a continuous range. Such discretizations happen, for example, if clipping is applied to a real-valued time series, or if methods based on ordinal patterns are used for its analysis.

All these (and possibly further) fields of time series analysis are covered by the intended Special Issue on "Discrete-valued Time Series". The aim is to bring together papers on:

  • Stochastic models for discrete-valued time series of any type;
  • Methods for analyzing discrete-valued or discretized time series (identification, estimation, validation, etc.);
  • Univariate or multivariate discrete-valued or discretized time series;
  • Applications of discrete time-series methods to forecasting, change-point detection, or statistical process control (among others).

Papers including real applications, and maybe covering historical aspects of discrete-valued time series analysis, are particularly welcome. The Special Issue is also open to interdisciplinary research or comprehensive survey papers (also aspects of teaching or software might be touched upon), as long as methods or models for discrete-valued or discretized time series constitute the core of the contribution.

Prof. Dr. Christian H. Weiss
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

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Published Papers (11 papers)

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Editorial

Jump to: Research, Review

4 pages, 196 KiB  
Editorial
Discrete-Valued Time Series
by Christian H. Weiß
Entropy 2023, 25(12), 1576; https://doi.org/10.3390/e25121576 - 23 Nov 2023
Cited by 1 | Viewed by 1092
Abstract
Time series are sequentially observed data in which important information about the phenomenon under consideration is contained not only in the individual observations themselves, but also in the way these observations follow one another [...] Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)

Research

Jump to: Editorial, Review

21 pages, 994 KiB  
Article
State Space Modeling of Event Count Time Series
by Sidratul Moontaha, Bert Arnrich and Andreas Galka
Entropy 2023, 25(10), 1372; https://doi.org/10.3390/e25101372 - 23 Sep 2023
Cited by 2 | Viewed by 1536
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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30 pages, 1502 KiB  
Article
An Observation-Driven Random Parameter INAR(1) Model Based on the Poisson Thinning Operator
by Kaizhi Yu and Tielai Tao
Entropy 2023, 25(6), 859; https://doi.org/10.3390/e25060859 - 27 May 2023
Cited by 3 | Viewed by 1497
Abstract
This paper presents a first-order integer-valued autoregressive time series model featuring observation-driven parameters that may adhere to a particular random distribution. We derive the ergodicity of the model as well as the theoretical properties of point estimation, interval estimation, and parameter testing. The [...] Read more.
This paper presents a first-order integer-valued autoregressive time series model featuring observation-driven parameters that may adhere to a particular random distribution. We derive the ergodicity of the model as well as the theoretical properties of point estimation, interval estimation, and parameter testing. The properties are verified through numerical simulations. Lastly, we demonstrate the application of this model using real-world datasets. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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25 pages, 424 KiB  
Article
Ruin Analysis on a New Risk Model with Stochastic Premiums and Dependence Based on Time Series for Count Random Variables
by Lihong Guan and Xiaohong Wang
Entropy 2023, 25(4), 698; https://doi.org/10.3390/e25040698 - 21 Apr 2023
Cited by 1 | Viewed by 2092
Abstract
In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among [...] Read more.
In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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15 pages, 500 KiB  
Article
Time Series of Counts under Censoring: A Bayesian Approach
by Isabel Silva, Maria Eduarda Silva, Isabel Pereira and Brendan McCabe
Entropy 2023, 25(4), 549; https://doi.org/10.3390/e25040549 - 23 Mar 2023
Viewed by 1601
Abstract
Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The [...] Read more.
Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The aim of this work is to contribute to the modelling of time series of counts under censoring using convolution closed infinitely divisible (CCID) models. The emphasis is on estimation and inference problems, using Bayesian approaches with Approximate Bayesian Computation (ABC) and Gibbs sampler with Data Augmentation (GDA) algorithms. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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12 pages, 484 KiB  
Article
Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts
by Manuel Cabral Morais
Entropy 2023, 25(3), 444; https://doi.org/10.3390/e25030444 - 2 Mar 2023
Viewed by 1340
Abstract
The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive [...] Read more.
The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 (TP2) transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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18 pages, 341 KiB  
Article
A Modified Multiplicative Thinning-Based INARCH Model: Properties, Saddlepoint Maximum Likelihood Estimation, and Application
by Yue Xu, Qi Li and Fukang Zhu
Entropy 2023, 25(2), 207; https://doi.org/10.3390/e25020207 - 21 Jan 2023
Cited by 3 | Viewed by 1405
Abstract
In this article, we propose a modified multiplicative thinning-based integer-valued autoregressive conditional heteroscedasticity model and use the saddlepoint maximum likelihood estimation (SPMLE) method to estimate parameters. A simulation study is given to show a better performance of the SPMLE. The application of the [...] Read more.
In this article, we propose a modified multiplicative thinning-based integer-valued autoregressive conditional heteroscedasticity model and use the saddlepoint maximum likelihood estimation (SPMLE) method to estimate parameters. A simulation study is given to show a better performance of the SPMLE. The application of the real data, which is concerned with the number of tick changes by the minute of the euro to the British pound exchange rate, shows the superiority of our modified model and the SPMLE. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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16 pages, 5695 KiB  
Article
A Conway–Maxwell–Poisson-Binomial AR(1) Model for Bounded Time Series Data
by Huaping Chen, Jiayue Zhang and Xiufang Liu
Entropy 2023, 25(1), 126; https://doi.org/10.3390/e25010126 - 7 Jan 2023
Cited by 2 | Viewed by 2370
Abstract
Binomial autoregressive models are frequently used for modeling bounded time series counts. However, they are not well developed for more complex bounded time series counts of the occurrence of n exchangeable and dependent units, which are becoming increasingly common in practice. To fill [...] Read more.
Binomial autoregressive models are frequently used for modeling bounded time series counts. However, they are not well developed for more complex bounded time series counts of the occurrence of n exchangeable and dependent units, which are becoming increasingly common in practice. To fill this gap, this paper first constructs an exchangeable Conway–Maxwell–Poisson-binomial (CMPB) thinning operator and then establishes the Conway–Maxwell–Poisson-binomial AR (CMPBAR) model. We establish its stationarity and ergodicity, discuss the conditional maximum likelihood (CML) estimate of the model’s parameters, and establish the asymptotic normality of the CML estimator. In a simulation study, the boxplots illustrate that the CML estimator is consistent and the qqplots show the asymptotic normality of the CML estimator. In the real data example, our model takes a smaller AIC and BIC than its main competitors. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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21 pages, 482 KiB  
Article
Partial Autocorrelation Diagnostics for Count Time Series
by Christian H. Weiß, Boris Aleksandrov, Maxime Faymonville and Carsten Jentsch
Entropy 2023, 25(1), 105; https://doi.org/10.3390/e25010105 - 4 Jan 2023
Cited by 6 | Viewed by 1885
Abstract
In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, [...] Read more.
In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule–Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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21 pages, 547 KiB  
Article
Adaptation of Partial Mutual Information from Mixed Embedding to Discrete-Valued Time Series
by Maria Papapetrou, Elsa Siggiridou and Dimitris Kugiumtzis
Entropy 2022, 24(11), 1505; https://doi.org/10.3390/e24111505 - 22 Oct 2022
Cited by 3 | Viewed by 2979
Abstract
A causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is found appropriate for the causality analysis of continuous-valued time [...] Read more.
A causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is found appropriate for the causality analysis of continuous-valued time series, even of high dimension, as it applies a dimension reduction by selecting the most relevant lag variables of all the observed variables to the response, using conditional mutual information (CMI). The presence of lag components of the driving variable in this vector implies a direct causal (driving-response) effect. In this study, the PMIME is appropriately adapted to discrete-valued multivariate time series, called the discrete PMIME (DPMIME). An appropriate estimation of the discrete probability distributions and CMI for discrete variables is implemented in the DPMIME. Further, the asymptotic distribution of the estimated CMI is derived, allowing for a parametric significance test for the CMI in the DPMIME, whereas for the PMIME, there is no parametric test for the CMI and the test is performed using resampling. Monte Carlo simulations are performed using different generating systems of discrete-valued time series. The simulation suggests that the parametric significance test for the CMI in the progressive algorithm of the DPMIME is compared favorably to the corresponding resampling significance test, and the accuracy of the DPMIME in the estimation of direct causality converges with the time-series length to the accuracy of the PMIME. Further, the DPMIME is used to investigate whether the global financial crisis has an effect on the causality network of the financial world market. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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Review

Jump to: Editorial, Research

27 pages, 514 KiB  
Review
A Systematic Review of INGARCH Models for Integer-Valued Time Series
by Mengya Liu, Fukang Zhu, Jianfeng Li and Chuning Sun
Entropy 2023, 25(6), 922; https://doi.org/10.3390/e25060922 - 11 Jun 2023
Cited by 10 | Viewed by 3147
Abstract
Count time series are widely available in fields such as epidemiology, finance, meteorology, and sports, and thus there is a growing demand for both methodological and application-oriented research on such data. This paper reviews recent developments in integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) [...] Read more.
Count time series are widely available in fields such as epidemiology, finance, meteorology, and sports, and thus there is a growing demand for both methodological and application-oriented research on such data. This paper reviews recent developments in integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models over the past five years, focusing on data types including unbounded non-negative counts, bounded non-negative counts, Z-valued time series and multivariate counts. For each type of data, our review follows the three main lines of model innovation, methodological development, and expansion of application areas. We attempt to summarize the recent methodological developments of INGARCH models for each data type for the integration of the whole INGARCH modeling field and suggest some potential research topics. Full article
(This article belongs to the Special Issue Discrete-Valued Time Series)
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