Special Issue "Time Series Modelling"

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 (30 April 2021).

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 Issue Information

Dear Colleagues,

Time series consist of data observed sequentially in time, and they are assumed to stem from an underlying stochastic process. The scope of time series approaches thus covers models for stochastic processes as well as inferential procedures for model fitting, model diagnostics, forecasting, and various other applications. While time series data have been collected for a relatively long time in history (one may recall the famous time series on sunspot numbers), the development of methods and stochastic models for such time series is more recent. For example, the correlogram or the autoregressive and moving-average models for time series, which are currently part of any course on time series analysis and are covered by any statistical software, date back to only the 1920s, and thus will celebrate their 100th birthday in the upcoming years. Furthermore, the first comprehensive textbook on time series was published exactly 50 years ago by G.E.P. Box and G.M. Jenkins in 1970. Thus, since 2020 constitutes a two-fold "anniversary year" in some sense, it is reasonable to use this opportunity to publishing a Special Issue on "Time Series Modelling".

The aim is to bring together papers from the following areas related to time series:

  • stochastic models for time series, as well as methods for analyzing time series (estimation, diagnostics);
  • univariate or multivariate real-valued time series, as well as discrete-valued time series (such as count time series or categorical time series); and
  • applications of time series methods for forecasting, change-point detection, or statistical process control, among others.

Papers including real applications, also those covering historical aspects of time series analysis, are particularly welcome. The Special Issue is also open to interdisciplinary research, comprehensive survey papers, as well as papers with aspects of teaching and software, with core contributions including methods or models for time series.

Prof. Dr. Christian H. Weiss
Guest Editor

Manuscript Submission Information

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Published Papers (15 papers)

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Research

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Article
Statistical Inference for Periodic Self-Exciting Threshold Integer-Valued Autoregressive Processes
Entropy 2021, 23(6), 765; https://doi.org/10.3390/e23060765 - 17 Jun 2021
Viewed by 186
Abstract
This paper considers the periodic self-exciting threshold integer-valued autoregressive processes under a weaker condition in which the second moment is finite instead of the innovation distribution being given. The basic statistical properties of the model are discussed, the quasi-likelihood inference of the parameters [...] Read more.
This paper considers the periodic self-exciting threshold integer-valued autoregressive processes under a weaker condition in which the second moment is finite instead of the innovation distribution being given. The basic statistical properties of the model are discussed, the quasi-likelihood inference of the parameters is investigated, and the asymptotic behaviors of the estimators are obtained. Threshold estimates based on quasi-likelihood and least squares methods are given. Simulation studies evidence that the quasi-likelihood methods perform well with realistic sample sizes and may be superior to least squares and maximum likelihood methods. The practical application of the processes is illustrated by a time series dataset concerning the monthly counts of claimants collecting short-term disability benefits from the Workers’ Compensation Board (WCB). In addition, the forecasting problem of this dataset is addressed. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Multivariate Count Data Models for Time Series Forecasting
Entropy 2021, 23(6), 718; https://doi.org/10.3390/e23060718 - 05 Jun 2021
Viewed by 388
Abstract
Count data appears in many research fields and exhibits certain features that make modeling difficult. Most popular approaches to modeling count data can be classified into observation and parameter-driven models. In this paper, we review two models from these classes: the log-linear multivariate [...] Read more.
Count data appears in many research fields and exhibits certain features that make modeling difficult. Most popular approaches to modeling count data can be classified into observation and parameter-driven models. In this paper, we review two models from these classes: the log-linear multivariate conditional intensity model (also referred to as an integer-valued generalized autoregressive conditional heteroskedastic model) and the non-linear state-space model for count data. We compare these models in terms of forecasting performance on simulated data and two real datasets. In simulations, we consider the case of model misspecification. We find that both models have advantages in different situations, and we discuss the pros and cons of inference for both models in detail. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
A New First-Order Integer-Valued Autoregressive Model with Bell Innovations
Entropy 2021, 23(6), 713; https://doi.org/10.3390/e23060713 - 04 Jun 2021
Viewed by 435
Abstract
A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter [...] Read more.
A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule–Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates’ performances. Real data examples are provided. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
A New Overdispersed Integer-Valued Moving Average Model with Dependent Counting Series
Entropy 2021, 23(6), 706; https://doi.org/10.3390/e23060706 - 02 Jun 2021
Viewed by 387
Abstract
A new integer-valued moving average model is introduced. The assumption of independent counting series in the model is relaxed to allow dependence between them, leading to the overdispersion in the model. Statistical properties were established for this new integer-valued moving average model with [...] Read more.
A new integer-valued moving average model is introduced. The assumption of independent counting series in the model is relaxed to allow dependence between them, leading to the overdispersion in the model. Statistical properties were established for this new integer-valued moving average model with dependent counting series. The Yule–Walker method was applied to estimate the model parameters. The estimator’s performance was evaluated using simulations, and the overdispersion test of the INMA(1) process was applied to examine the dependence between counting series. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Ordinal Pattern Dependence in the Context of Long-Range Dependence
Entropy 2021, 23(6), 670; https://doi.org/10.3390/e23060670 - 26 May 2021
Viewed by 543
Abstract
Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This [...] Read more.
Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Count Data Time Series Modelling in Julia—The CountTimeSeries.jl Package and Applications
Entropy 2021, 23(6), 666; https://doi.org/10.3390/e23060666 - 25 May 2021
Viewed by 349
Abstract
A new software package for the Julia language, CountTimeSeries.jl, is under review, which provides likelihood based methods for integer-valued time series. The package’s functionalities are showcased in a simulation study on finite sample properties of Maximum Likelihood (ML) estimation and three real-life data [...] Read more.
A new software package for the Julia language, CountTimeSeries.jl, is under review, which provides likelihood based methods for integer-valued time series. The package’s functionalities are showcased in a simulation study on finite sample properties of Maximum Likelihood (ML) estimation and three real-life data applications. First, the number of newly infected COVID-19 patients is predicted. Then, previous findings on the need for overdispersion and zero inflation are reviewed in an application on animal submissions in New Zealand. Further, information criteria are used for model selection to investigate patterns in corporate insolvencies in Rhineland-Palatinate. Theoretical background and implementation details are described, and complete code for all applications is provided online. The CountTimeSeries package is available at the general Julia package registry. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Entropy Based Student’s t-Process Dynamical Model
Entropy 2021, 23(5), 560; https://doi.org/10.3390/e23050560 - 30 Apr 2021
Viewed by 338
Abstract
Volatility, which represents the magnitude of fluctuating asset prices or returns, is used in the problems of finance to design optimal asset allocations and to calculate the price of derivatives. Since volatility is unobservable, it is identified and estimated by latent variable models [...] Read more.
Volatility, which represents the magnitude of fluctuating asset prices or returns, is used in the problems of finance to design optimal asset allocations and to calculate the price of derivatives. Since volatility is unobservable, it is identified and estimated by latent variable models known as volatility fluctuation models. Almost all conventional volatility fluctuation models are linear time-series models and thus are difficult to capture nonlinear and/or non-Gaussian properties of volatility dynamics. In this study, we propose an entropy based Student’s t-process Dynamical model (ETPDM) as a volatility fluctuation model combined with both nonlinear dynamics and non-Gaussian noise. The ETPDM estimates its latent variables and intrinsic parameters by a robust particle filtering based on a generalized H-theorem for a relative entropy. To test the performance of the ETPDM, we implement numerical experiments for financial time-series and confirm the robustness for a small number of particles by comparing with the conventional particle filtering. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Asymptotic Properties of Estimators for Seasonally Cointegrated State Space Models Obtained Using the CVA Subspace Method
Entropy 2021, 23(4), 436; https://doi.org/10.3390/e23040436 - 08 Apr 2021
Viewed by 383
Abstract
This paper investigates the asymptotic properties of estimators obtained from the so called CVA (canonical variate analysis) subspace algorithm proposed by Larimore (1983) in the case when the data is generated using a minimal state space system containing unit roots at the seasonal [...] Read more.
This paper investigates the asymptotic properties of estimators obtained from the so called CVA (canonical variate analysis) subspace algorithm proposed by Larimore (1983) in the case when the data is generated using a minimal state space system containing unit roots at the seasonal frequencies such that the yearly difference is a stationary vector autoregressive moving average (VARMA) process. The empirically most important special cases of such data generating processes are the I(1) case as well as the case of seasonally integrated quarterly or monthly data. However, increasingly also datasets with a higher sampling rate such as hourly, daily or weekly observations are available, for example for electricity consumption. In these cases the vector error correction representation (VECM) of the vector autoregressive (VAR) model is not very helpful as it demands the parameterization of one matrix per seasonal unit root. Even for weekly series this amounts to 52 matrices using yearly periodicity, for hourly data this is prohibitive. For such processes estimation using quasi-maximum likelihood maximization is extremely hard since the Gaussian likelihood typically has many local maxima while the parameter space often is high-dimensional. Additionally estimating a large number of models to test hypotheses on the cointegrating rank at the various unit roots becomes practically impossible for weekly data, for example. This paper shows that in this setting CVA provides consistent estimators of the transfer function generating the data, making it a valuable initial estimator for subsequent quasi-likelihood maximization. Furthermore, the paper proposes new tests for the cointegrating rank at the seasonal frequencies, which are easy to compute and numerically robust, making the method suitable for automatic modeling. A simulation study demonstrates by example that for processes of moderate to large dimension the new tests may outperform traditional tests based on long VAR approximations in sample sizes typically found in quarterly macroeconomic data. Further simulations show that the unit root tests are robust with respect to different distributions for the innovations as well as with respect to GARCH-type conditional heteroskedasticity. Moreover, an application to Kaggle data on hourly electricity consumption by different American providers demonstrates the usefulness of the method for applications. Therefore the CVA algorithm provides a very useful initial guess for subsequent quasi maximum likelihood estimation and also delivers relevant information on the cointegrating ranks at the different unit root frequencies. It is thus a useful tool for example in (but not limited to) automatic modeling applications where a large number of time series involving a substantial number of variables need to be modelled in parallel. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Monitoring the Zero-Inflated Time Series Model of Counts with Random Coefficient
Entropy 2021, 23(3), 372; https://doi.org/10.3390/e23030372 - 20 Mar 2021
Cited by 1 | Viewed by 617
Abstract
In this research, we consider monitoring mean and correlation changes from zero-inflated autocorrelated count data based on the integer-valued time series model with random survival rate. A cumulative sum control chart is constructed due to its efficiency, the corresponding calculation methods of average [...] Read more.
In this research, we consider monitoring mean and correlation changes from zero-inflated autocorrelated count data based on the integer-valued time series model with random survival rate. A cumulative sum control chart is constructed due to its efficiency, the corresponding calculation methods of average run length and the standard deviation of the run length are given. Practical guidelines concerning the chart design are investigated. Extensive computations based on designs of experiments are conducted to illustrate the validity of the proposed method. Comparisons with the conventional control charting procedure are also provided. The analysis of the monthly number of drug crimes in the city of Pittsburgh is displayed to illustrate our current method of process monitoring. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Robust Estimation for Bivariate Poisson INGARCH Models
Entropy 2021, 23(3), 367; https://doi.org/10.3390/e23030367 - 19 Mar 2021
Cited by 1 | Viewed by 437
Abstract
In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the [...] Read more.
In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the minimum density power divergence estimator. We demonstrate the proposed estimator is consistent and asymptotically normal under certain regularity conditions. Monte Carlo simulations are conducted to evaluate the performance of the estimator in the presence of outliers. Finally, a real data analysis using monthly count series of crimes in New South Wales and an artificial data example are provided as an illustration. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Comparative Analysis of Different Univariate Forecasting Methods in Modelling and Predicting the Romanian Unemployment Rate for the Period 2021–2022
Entropy 2021, 23(3), 325; https://doi.org/10.3390/e23030325 - 09 Mar 2021
Viewed by 711
Abstract
Unemployment has risen as the economy has shrunk. The coronavirus crisis has affected many sectors in Romania, some companies diminishing or even ceasing their activity. Making forecasts of the unemployment rate has a fundamental impact and importance on future social policy strategies. The [...] Read more.
Unemployment has risen as the economy has shrunk. The coronavirus crisis has affected many sectors in Romania, some companies diminishing or even ceasing their activity. Making forecasts of the unemployment rate has a fundamental impact and importance on future social policy strategies. The aim of the paper is to comparatively analyze the forecast performances of different univariate time series methods with the purpose of providing future predictions of unemployment rate. In order to do that, several forecasting models (seasonal model autoregressive integrated moving average (SARIMA), self-exciting threshold autoregressive (SETAR), Holt–Winters, ETS (error, trend, seasonal), and NNAR (neural network autoregression)) have been applied, and their forecast performances have been evaluated on both the in-sample data covering the period January 2000–December 2017 used for the model identification and estimation and the out-of-sample data covering the last three years, 2018–2020. The forecast of unemployment rate relies on the next two years, 2021–2022. Based on the in-sample forecast assessment of different methods, the forecast measures root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE) suggested that the multiplicative Holt–Winters model outperforms the other models. For the out-of-sample forecasting performance of models, RMSE and MAE values revealed that the NNAR model has better forecasting performance, while according to MAPE, the SARIMA model registers higher forecast accuracy. The empirical results of the Diebold–Mariano test at one forecast horizon for out-of-sample methods revealed differences in the forecasting performance between SARIMA and NNAR, of which the best model of modeling and forecasting unemployment rate was considered to be the NNAR model. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
A New Extension of Thinning-Based Integer-Valued Autoregressive Models for Count Data
Entropy 2021, 23(1), 62; https://doi.org/10.3390/e23010062 - 31 Dec 2020
Cited by 3 | Viewed by 537
Abstract
The thinning operators play an important role in the analysis of integer-valued autoregressive models, and the most widely used is the binomial thinning. Inspired by the theory about extended Pascal triangles, a new thinning operator named extended binomial is introduced, which is a [...] Read more.
The thinning operators play an important role in the analysis of integer-valued autoregressive models, and the most widely used is the binomial thinning. Inspired by the theory about extended Pascal triangles, a new thinning operator named extended binomial is introduced, which is a general case of the binomial thinning. Compared to the binomial thinning operator, the extended binomial thinning operator has two parameters and is more flexible in modeling. Based on the proposed operator, a new integer-valued autoregressive model is introduced, which can accurately and flexibly capture the dispersed features of counting time series. Two-step conditional least squares (CLS) estimation is investigated for the innovation-free case and the conditional maximum likelihood estimation is also discussed. We have also obtained the asymptotic property of the two-step CLS estimator. Finally, three overdispersed or underdispersed real data sets are considered to illustrate a superior performance of the proposed model. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Modeling Spectral Properties in Stationary Processes of Varying Dimensions with Applications to Brain Local Field Potential Signals
Entropy 2020, 22(12), 1375; https://doi.org/10.3390/e22121375 - 05 Dec 2020
Cited by 1 | Viewed by 487
Abstract
In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, [...] Read more.
In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, a frequency specific spectral ratio (FS-ratio) statistic is proposed and its asymptotic properties are derived. The FS-ratio is blind to the dimension of the stationary process and captures the proportion of spectral power in various frequency bands. Here we develop a technique to automatically identify frequency bands that carry significant spectral power. We apply our method to track changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. At every epoch (a distinct time segment from the duration of the experiment), the nonstationary LFP signal is decomposed into stationary and nonstationary latent sources and the complexity is analyzed through these latent stationary sources and their dimensions that can change across epochs. The analysis indicates that spectral information in the Beta frequency band (12–30 Hertz) demonstrated the greatest change in structure and complexity due to the stroke. Full article
(This article belongs to the Special Issue Time Series Modelling)
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Article
Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection
Entropy 2020, 22(12), 1363; https://doi.org/10.3390/e22121363 - 30 Nov 2020
Viewed by 761
Abstract
We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be [...] Read more.
We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier approach”). We refer to these approaches as Functional Kernel Density Estimation for Anomaly Detection as they both yield functionals that can score a time series for how anomalous it is. Both methods naturally handle missing data and apply to a variety of settings, performing well when compared with an outlyingness score derived from a boxplot method for functional data, with a Principal Component Analysis approach for functional data, and with the Functional Isolation Forest method. We illustrate the use of the proposed methods with aviation safety report data from the International Air Transport Association (IATA). Full article
(This article belongs to the Special Issue Time Series Modelling)
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Review

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Review
A Systematic Review of Statistical and Machine Learning Methods for Electrical Power Forecasting with Reported MAPE Score
Entropy 2020, 22(12), 1412; https://doi.org/10.3390/e22121412 - 15 Dec 2020
Cited by 2 | Viewed by 657
Abstract
Electric power forecasting plays a substantial role in the administration and balance of current power systems. For this reason, accurate predictions of service demands are needed to develop better programming for the generation and distribution of power and to reduce the risk of [...] Read more.
Electric power forecasting plays a substantial role in the administration and balance of current power systems. For this reason, accurate predictions of service demands are needed to develop better programming for the generation and distribution of power and to reduce the risk of vulnerabilities in the integration of an electric power system. For the purposes of the current study, a systematic literature review was applied to identify the type of model that has the highest propensity to show precision in the context of electric power forecasting. The state-of-the-art model in accurate electric power forecasting was determined from the results reported in 257 accuracy tests from five geographic regions. Two classes of forecasting models were compared: classical statistical or mathematical (MSC) and machine learning (ML) models. Furthermore, the use of hybrid models that have made significant contributions to electric power forecasting is identified, and a case of study is applied to demonstrate its good performance when compared with traditional models. Among our main findings, we conclude that forecasting errors are minimized by reducing the time horizon, that ML models that consider various sources of exogenous variability tend to have better forecast accuracy, and finally, that the accuracy of the forecasting models has significantly increased over the last five years. Full article
(This article belongs to the Special Issue Time Series Modelling)
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