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There are several real applications where the categories behind compositional data (CoDa) exhibit a natural order, which, however, is not accounted for by existing CoDa methods. For various application areas, it is demonstrated that appropriately developed methods for ordinal CoDa provide valuable additional insights and are, thus, recommended to complement existing CoDa methods. The potential benefits are demonstrated for the (visual) descriptive analysis of ordinal CoDa, for statistical inference based on CoDa samples, for the monitoring of CoDa processes by means of control charts, and for the analysis and modelling of compositional time series. The novel methods are illustrated by a couple of real-world data examples. [ABSTRACT FROM AUTHOR]

Several models for count time series have been developed during the last decades, often inspired by traditional autoregressive moving average (ARMA) models for real‐valued time series, including integer‐valued ARMA (INARMA) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models. Both INARMA and INGARCH models exhibit an ARMA‐like autocorrelation function (ACF). To achieve negative ACF values within the class of INGARCH models, log and softplus link functions are suggested in the literature, where the softplus approach leads to conditional linearity in good approximation. However, the softplus approach is limited to the INGARCH family for unbounded counts, that is, it can neither be used for bounded counts, nor for count processes from the INARMA family. In this paper, we present an alternative solution, named the Tobit approach, for achieving approximate linearity together with negative ACF values, which is more generally applicable than the softplus approach. A Skellam–Tobit INGARCH model for unbounded counts is studied in detail, including stationarity, approximate computation of moments, maximum likelihood and censored least absolute deviations estimation for unknown parameters and corresponding simulations. Extensions of the Tobit approach to other situations are also discussed, including underlying discrete distributions, INAR models, and bounded counts. Three real‐data examples are considered to illustrate the usefulness of the new approach. [ABSTRACT FROM AUTHOR]

In the past four decades, research on count time series has made significant progress, but research on ℤ$$ \mathbb{Z} $$‐valued time series is relatively rare. Existing ℤ$$ \mathbb{Z} $$‐valued models are mainly of autoregressive structure, where the use of the rounding operator is very natural. Because of the discontinuity of the rounding operator, the formulation of the corresponding model identifiability conditions and the computation of parameter estimators need special attention. It is also difficult to derive closed‐form formulae for crucial stochastic properties. We rediscover a stochastic rounding operator, referred to as mean‐preserving rounding, which overcomes the above drawbacks. Then, a novel class of ℤ$$ \mathbb{Z} $$‐valued ARMA models based on the new operator is proposed, and the existence of stationary solutions of the models is established. Stochastic properties including closed‐form formulae for (conditional) moments, autocorrelation function, and conditional distributions are obtained. The advantages of our novel model class compared to existing ones are demonstrated. In particular, our model construction avoids identifiability issues such that maximum likelihood estimation is possible. A simulation study is provided, and the appealing performance of the new models is shown by several real‐world data sets. [ABSTRACT FROM AUTHOR]

A new and flexible class of ARMA‐like (autoregressive moving average) models for nominal or ordinal time series is proposed, which are characterized by using so‐called weighting operators and are, thus, referred to as weighted discrete ARMA (WDARMA) models. By choosing an appropriate type of weighting operator, one can model, for example, nominal time series with negative serial dependencies, or ordinal time series where transitions to neighboring states are more likely than sudden large jumps. Essential stochastic properties of WDARMA models are derived, such as the existence of a stationary, ergodic, and φ‐mixing solution as well as closed‐form formulae for marginal and bivariate probabilities. Numerical illustrations as well as simulation experiments regarding the finite‐sample performance of maximum likelihood estimation are presented. The possible benefits of using an appropriate weighting scheme within the WDARMA class are demonstrated by a real‐world data application. [ABSTRACT FROM AUTHOR]

We provide a new testing procedure to detect serial dependence in time series. Our method is based solely on the ordinal structure of the data. We explicitly allow for ties in the data windows we consider. Consequently, we use generalised ordinal patterns, that is, Cayley permutations. Unlike in the classical case, the pattern distribution is not uniform under the null hypothesis of serial independence. In our new framework, the underlying distribution has to be taken into account and we overcome this problem by a bootstrap procedure. The applicability of our method is supported by a simulation study and two real-world data examples. [ABSTRACT FROM AUTHOR]

The soft-clipping binomial INGARCH (scBINGARCH) models are proposed as time series models for bounded counts, which have a nearly linear structure and also allow for negative autocor-relations. Conditions that guarantee the existence and certain mixing properties of the scBINGARCH process are derived, and further stochastic properties are discussed. The consistency and asymptotic nor-mality of maximum likelihood estimators are established, and finite-sample properties are studied with simulations. The practical relevance of the scBINGARCH model's ability to allow for negative parameter and ACF values is demonstrated by some real-data examples. [ABSTRACT FROM AUTHOR]

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule–Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated. [ABSTRACT FROM AUTHOR]

The article discusses the analysis and modeling of time series data, with a focus on discrete-valued time series. It highlights the increasing research interest in this area and the various types of discrete-valued time series, such as count time series and integer-valued time series. The article also presents different models and methods for analyzing discrete-valued time series, including INARMA models and INGARCH models. Additionally, it mentions applications of these models in areas such as insurance risk analysis and statistical process monitoring. The article concludes by summarizing the contributions to a special issue on discrete-valued time series. [Extracted from the article]

We consider the problem of monitoring for the existence of serial dependence in real-valued and continuously distributed processes. These can exist when a process goes out-of-control. Besides, as many control charts are designed under the assumption of independent and identically distributed observations, the validity of this assumption needs to be checked. In the literature, the majority of studies handled this problem by using a specific case of autoregressive moving-average time series models. Also a moving-window approach is often considered, which leads to the drawback that at least a window length of observations need to be collected before an out-of-control situation can be detected. Here, we use ordinal patterns and propose charts that are fully nonparametric and distribution-free, have a unique chart design, and can be used almost instantaneously at the start of process monitoring. The proposed control charts do not require any model fitting, thus, eliminating the problems associated with estimation errors, but might be used as part of a Phase-I study. Through a comprehensive performance comparison study under various out-of-control scenarios, the effectiveness of the charts in uncovering serial dependence is shown and recommendations for selection are given. Implementations of the proposed charts are illustrated by using batch yield data from a chemical process. [ABSTRACT FROM AUTHOR]

Qualitative, more specifically, ordinal data generating processes are common in real-world process control implementations. In this study, a survey of control charts for the sample-based monitoring of independent and identically distributed ordinal data is provided together with critical comparisons of the control statistics, for memory-less Shewhart-type and for memory-utilizing exponentially weighted moving average (EWMA) and cumulative-sum types of control charts. New results and proposals are also provided for process monitoring. Using some real-world quality scenarios from the literature, a simulation study for performance comparisons is conducted, covering sixteen different types of control chart. It is shown that demerit-type charts used in combination with EWMA smoothing generally perform better than the other charts, which may rely on quite sophisticated derivations. A real-world data example for monitoring flashes in electric toothbrush manufacturing is discussed to illustrate the application and interpretation of the control charts in the study. [ABSTRACT FROM AUTHOR]

Existing integer-valued generalised autoregressive conditional heteroskedasticity (INGARCH) models for spatio-temporal counts do not allow for negative parameter and autocorrelation values. Using approximately linear INGARCH models, the unified and flexible spatio-temporal (B)INGARCH framework for modelling unbounded (bounded) counts is proposed. These models combine negative dependencies with kinds of a long memory. They are easily adapted to special marginal features or cross-dependencies: When modelling precipitation data (counts of rainy hours), we account for zero-inflation, while for cloud-coverage data (counts of okta), we deal with missing data and additional cross-correlation. A copula related to the spatial error model shows an appealing performance. [ABSTRACT FROM AUTHOR]