Ordinal Time Series: Modeling, Forecasting, and Control

 
Three-years project, funded by Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 516522977.

Project aims:

An ordinal time series is a temporal sequence of discrete-valued observations, the range of which is qualitative and consists of a finite number of ordered categories. Ordinal time series arise in many different situations in economics and related fields. They can have various forms with respect to their dependence structure or their marginal distribution. These characteristics can be worked out by using analytical tools that have been recently developed for ordinal time series. Afterwards, an adequate modeling of the ordinal time series would be necessary, which could be used as a base for the forecasting of the time series or for the statistical control of its further course. This is the starting point of the planned research project, because neither for modeling nor for forecasting or controlling ordinal time series, there has been a sufficient repertoire of tailor-made methods to date. Instead, approaches for nominal time series are mostly used (which then, however, disregard the natural ordering of the categories), or those for quantitative time series (which then, however, implicitly assume a metric structure).

The aim of the planned research project is to develop a comprehensive package of methods for the stochastic modeling, forecasting and control of ordinal time series. In this context, existing procedures are to be taken up, and the areas that are still open are to be closed by novel own contributions. The first step would be to develop a toolbox of as diverse models as possible, covering a wide range of stochastic properties. For all resulting model types, in addition to the actual model definition and the stochastic model properties, the question of model fitting (identification, estimation, validation) must always be considered. This would be followed by the sub-projects on forecasting and control, which should already incorporate the newly developed models. Concerning forecasting, adequate criteria for assessing the quality of prediction shall be derived (i.e., which account for the ordinal nature of the data) in order to then use them to empirically investigate the performance of the various forecast approaches (point, interval and PMF predictions) in detail. With regard to monitoring, control charts for serially dependent ordinal processes are to be developed and analyzed, where, in addition to sample-based charts, the focus is primarily on memory-type individuals charts, which have been completely lacking in the literature to date. For all proposed methods, the performance and applicability will investigated in detail, both through comprehensive comparative simulation studies and through the application to real-world data examples being relevant in economics.

Project duration:

April 2023 – March 2026.

Project results:

• Weiß, C.H., Schnurr, A. (2023):
  Generalized ordinal patterns in discrete-valued time series: nonparametric testing for serial dependence.
  Accepted for publication in Journal of Nonparametric Statistics (open access).
   
  Abstract: 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 generalized 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.
   
• Weiß, C.H. (2023):
  Ordinal compositional data and time series.
  Accepted for publication in Statistical Modelling (open access).
   
  Abstract: 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.
   
• Jahn, M., Weiß, C.H. (2023):
  Nonlinear GARCH-type models for ordinal time series.
  Accepted for publication in Stochastic Environmental Research and Risk Assessment (open access).
   
  Abstract: Despite their relevance in various areas of application, only few stochastic models for ordinal time series are discussed in the literature. To allow for a flexible serial dependence structure, different ordinal GARCH-type models are proposed, which can handle nonlinear dependence as well as kinds of an intensified memory. The (logistic) ordinal GARCH model accounts for the natural order among the categories by relying on the conditional cumulative distributions. As an alternative, a conditionally multinomial model is developed which uses the softmax response function. The resulting softmax GARCH model incorporates the ordinal information by considering the past (expected) categories. It is shown that this latter model is easily combined with an artificial neural network response function. This introduces great flexibility into the resulting neural softmax GARCH model, which turns out to be beneficial in three real-world time series applications (air quality levels, fear states, cloud coverage).
   
• Weiß, C.H., Swidan, O. (2024):
  Weighted Discrete ARMA Models for Categorical Time Series.
  Accepted for publication in Journal of Time Series Analysis (open access).
   
  Abstract: 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 neighbouring 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 phi-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.
   
• Weiß, C.H., Swidan, O. (2024):
  Hidden-Markov Models for Ordinal Time Series.
  Accepted for publication in AStA Advances in Statistical Analysis, 2024 (open access).
   
  Abstract: A common approach for modeling categorical time series are Hidden-Markov models (HMMs), where the actual observations are assumed to depend on hidden states in their behavior and transitions. Such categorical HMMs are even applicable to nominal data, but suffer from a large number of model parameters.
  In the ordinal case, however, the natural order among the categorical outcomes offers the potential to reduce the number of parameters while improving their interpretability at the same time. The class of ordinal HMMs proposed in this article link a latent-variable approach with categorical HMMs. They are characterized by parametric parsimony and allow the easy calculation of relevant stochastic properties, such as marginal and bivariate probabilities. These points are illustrated by numerical examples and simulation experiments, where the performance of maximum likelihood estimation is analyzed in finite samples. The developed methodology is applied to real-world data from a health application.
   
• to be continued!