Ordinal Time Series: Modeling, Forecasting and Control

有序时间序列是离散值观测的时间序列，其范围是定性的，由有限数量的有序类别组成。有序时间序列在经济学及相关领域的许多不同情况下都会出现。根据它们的依赖结构或边缘分布，它们可以有各种形式。这些特征可以通过使用最近开发的用于有序时间序列的分析工具来计算。此后，需要对有序时间序列进行适当的建模，以此作为预测时间序列或对其进一步进程进行统计控制的基础。这是计划中的研究项目的起点，因为无论是为了建模，还是为了预测或控制有序时间序列，到目前为止都有足够的量身定做的方法。取而代之的是，主要使用名义时间序列的方法(然而，这会忽略类别的自然顺序)，或者使用定量时间序列的方法(然而，这些方法隐含地假设了度量结构)。计划中的研究项目的目的是开发一套全面的方法，用于有序时间序列的随机建模、预测和控制。在这方面，将采用现有的程序，仍然开放的领域将由新的自己的贡献来结束。第一步将是开发一个尽可能多样化的模型工具箱，涵盖广泛的随机属性。对于所有由此产生的模型类型，除了实际的模型定义和随机模型属性之外，必须始终考虑模型拟合问题(识别、估计、验证)。紧随其后的是关于预测和控制的次级项目，这些项目应该已经纳入了新开发的模型。关于预测，应得出评估预测质量的适当标准(即说明数据的序数性质)，以便使用这些标准对各种预测方法(点预测、区间预测和PMF预测)的性能进行详细的经验性调查。关于监测，将开发和分析顺序过程的控制图，除了基于样本的控制图外，重点主要放在记忆型个人控制图上，这是迄今为止文献中完全缺乏的。对于所有提出的方法，将通过全面的比较模拟研究和对与经济学相关的真实世界数据实例的应用，详细研究其性能和适用性。

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.