Statistical Methods and Models for Interdependent Categorical, particularly Ordinal Data

有各种统计方法可用于分析和建模高维的、相互依赖的变量，例如图形模型或主成分分析。然而，这些方法通常需要连续或度量的数据。对于高维分类数据，特别是有序数据，相应的方法相当有限，尽管这类数据经常出现在各种应用中。因此，该项目的目标是通过开发适当的方法，如正则化图形模型和序数变量的主成分分析，填补统计方法上的这一空白。除了规模级别-名义与顺序-我们还将区分彼此排序的变量，例如随着时间的推移，以及没有特定结构的变量，这些变量意味着某种特定的关联模式。对于第一类数据的建模，我们还将借用和扩展功能数据分析的方法，例如，引入“分类功能数据的最佳比例”或“离散功能数据的图形模型”。一方面，将要开发的方法将受到真实世界数据问题的激励，并针对真实世界的数据问题而量身定做，例如感官质量控制。另一方面，所提议的方法一般不应局限于特定的适用领域，而应尽可能广泛地适用。所有数据分析都将与相应的合作者密切合作。

There are various statistical methods available for analyzing and modeling high-dimensional, interdependent variables, such as graphical models or principal component analysis. Those methods, however, usually require continuous or metrically scaled data. Corresponding methods for high-dimensional categorical, particularly ordinal data are rather limited, although those kind of data is frequently found in various applications. Therefore, the goal of the project is to fill this gap in statistical methodology by developing appropriate methods, such as regularized graphical models and principal component analysis for ordinal variables. Besides scale level – nominal vs. ordinal – we will distinguish between variables that are ordered among each other, e.g. over time, and variables without a specific structure that implies some specific association pattern. For modeling the first type of data, we will also borrow and extend methods from functional data analysis, leading, e.g., to “optimal scaling for categorical functional data” or “graphical models for discrete functional data”. On the one hand, the methods to be developed will be motivated by and tailored to real world data problems, such as sensory quality control. On the other hand, the methods proposed shall, in general, not be restricted to a specific field of application, but be applicable as broadly as possible. All data analyses will be done in close cooperation with the corresponding collaborators.