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，相互依赖的变量，如图形模型或主成分分析。然而，这些方法通常需要连续的或度量尺度的数据。尽管高维分类数据在各种应用中经常出现，但对于高维分类数据，特别是有序数据，相应的方法相当有限。因此，该项目的目标是通过开发适当的方法，如正规化图形模型和有序变量的主成分分析，填补统计方法中的这一空白。除了标度水平-名义与有序-我们将区分变量之间的顺序，例如随着时间的推移，和变量之间没有一个特定的结构，这意味着一些特定的关联模式。对于第一种类型的数据建模，我们还将借用和扩展函数数据分析的方法，例如，到“分类函数数据的最佳缩放”或“离散函数数据的图形模型”。一方面，要开发的方法将受到真实的世界数据问题的激励，并针对这些问题量身定制，例如感官质量控制。另一方面，所提议的方法一般不应限于某一特定的应用领域，而应尽可能广泛地适用。所有数据分析将与相应的合作者密切合作进行。

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.