In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats. Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. We call such data non-standard data. To define depth functions for non-standard data, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing these depth functions, we give a systematic basis of depth functions for non-standard using formal concept analysis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data. Thereby, we increase the number of spaces in which centrality can be discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.

翻译：本文提出了一种针对非标准统计数据类型的数据深度函数概念。数据深度函数在赋范向量空间中已得到深入研究，但针对无法预设特定数据结构的数据的深度函数讨论尚属空白。我们将此类数据称为非标准数据。为定义非标准数据的深度函数，我们通过形式概念分析实现数据统一表征。除引入这些深度函数外，我们通过引入结构性质，为基于形式概念分析的非标准数据深度函数建立了系统基础。进一步地，我们将广义Tukey深度嵌入到我们的数据深度概念中，并利用所引入的结构性质对其进行分析。因此，本文为非标准数据的中心性和异常性提供了数学形式化定义，从而拓展了可用于讨论中心性的空间范畴。特别地，这为定义非标准数据的其他深度函数及统计推断方法奠定了基础。