Data depth functions for non-standard data by use of formal concept analysis

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. In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats and therefore we do not have one specific data structure. We call such data in general non-standard data. To achieve this, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing depth functions for non-standard data using formal concept analysis, we give a systematic basis 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 and therefore increases the spaces centrality is currently discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.