Exploring Scale-Measures of Data Sets

An inevitable step of any data-based knowledge discovery process is measurement [24] and the associated (explicit or implicit) scaling of the data [27]. The latter is particularly constrained by the underlying mathematical formulation of the data representation, e.g., real-valued vector spaces or weighted graphs, the requirements of the data procedures, e.g., the presence of a distance function, and, more recently, the need for human understanding of the results. Considering the scaling of data as part of the analysis itself, in particular formalizing it and thus making it controllable, is a salient feature of formal concept analysis (FCA) [7]. This field of research has spawned a variety of specialized scaling methods, such as logical scaling [25], and in the form of scale-measures links the scaling process with the study of continuous mappings between closure systems. Recent results by the authors [13] revealed that the set of all scale-measures for a given data set constitutes a lattice. Furthermore, it was shown that any scale-measure can be expressed in simple propositional terms using disjunction, conjunction and negation. Among other things, the previous results allow a computational transition between different scale-measures, which we may call scalemeasure navigation, as well as their interpretability by humans.