On the Lattice of Conceptual Measurements

TL;DR

This paper introduces a new lattice-based framework for data set scaling using formal concept analysis, providing canonical forms, lattice structures, and logical representations to enhance data analysis methods.

Contribution

It develops a novel lattice-theoretic approach to data set scaling with formal concept analysis, including canonical and logical representations, and proves key structural properties.

Findings

Scale-measures form a lattice ordered by closure systems

Lattice of scale-measures is isomorphic to sub-closure systems

Provides propositional logic representation of scale-measures

Abstract

We present a novel approach for data set scaling based on scale-measures from formal concept analysis, i.e., continuous maps between closure systems, and derive a canonical representation. Moreover, we prove said scale-measures are lattice ordered with respect to the closure systems. This enables exploring the set of scale-measures through by the use of meet and join operations. Furthermore we show that the lattice of scale-measures is isomorphic to the lattice of sub-closure systems that arises from the original data. Finally, we provide another representation of scale-measures using propositional logic in terms of data set features. Our theoretical findings are discussed by means of examples.