On the lattice of conceptual measurements

Scaling Dimension

“…As shown by Proposition 1 there is a close tie between contextual views and canonical representation of scale-measures as proposed in Proposition 10 of Hanika and Hirth [7]. Said representations provide for a scale-measure σ of K into S an equivalent scale-measure based on the identity map of K into a context of the form (G, A, ∈) where A ⊆ Ext(K).…”

“…An investigation on how the scaling dimension relates to other measures of dimension within the realm of FCA [9,15] is therefore deemed future work. Due to novel insights into the computational tractability of recognizing scalemeasures [7] (that is in preparation and will be made public later this year) we have little hope that the scaling dimension and interordinal scaling dimension can be decided in polynomial time. Despite that, efficient algorithms for real-world data that compute the scaling dimension and its specific versions, i.e., ordinal, interordinal, nominal, etc, may be developed.…”

“…In particular, we introduced the notion of pre-scaling to formalize background knowledge on attribute domains, e.g., underlying order relations, that can be used and extended to scales in the process of conceptual scaling. To deal with the complexity of scalings selection or logical compression methods have been proposed to reflect parts of the conceptual structure [7]. Furthermore, we introduced the notions of conceptual and contextual views to characterize these methods and provided a first formal definition.…”

“…To prove properties of the scaling dimension, we need results about scale measures which were published elsewhere [3,7]. The following lemma follows from Propositions 120 and 122 from Ganter and Wille [3].…”

“…Another important aspect is the complexity of the conceptual structure created by scaling. Several authors have suggested to restrict to important attribute combinations [5,7,12]. We formalize this, speaking of conceptual views of the data.…”