Granular computing: Granular classifiers and missing values

Granular Computing is a paradigm destined to study how to compute with granules of knowledge that are collective objects formed from individual objects by means of a similarity measure. The idea of granulation was put forth by Lotfi Zadeh: granulation is inculcated in fuzzy set theory by the very definition of a fuzzy set and inverse values of fuzzy membership functions are elementary forms of granules. Similarly, rough sets admit granules defined naturally as classes of indiscernibitity relations; the search for more flexible granules has led to granules based on blocks (Grzymala-Busse), templates (H.S.Nguyen), rough inclusions (Polkowski, Skowron), and tolerance or similarity relations, and more generally, binary relations (T.Y. Lin, Y. Y. Yao). Granulation is an essential ingredient of humane thinking and it is playing a vital role in cognitive processes which are studied in Cognitive Informatics as emulations by computing machines of real cognitive processes in humane thinking. Rough inclusions establish a form of similarity relations that are reflexive but not necessarily symmetric; in applications presented in this work, we restrict ourselves to symmetric rough inclusions based on the set DIS(u, v) = {a E A : a(u) :7 a(v)} of attributes discerning between given objects u,v without any additional parameters. Our rough inclusions are induced in their basic forms in a unified framework of continuous t-norms; in this work we apply the rough inclusion mu L induced from the Lukasiewicz t-norm L(x,y) max{0,x+y-1} by means of the formula g(vertical bar DIS(u,v)vertical bar/vertical bar A vertical bar) = vertical bar IND(u,v)vertical bar/vertical bar A vertical bar, where g is the function that occurs in the functional representation of L and IND(u, v) = U x U \ DIS(u, v). Granules of knowledge induced by rough inclusions are formed as neighborhoods of given radii of objects by means of the class operator of mereology (see below). L.Polkowski in his feature talks at conferences 2005, 2006 IEEE GrC, put forth the hypothesis that similarity of objects in a granule should lead to closeness of sufficiently many attribute values on objects in the granule and thus averaging in a sense values of attributes on objects in a granule should lead to a new data set, the granular one, which should preserve information encoded in the original data set to a satisfactory degree. This hypothesis is borne out in this work with tests on real data sets. We also address the problem of missing values in data sets; this problem has been addressed within rough set theory by many authors, e.g., Grzymala-Busse, Kryszkiewicz, Rybinski. We propose a novel approach to this problem: an object with missing values is absorbed in a granule and takes part in determining a granular object; then, at classification stage, objects with missing values are matched against closest granular objects. We present details of this approach along with tests on real data sets. This paper is a companion to [22] where theoretical principles of granule formation are emphasized.