Some aspects of qualitative data analysis

Abstract

Let n ⩾ 1 be an arbitrary natural number and let X, X 1,\\h., X n be arbitrary (qualitative) random variables. Then it is the main aim of this paper to study and to characterize real-valued functions S( X) or S( X, X 1,\\h., X n ) which are comparable with the degree up to which the variables X, X 1,\\h., X n are known. Functions S( X) appear for example as expectations or as goodness criteria for a homogeneity or heterogeneity function X in cluster analysis, while functions S( X, X 1) may appear as similarity or dissimilarity coefficients (correlation coefficients) or as stress measures in multi-dimensional scaling (MDS) if X and X 1 represent similarity or dissimilarity coefficients. In addition, if X 1,\\h., X n are observed random variables, then S( X, X 1,\\h., X n ) may be an objective function which has to be optimized in order to estimate the unknown variable X.