Logical basis of dimensionality

Abstract

The isolation of dimensions from a data matrix has been traditionally formulated in terms of an algebraic or geometric model. Order analysis was developed as a method of multidimensional analysis and scaling based on the theory of Boolean algebra. The order analytic algorithm utilizes functions of the propositional calculus in lieu of eigenvalues and eigenvectors of the general linear model. Also, the graphic presentation of latent space in coordinates of the Euclidian space is paralleled in ordering-theoretic models by dendrograms of the test space. A conceptual outline of order analysis is presented, followed by an empirical comparison of factor and order analysis solutions of a sample data problem. Resulting factor and order analytic structures are evaluated in terms of meeting criteria of simple structure and correct reflection of broad cognitive categories. In addition, the relations of proximity and dominance are discussed from the perspectives of both Cartesian and Leibnitzian theories of dimensionality as pertaining to problems of multivariate analysis and scaling.