Sentence-frame Data: Appendices to Chapter 5 of Class and Hierarchy

Abstract

Parametric mapping is a type of scaling that treats both the data and the solution as being metric. Thus it differs from quasi-non-metric scaling methods such as MINISSA, where only rank order comparisons are made among the data elements. There are other metric scaling methods. For example, INDSCAL is most frequently used in its metric version. Like other metric scaling methods, parametric mapping (PARAMAP) takes a set of distances between stimulus points as data, and seeks to find a corresponding set of distances in a ‘solution space’ of low dimensionality. Most other scaling methods try to find a solution space such that the inter-point distances in the data are as close as possible to a linear or monotonic function of the corresponding distances, or dissimilarities in the solution. PARAMAP is different. It tries to find a solution space such that the function by which distances in the ‘data space’ can be predicted from corresponding distances in the solution space is as continuous, or ‘smooth’ as possible. The function relating distances in the data to distances in the solution need not be monotonic, so long as it is ‘smooth’. As with other scaling methods, an iterative ‘steepest descent’ algorithm finds stimulus coordinates in a solution space of given dimensionality. These stimulus coordinates serve to generate Euclidean distances in the solution space, and these are found so as to minimise an index of departure from continuity or ‘smoothness’, of the function relating distances between pairs of points in the data space to corresponding distances in the solution space.