Q-analysis

Abstract: At the basis of Q-analysis are the notions of well-defined sets, and of hierarchies constructed out of the power sets of such sets. It is here argued that such set-theoretic hierarchies are in general inadequate in themselves for representing the structure of empirical phenomena, because sets defined as collections of individual objects or of other sets cannot, on the whole, be made to correspond directly to specific empirical entities. This is because empirical concepts correspond to more or less complex structures, rather than collections, of other concepts, whereas a set is, by definition, an unstructured collection of elements. This point is illustrated through several examples of sets of ‘things’ that purport to represent certain higher-order ‘things’, but in fact do not. There follows an analysis of the problem that draws from mathematical philosophy and logic on the one hand, and from the theory of Boolean algebras on the other. Both these perspectives point to the same conclusion, namely, that Q-a...

Abstract: A new star-hub structure of binary relations is discussed in the context of the methodology of Q-analysis, and parallels are drawn with maximal rectangles and Galois lattice structures. Although these structures generalize those of Q-analysis, there remain problems due to the very large number of star-hub pairs generated by fairly modest data sets. It is argued that more theory is necessary, and some possibilities are discussed. It is suggested that the criteria for defining new structures will come most fruitfully from the study of the relationship between backcloth and the ways it constrains traffic. Finally, it is argued that these combinatorial structures are still not sufficient to fully describe complex systems and that for this one needs to consider polyhedra in the context of N-ary relations.

Abstract: Some of the methodological consequences of using the basic notion of set-membership to define “scientific” or “hard” data are discussed in this chapter. Chiefly because of the Russell theory of types (but not exclusively because of that), a need exists for a well-defined hierarchy of data sets. Such a hierarchy is defined in terms of cover sets (rather than by partitions) and is expressed by a set of mathematical relations between the finite data sets. The structure of the data is then identified by the simplicial complexes that represent these relations, and these will contain the static backcloth, S(N),for that data. Some simple connectivity properties of a typical complex are listed, and an illustration of the methodological technique is provided by examples taken from an earlier study of an area in the town of Southend-on-Sea and from a current regional project. The importance of the structure (q-connectivities) for the dynamics of (generalized) traffic on the backcloth, S(N), is then examined and critically compared with regression analysis.

Abstract: A novel method called Visual Q-Analysis (VQA) is proposed to analyze structures of complex systems. This method is based upon Atkin's Q-analysis where the structure of a system is represented by simplicial complex in topology and analyzed in terms of q-connectivity. Two different types of hierarchies, Q-hierarchy and F-hierarchy, are introduced and algorithms to obtain these are given. In order to draw these hierarchies in a visually understandable form SKETCH system developed by the authors are used. The Q-hierarchy visualizes a hierarchical q-connectivity structure among all the simplices and the F-hierarchy expresses a structure of face-sharing among the simplices in the complex. By inspecting their drawings we can grasp the structural information embedded in the complex. This method is applied to a structural study of technological development of future computers of Japan in terms of relationships between social needs and technological requirements (seeds). Results of the application not only show the effectiveness of VQA to support in planning technological developments but also suggest wide applicabilities of VQA to various other fields.