Environment and Planning B: Urban Analytics and City Science

The idea of a surprise is developed from that of a p-event. It is compared with probabilistic ideas of expected and unexpected, and an algebra of surprises is shown to contain elementary probability algebra as a trivial case. Some indication is given of how the ideas might be used to deal with problems of ‘low risk with disastrous consequences’—such as the dangers associated with atomic power stations.

Structures of graph theory are compared with those of Q-analysis and there are many similarities. The graph and simplicial complex defined by a relation are equivalent in terms of the information they represent, so that the choice between graph theory and Q-analysis depends on which gives the most natural and complete description of a system. The higher dimensional graphs are shown to be simplicial families or complexes. Although network theory is very successful in those physical science applications for which it was developed, it is argued that Q-analysis gives a better description of human network systems as patterns of traffic on a backcloth of simplicial complexes. The q-nearness graph represents the q-nearness of pairs of simplices for a given q-value. It is concluded that known results from graph theory could be applied to the q-nearness graph to assist in the investigation of q-connectivity, to introduce the notion of connection defined by graph cuts, and to assist in computation. The application of the q-nearness graph to q-transmission and shomotopy is investigated.

The plans of a small sample of houses built in the bungalow style have previously been characterized through a parametric shape grammar which provides a constructive description for each plan, that is, a set of operations by which the plan can be generated from some initial shape. The relation between plans and their constructive descriptions is investigated by means of Q-analysis. The result is a classification of the given plans which appears superior to intuitive classifications. The method employed is flexible and general enough to be applied to larger and more diverse samples of plans. Parametric shape grammars appear as a promising new area of application for Q-analysis.

Relations between highly complex human and physical systems are of increasing concern today. Many of these are exemplified by problems associated with the failure of electrical power transmission systems which are not being solved within the current paradigm. Traditional mathematical forms are unable to express the hierarchical relations characterizing the physical and human systems. It is suggested that Q-analysis provides a natural and unconstrained language for describing both the technical and the management structures, and that many traditional problems are easily translated into more appropriate statements rooted in real data sets.

The Q-analysis algorithm is sometimes used as a clustering procedure, sometimes inappropriately. The data for clustering a set of elements in terms of a set of descriptors are assumed to be weighted in a scale. This paper shows the scale requires no structure beyond an ordinal order relation to allow a refinement of the q-connected component partitions by Q-discrimination analysis. This requires a clear understanding of the meaning of the scale values when data are being collected, and gives a precise meaning to the slicing procedure of Q-analysis. The relevant definitions of Q-analysis are presented through examples and the paper is mathematically self-contained. It concludes with two illustrative examples and discusses the questions they raise. Q-discrimination analysis is presented as part of the methodology of Q-analysis, and to the extent that clustering is essentially concerned with set definition its application represents the beginning of a scientific enquiry rather than an end result.

Phillip's repertory grid data on children's perception of designed objects had been collected during an experiment devised within Kelly's personal construct theory. Some of the data returned by the children cannot be processed by the techniques usually applied to repertory grids. The slicing method of Q-discrimination analysis, a part of Q-analysis, allows these ‘illegal’ data to be processed without distortion of the children's replies. Although the Q-analysis shows the data are incomplete, it allows some tentative structural observations to be made. It was found that the children's recorded perceptions varied considerably, that some of their constructs were more logical or useful than others, but that there was considerable consensus over what they liked. The teachers' judgements on good or bad design were not identical, but surprisingly the group new to design lessons estimated their teachers' judgements best. The other group with two years previous design lessons used simpler constructs and these caused them less problems. Given a stimulating environment suited to developing new perspectives and encouraging creativity, the procedural aspect of design is argued to be best served by guiding children towards using more precise and widely applicable constructs.

A framework for generating social networks is first presented and then illustrated by means of the attitudes towards urban systems analysis elicited from a group of experts in this field. The framework is based upon the notion of constructing networks from relations on two different sets of elements, and from this approach emerges the idea of primal and dual social networks. Such networks have close parallels in the structural analysis of social relations known as Q-analysis developed by Atkin (1974), and thus it is suggested that this analysis be used to implement the formal theory in empirical terms. The data set used was collected at a meeting of urban systems analysts held in Oxford in 1980, and this consists of the attitudes of a set of sixteen analysts towards the importance of various factors and fields affecting the development of this area. From the data, it is possible to derive a variety of networks: primal networks relating actors to one another, and duals relating factors and fields. Q-analysis is used to examine the resulting structures and this reveals that the well-known hypothesis of strong consensus in such expert groups is borne out. There is little structural differentiation between actors, but considerable differentiation among factors and fields. In conclusion, certain directions for future research are established.