The identification of many-dimensional relations
Robert Francis Madden, William Ross Ashby · International Journal of Systems Science · 1972
Any system, no matter how general or complex, can be described by a relationship amongst nominal variables. In this paper we investigate the problem of identifying snob a relationship, and the properties of the system corresponding to it, when the data descriptive of the system have been gathered from diverse and uncorrelated sources. We represent this problem theoretically as one of reconstructing and identifying an n-dimensional relation from its projections, and discuss the practical implications of this representation. We show that such reconstruction and identification is possible only if certain basic features of the corresponding system are constrained in ft very definite manner. For example, if the relation is to be identifiable, i.e. if the sot of relevant variables is to be ’ perfectly constrained ’, then the system must contain no more than a certain number of functionally determined variables, and no more than a certain number of independent variables. Furthermore, as the number of variables becomes large then, in theory at least, only a vanishingly small fraction of relations are identifiable. The treatment throughout is confined to nominal variables, and is intended to contribute to the constraint analysis of large and complex systems.