Dissimilarity cumulation theory in smoothly connected spaces

This is the third paper in the series introducing the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling. The previously developed dissimilarity-based theory of path length is used to construct the notion of a smooth path, defined by the property that the ratio of the dissimilarity between its points to the length of the subtended fragment of the path tends to unity as the points get closer to each other. We consider a class of stimulus spaces in which for every path there is a series of piecewise smooth paths converging to it pointwise and in length; and a subclass of such spaces where any two sufficiently close points can be connected by a smooth “geodesic in the small”. These notions are used to construct a broadly understood Finslerian geometry of stimulus spaces representable by regions of Euclidean n-spaces. With an additional assumption of comeasurability in the small between the canonical psychometric increments of the first and second kind, this establishes a link between Universal Fechnerian Scaling and Multidimensional Fechnerian Scaling in Euclidean n-spaces. The latter was a starting point for our theoretical program generalizing Fechner’s idea that sensation magnitudes can be computed by integration of a local discriminability measure.