Dissimilarity cumulation theory and subjective metrics

摘要

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a not equal double right arrow Dab>0; (2) Daa = 0; (3) If Da(n)a'(n) -> 0 and Db(n)b(n)(') -> 0, then Da(n)(')b(n)(') - Da(n)b(n) -> 0; and (4) for any sequence {a(n)X(n)b}n epsilon N where X-n is a chain of stimuli, Da(n)X(n)b(n) -> 0 Da(n)b(n) -> 0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb + DbYa across all possible chains X and Y inserted between a and b. (C) 2007 Elsevier Inc. All rights reserved.