Axiomatization of an exponential similarity function

An individual is asked to assess a real-valued variable y based on certain characteristics x = (x(1),..,x(m)), and on a database consisting of n observations of (x1,...,x(m),y). A possible approach to combine past observations of x and y with the current values of x to generate an assessment of y is similarity-weighted averaging. It suggests that the predicted value of y,y(n+1)(s), be the weighted average of all previously observed values y(i), where the weight,n of y(i) is the similarity between the vector x(n+1)(1),...,x(n+1)(m), associated with y(n+1), and the previously observed vector, x(i)(1),..., x(i)(m). This paper axiornatizes, in terms of the prediction y(n+1), a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared. (c) 2007 Elsevier B.V. All rights reserved.