Extrapolation when very little is known about the source

Abstract

Prompted by the inadequacies of the now traditional characterization of chance and uncertainty through the Kolmogorov axioms for probability and the relative frequency interpretation of probability, we propose and examine a nonstatistical approach to extrapolation. The basic problem is the association of a real number y to a sequence of real numbers x in such a manner that the pair ( x , y ) conforms with a set of data sequences D = ( x i , y i ), {iti} = 1, M , our prior knowledge of the data source, and our objectives. Our aim is to so define the activity of extrapolation that we can derive extrapolations with only minimal assumptions about the data source. While we are free to define the human activity of extrapolation to suit ourselves, the data source functions independently of our wishful or metaphysical thinking. The basic principle we adhere to is that the extrapolation of x is a function of only those y i for which x is similar to or close to x i ; extrapolate the output of a system by examination of the outputs of similar systems. This vague sentiment is clarified and formalized through ten axioms and leads to an optimal extrapolation function π *( x ; D ). The performance of π * is then studied, both for very large and very small sample sizes ( M ), when the sequences ( x , y ), ( x i , y i ) are, in fact, independent and identically distributed random vectors.