Quasilinear Structures in Stochastic Arithmetic and their Application

Abstract

Stochastic arithmetic has been developed as a model for computing with imprecise numbers. In this model, numbers are representedby independent Gaussian variables with known mean value and standarddeviation and are called stochastic numbers. The algebraic properties of stochastic numbers have already been studied byseveral authors. Anyhow, in most life problems the variables are not independentand a direct application of the model to estimate the standard deviation on the result of a numerical computation may lead to some overestimation of the correct value. In this work “quasilinear” algebraic structures based on standard stochastic arithmeticare studied and, from pure abstract algebraic considerations, new arithmetic operationscalled “inner stochastic addition and subtraction” are introduced. They appear to be stochastic analogues to the inner interval addition and subtractionused in interval arithmetic. The algebraic properties of these operations and the involved algebraic structures are then studied. Finally, the connection of these inner operations to the correlation coefficient ofthe variables is developed and it is shown that they allow the computation with non-independent variables. The corresponding methodology for the practicalapplication of the new structures in relation to problems analogous to “dependency problems”in interval arithmetic is given and some numerical experiments showing the interest ofthese new operations are presented.ACM Computing Classification System (1998): D.2.4, G.3, G.4.