### 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 represented

by independent Gaussian variables with known mean value and standard

deviation and are called stochastic numbers.

The algebraic properties of stochastic numbers have already been studied by

several authors. Anyhow, in most life problems the variables are not independent

and 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 arithmetic

are studied and, from pure abstract algebraic considerations, new arithmetic operations

called “inner stochastic addition and subtraction” are introduced.

They appear to be stochastic analogues to the inner interval addition and subtraction

used 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 of

the variables is developed and it is shown that they allow the computation with

non-independent variables. The corresponding methodology for the practical

application of the new structures in relation to problems analogous to “dependency problems”

in interval arithmetic is given and some numerical experiments showing the interest of

these new operations are presented.

ACM Computing Classification System (1998): D.2.4, G.3, G.4.

#### Keywords

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ISSN 1314-7897 - Online

ISSN 1312-6555 - Print