Maximum consistency method for data fitting under interval uncertainty

For the linear regression model , we consider the problem of data fitting under interval uncertainty. Let an interval × -matrix = ( ) and an interval -vector = ( ) represent the input data and output responses of the model respectively, such that , , ... , , in the -th experiment, = 1, 2, ... , . It is necessary to find the coefficients that best fit the above linear relation for the data given. A family of values of the parameters is called consistent with the interval data ( , , , ), , = 1, 2, ... , , if, for every index i, there exist such point representatives , , ... , , bi bi that ai1x1 + ai2 x2 + ... + ain xn = bi . The set of all the parameter values consistent with the data given form a parameter uncertainty set. As an estimate of the parameters, it makes sense to take a point from the parameter uncertainty set providing that it is nonempty. Otherwise, if the parameter uncertainty set is empty, then the estimate should be a point where maximal “consistency” (in a prescribed sense) with the data is achieved. The parameter uncertainty set is nothing but the solution set ( , ) to the interval system of linear equations x = defined in interval analysis: ( , ) = { x | x = for some from and from }. For the above data fitting problem, we propose, as the consistency measure, the values of the recognizing functional of the solution set ( , ), which is defined as