Nested Intervals and Sets: Concepts, Relations to Fuzzy Sets, and Applications

In data processing, we often encounter the following problem: Suppose that we have processed the measurement results x̃1, ..., x̃n, and, from this processing, have obtained an estimate ỹ = f(x̃1, ..., x̃n) for a quantity y = f(x1, ..., xn); we know the intervals xi of possible values of xi, and we want to find the interval y of possible values of y. Interval computations are one of the main techniques for solving this problem. In some cases, for each i, in addition to the guaranteed interval xi of possible values, we have a smaller interval that an expert believes to contain xi. There may be several such nested intervals. In these cases, in addition to the guaranteed interval y, it is desirable to know the possible intervals of y that correspond to the opinions of different experts. Techniques of such nested interval computations and real-life applications of these techniques are described in this paper. ∗This work was partially carried out while Hung T. Nguyen was on sabbatical leave at the University of Southern California, Los Angeles, Spring 1995