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 $${\tilde x_1},...,{\tilde x_n}$$, and, from this processing, have obtained an estimate $$\tilde y = f({\tilde x_1},...,{\tilde x_n})$$ for a quantity y = f(xi,…,xn); we know the intervals x_i of possible values of x_i, 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 x_i of possible values, we have a smaller interval that an expert believes to contain x_i. 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.