Fast Algorithms for Computing Statistics Under Interval and Fuzzy Uncertainty, and Their Applications

In many engineering applications, we have to combine probabilistic, interval, and fuzzy uncertainty. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. In environmental measurements, we often only measure the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data. In this paper, we provide a brief survey of algorithms for computing various statistics under interval (and fuzzy) uncertainty and of their applications, including applications to the seismic inverse problem in geosciences, to chip design in computer engineering, and to radar data processing. 1 Formulation of the Problem Computing Statistics is Important. In many engineering applications, we are interested in computing statistics. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. For each of these characteristics C, there is an expression C(x1, . . . , xn) that enables us to provide an estimate for C based on the observed values x1, . . . , xn. For example, a reasonable statistic for estimating the mean value of a probability distribution is the population average E(x1, . . . , xn) = 1 n · (x1 + . . . + xn); a reasonable statistic for estimating the variance V is the population variance V (x1, . . . , xn) = 1