Interval versions of statistical techniques with applications to environmental analysis, bioinformatics, and privacy in statistical databases

Abstract

In many areas of science and engineering, it is desirable to estimate statistical characteristics (mean, variance, covariance, etc.) under interval uncertainty. For example, we may want to use the measured values x ( t ) of a pollution level in a lake at different moments of time to estimate the average pollution level; however, we do not know the exact values x ( t ) —e.g., if one of the measurement results is 0, this simply means that the actual (unknown) value of x ( t ) can be anywhere between 0 and the detection limit (DL). We must, therefore, modify the existing statistical algorithms to process such interval data. Such a modification is also necessary to process data from statistical databases, where, in order to maintain privacy, we only keep interval ranges instead of the actual numeric data (e.g., a salary range instead of the actual salary). Most resulting computational problems are NP-hard—which means, crudely speaking, that in general, no computationally efficient algorithm can solve all particular cases of the corresponding problem. In this paper, we overview practical situations in which computationally efficient algorithms exist: e.g., situations when measurements are very accurate, or when all the measurements are done with one (or few) instruments. As a case study, we consider a practical problem from bioinformatics: to discover the genetic difference between the cancer cells and the healthy cells, we must process the measurements results and find the concentrations c and h of a given gene in cancer and in healthy cells. This is a particular case of a general situation in which, to estimate states or parameters which are not directly accessible by measurements, we must solve a system of equations in which coefficients are only known with interval uncertainty. We show that in general, this problem is NP-hard, and we describe new efficient algorithms for solving this problem in practically important situations.