When we have n results x1,..., xn of repeated measurement of the same quantity, traditional statistical approach usually starts with computing their sample average E and their sample variance V. Often, due to the inevitable measurement uncertainty, we do not know the exact values of the quantities, we only know the intervals xi of possible values of xi. In such situations, for different possible values xi ∈ xi, we get different values of the variance. We must therefore find the range V of possible values of V. It is known that in general, this problem is NP-hard. For the case when the measurements are sufficiently accurate, so that for some integer c, no sub-collection of> c “narrowed ” intervals of xi has a common intersection, it is known that we can compute the interval V in quadratic time O(n 2). For large amount of data, i.e., for large n, it is desirable to speed up the computations. In this paper, we describe a new algorithm for computing V that requires time O(n · log(n)) (which is much faster than O(n 2)). 1

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 n · n∑

Abstract. In many engineering applications, we have to combine probabilistic and interval errors. 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 know the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data. Such modification are described in this paper.

Why indirect measurements? In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure y indirectly. Specifically, we find some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem). Then, to estimate y, we first measure the values of the quantities x1,..., xn, and then we use the results �x1,..., �xn of these measurements to to compute an estimate �y for y as �y = f(�x1,..., �xn): �x1 �x2

Abstract. Statistical units described by interval-valued variables represent a special case of Symbolic Objects, where all descriptors are quantitative variables. In this context, the paper presents two different metrics in R p for interval-valued data that are based on the definition of the Hausdorff distance in R. Hausdorff distance in R p (for any p≥1) is a L ∞ norm between pairs of closed sets. However, when p> 1 the problem complexity leads towards the definition of L2 norms approximating as well as possible the Hausdorff distance. Given a set of n units described by p interval-valued variables, we compute and represent the distances over factorial planes that are defined by factorial analyses that are consistent with the two distance measure definitions.

Abstract—Since the 1960s, many algorithms have been designed to deal with interval uncertainty. In the last decade, there has been a lot of progress in extending these algorithms to the case when we have a combination of interval, probabilistic, and fuzzy uncertainty. We provide an overview of related algorithms, results, and remaining open problems. I. MAIN PROBLEM Why indirect measurements? In many real-life situations, we are interested in the value of a physical quantity y that is dif cult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure y indirectly. Speci cally, we nd some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem). Then, to estimate y, we rst measure the values of the quantities x1,..., xn, and then we use the results ˜x1,..., ˜xn of these measurements to to compute an estimate ˜y for y as ˜y = f(˜x1,..., ˜xn). ˜x1 ˜x2

Dean of the Graduate SchoolThis dissertation is dedicated to my deeply loved grandfather who passed away in 2003. His wish to pursue a graduate study could not be fulfilled due to World War II. And to my parents, my wife Qianyin and my son Kevin for their great love. FAST ALGORITHMS FOR COMPUTING STATISTICS

Int. J. Automation and Control, Vol. x, No. x, xxxx 1 Application-motivated combinations of fuzzy, interval, and probability approaches, with application to geoinformatics, bioinformatics,