Abstract. In many engineering applications, we have to combine probabilistic and interval 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 survey of algorithms for computing various statistics under interval uncertainty and their computational complexity. The survey includes both known and new algorithms.

Abstract. Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many real-life problems go beyond this paradigm. In some cases, we do not have an algorithm ¦, we only know some relation (constraints) between ©� � and. In other cases, in addition to knowing the intervals, we may know some relations between; we may have some information about the probabilities of different values of © � , and we may know the exact values of some of the inputs (e.g., we may know that © £ ���¨�� �). In this paper, we describe the approaches for solving these real-life problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers. 1

In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance V = 1 n · n∑ (xi − E) i=1

In many application areas, it is important to detect outliers. Traditional engineering approach to outlier detection is that we start with some "normal" values x1 ; : : : ; xn , compute the sample average E, the sample standard variation oe, and then mark a value x as an outlier if x is outside the k0-sigma interval [E \Gamma k0 \Delta oe; E+k0 \Delta oe] (for some pre-selected parameter k0 ). In real life, we often have only interval ranges [x i ; x i ] for the normal values x1 ; : : : ; xn . In this case, we only have intervals of possible values for the bounds E \Gamma k0 \Delta oe and E+k0 \Delta oe. We can therefore identify outliers as values that are outside all k0-sigma intervals.

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

Abstract. It is known that in general, statistical analysis of interval data is an NP-hard problem: even computing the variance of interval data is, in general, NP-hard. Until now, only one case was known for which a feasible algorithm can compute the variance of interval data: the case when all the measurements are accurate enough – so that even after the measurement, we can distinguish between different measured values �xi. In this paper, we describe several new cases in which feasible algorithms are possible – e.g., the case when all the measurements are done by using the same (not necessarily very accurate) measurement instrument – or at least a limited number of different measuring instruments. 1

We show how quantum computing can speed up computations related to processing probabilistic, interval, and fuzzy uncertainty.

Abstract. In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn). Measurements are never 100 % accurate; hence, the measured values �xi are different from xi, and the resulting estimate �y = f(�x1,..., �xn) is different from the desired value y = f(x1,..., xn). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different def measurement error ∆xi = �xi − xi. In many practical situations, we only know the upper bound ∆i for this error; hence, after the measurement, the only information that we have about xi is that it belongs def to the interval xi = [�xi − ∆i, �xi + ∆i]. In this case, it is important to find the range y of all possible values of y = f(x1,..., xn) when xi ∈ xi. We start with a brief overview of the corresponding interval computation problems. We then discuss what to do when, in addition to the upper bounds ∆i, we have some partial information about the probabilities of different values of ∆xi.

In this paper, we start research into using intervals to bound the impact of bounded measurement errors on the computation of bounds on finite population parameters (“descriptive statistics”). Specifically, we provide a feasible (quadratic time) algorithm for computing the lower bound σ 2 on the finite population variance function of interval data. We prove that the problem of computing the upper bound σ 2 is, in general, NP-hard. We provide a feasible algorithm that computes σ 2 under reasonable easily verifiable conditions, and provide preliminary results on computing other functions of finite populations. 1

In many real-life situations, we know the probability distribution of two random variables x1 and x2, but we have no information about the correlation between x1 and x2; what are the possible probability distributions for the sum x1+x2? This question was originally raised by A. N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x1 + x2; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n> 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x1 +... + xn? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not bestpossible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem.