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Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity
 Reliable Computing
, 2006
"... 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, ..."
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Cited by 41 (40 self)
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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.
Population Variance under Interval Uncertainty: A
 New Algorithm, Reliable Computing
, 2006
"... 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 ..."
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Cited by 20 (17 self)
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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
Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty
, 2007
"... This report addresses the characterization of measurements that include epistemic uncertainties in the form of intervals. It reviews the application of basic descriptive statistics to data sets which contain intervals rather than exclusively point estimates. It describes algorithms to compute variou ..."
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Cited by 20 (14 self)
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This report addresses the characterization of measurements that include epistemic uncertainties in the form of intervals. It reviews the application of basic descriptive statistics to data sets which contain intervals rather than exclusively point estimates. It describes algorithms to compute various means, the median and other percentiles, variance, interquartile range, moments, confidence limits, and other important statistics and summarizes the computability of these statistics as a function of sample size and characteristics of the intervals in the data (degree of overlap, size and regularity of widths, etc.). It also reviews the prospects for analyzing such data sets with the methods of inferential statistics such as outlier detection and regressions. The report explores the tradeoff between measurement precision and sample size in statistical results that are sensitive to both. It also argues that an approach based on interval statistics could be a reasonable alternative to current standard methods for evaluating, expressing and propagating measurement uncertainties.
Using expert knowledge in solving the seismic inverse problem
 In: Proceedings of the 24nd International Conference of the North American Fuzzy Information Processing Society NAFIPS’2005, Ann Arbor
, 2005
"... For many practical applications, it it important to solve the seismic inverse problem, i.e., to measure seismic travel times and reconstruct velocities at different depths from this data. The existing algorithms for solving the seismic inverse problem often take too long and/or produce unphysical r ..."
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Cited by 12 (11 self)
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For many practical applications, it it important to solve the seismic inverse problem, i.e., to measure seismic travel times and reconstruct velocities at different depths from this data. The existing algorithms for solving the seismic inverse problem often take too long and/or produce unphysical results – because they do not take into account the knowledge of geophysicist experts. In this paper, we analyze how expert knowledge can be used in solving the seismic inverse problem. Key words: seismic inverse problem, expert knowledge
Fast algorithm for computing the upper endpoint of sample variance for interval data: case of sufficiently accurate measurements
 Reliable Computing
, 2006
"... When we have n results x1,..., xn of repeated measurement of the same quantity, the 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 quantiti ..."
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Cited by 12 (7 self)
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When we have n results x1,..., xn of repeated measurement of the same quantity, the 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 NPhard. For the case when the measurements are sufficiently accurate (in some precise sense), it is known that we can compute the interval V in quadratic time O(n 2). 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
New Algorithms for Statistical Analysis of Interval Data
 Proceedings of the Workshop on StateoftheArt in Scientific Computing PARA’04
, 2004
"... It is known that in general, statistical analysis of interval data is an NPhard problem: even computing the variance of interval data is, in general, NPhard. Until now, only one case was known for which a feasible algorithm can compute the variance of interval data: the case when all the measureme ..."
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Cited by 11 (8 self)
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It is known that in general, statistical analysis of interval data is an NPhard problem: even computing the variance of interval data is, in general, NPhard. 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
Computing BestPossible Bounds for the Distribution of a Sum of Several Variables is NPhard
 International Journal of Approximate Reasoning
, 1997
"... In many reallife 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. Algori ..."
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Cited by 10 (4 self)
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In many reallife 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 bestpossible 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 bestpossible bounds for arbitrary n is an NPhard (computationally intractable) problem.
Fast Algorithms for Computing Statistics under Interval Uncertainty, with Applications to Computer Science and to Electrical and Computer Engineering
, 2007
"... 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 ..."
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Cited by 6 (3 self)
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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∑
Foundations of Statistical Processing of Setvalued Data: Towards Efficient Algorithms
 Proceedings of the Fifth International Conference on Intelligent Technologies InTech’04
, 2004
"... Abstract — Due to measurement uncertainty, often, instead of the actual values xi of the measured quantities, we only know the intervals xi = [�xi − ∆i, �xi + ∆i], where �xi is the measured value and ∆i is the upper bound on the measurement error (provided, e.g., by the manufacturer of the measuring ..."
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Cited by 5 (4 self)
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Abstract — Due to measurement uncertainty, often, instead of the actual values xi of the measured quantities, we only know the intervals xi = [�xi − ∆i, �xi + ∆i], where �xi is the measured value and ∆i is the upper bound on the measurement error (provided, e.g., by the manufacturer of the measuring instrument). These intervals can be viewed as random intervals, i.e., as samples from the intervalvalued random variable. In such situations, instead of the exact value of a sample statistic such as covariance Cx,y, we can only have an interval Cx,y of possible values of this statistic. In this paper, we extend the foundations of traditional statistics to statistics of such setvalued data, and describe how this foundation can lead to efficient algorithms for computing the corresponding setvalued statistics. I. STATISTICAL ESTIMATION:
Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 4 (1 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.