Results 1  10
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17
Computing Variance for Interval Data is NPHard
, 2002
"... When we have only interval ranges [x i ; x i ] of sample values x 1 ; : : : ; xn , what is the interval [V ; V ] of possible values for the variance V of these values? We prove that the problem of computing the upper bound V is NPhard. We provide a feasible (quadratic time) algorithm for computi ..."
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Cited by 61 (44 self)
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When we have only interval ranges [x i ; x i ] of sample values x 1 ; : : : ; xn , what is the interval [V ; V ] of possible values for the variance V of these values? We prove that the problem of computing the upper bound V is NPhard. We provide a feasible (quadratic time) algorithm for computing the lower bound V on the variance of interval data. We also provide a feasible algorithm that computes V under reasonable easily verifiable conditions.
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.
Outlier Detection Under Interval Uncertainty: Algorithmic Solvability and Computational Complexity
, 2003
"... 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 ..."
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Cited by 19 (12 self)
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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 k0sigma interval [E \Gamma k0 \Delta oe; E+k0 \Delta oe] (for some preselected 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 k0sigma intervals.
Exact bounds on finite populations of interval data
 Reliable Computing
, 2001
"... 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 fin ..."
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Cited by 14 (10 self)
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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, NPhard. 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
Exact Bounds on Sample Variance of Interval Data
, 2002
"... We provide a feasible (quadratic time) algorithm for computing the lower bound V on the sample variance of interval data. The problem of computing the upper bound V is, in general, NPhard. We provide a feasible algorithm that computes V for many reasonable situations. ..."
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Cited by 11 (8 self)
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We provide a feasible (quadratic time) algorithm for computing the lower bound V on the sample variance of interval data. The problem of computing the upper bound V is, in general, NPhard. We provide a feasible algorithm that computes V for many reasonable situations.
OnLine Algorithms for Computing Mean and Variance of Interval Data, and Their Use in Intelligent Systems
 OF INTERVAL DATA, AND THEIR USE IN INTELLIGENT SYSTEMS, INFORMATION SCIENCES
, 2003
"... When we have only interval ranges [x i ; x i ] of sample values x1 ; : : : ; xn , what is the interval [V ; V ] of possible values for the variance V of these values? There are quadratic time algorithms for computing the exact lower bound V on the variance of interval data, and for computing V und ..."
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Cited by 8 (4 self)
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When we have only interval ranges [x i ; x i ] of sample values x1 ; : : : ; xn , what is the interval [V ; V ] of possible values for the variance V of these values? There are quadratic time algorithms for computing the exact lower bound V on the variance of interval data, and for computing V under reasonable easily verifiable conditions. The problem is that in real life, we often make additional measurements. In traditional statistics, if we have a new measurement result, we can modify the value of variance in constant time. In contrast, previously known algorithms for processing interval data required that, once a new data point is added, we start from the very beginning.
Outlier Detection Under Interval and Fuzzy Uncertainty: Algorithmic Solvability and Computational Complexity
 COMPUTATIONAL COMPLEXITY, PROCEEDINGS OF THE 22ND INTERNATIONAL CONFERENCE OF THE NORTH AMERICAN FUZZY INFORMATION PROCESSING SOCIETY NAFIPS’2003
, 2003
"... In many application areas, it is important to detect outliers. Traditional engineering approach to outlier detection is that we start with some "normal" values x 1 ; : : : ; xn , compute the sample average E, the sample standard variation oe, and then mark a value x as an outlier if x is outside th ..."
Abstract

Cited by 7 (6 self)
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In many application areas, it is important to detect outliers. Traditional engineering approach to outlier detection is that we start with some "normal" values x 1 ; : : : ; 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 k 0 sigma interval [E \Gamma k 0 \Delta oe; E + k 0 \Delta oe] (for some preselected parameter k 0 ). In real life, we often have only interval ranges [x i ; x i ] for the normal values x 1 ; : : : ; xn . In this case, we only have intervals of possible values for the bounds E \Gamma k 0 \Delta oe and E + k 0 \Delta oe. We can therefore identify outliers as values that are outside all k 0 sigma intervals.
Realtime algorithms for statistical analysis of interval data
 Proceedings of the International Conference on Information Technology InTech’03, Chiang Mai
, 2003
"... When we have only interval ranges [x i, xi] of sample values x1,..., xn, what is the interval [V, V] of possible values for the variance V of these values? There are quadratic time algorithms for computing the exact lower bound V on the variance of interval data, and for computing V under reasonable ..."
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Cited by 7 (4 self)
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When we have only interval ranges [x i, xi] of sample values x1,..., xn, what is the interval [V, V] of possible values for the variance V of these values? There are quadratic time algorithms for computing the exact lower bound V on the variance of interval data, and for computing V under reasonable easily verifiable conditions. The problem is that in real life, we often make additional measurements. In traditional statistics, if we have a new measurement result, we can modify the value of variance in constant time. In contrast, previously known algorithms for processing interval data required that, once a new data point is added, we start from the very beginning. In this paper, we describe new algorithms for statistical processing of interval data, algorithms in which adding a data point requires only O(n) computational steps.
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: