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Computing Population Variance and Entropy under Interval Uncertainty: LinearTime Algorithms
, 2006
"... In statistical analysis of measurement results it is often necessary to compute the range [V, V] of the population variance V = 1 n · n∑ (xi − E) 2 where E = 1 n · n∑ xi when we only know the intervals i=1 [˜xi − ∆i, ˜xi + ∆i] of possible values of the xi. While V can be computed efficiently, the pr ..."
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In statistical analysis of measurement results it is often necessary to compute the range [V, V] of the population variance V = 1 n · n∑ (xi − E) 2 where E = 1 n · n∑ xi when we only know the intervals i=1 [˜xi − ∆i, ˜xi + ∆i] of possible values of the xi. While V can be computed efficiently, the problem of computing V is, in general, NPhard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm ” (Reliable Computing, 2006, Vol. 12, No. 4, pp. 273–280) we showed that in
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∑
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 6 (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.
Decision Making under Interval Uncertainty
"... To make a decision, we must find out the user’s preference, and help the user select an alternative which is the best – according to these preferences. Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them ..."
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Cited by 2 (2 self)
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To make a decision, we must find out the user’s preference, and help the user select an alternative which is the best – according to these preferences. Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is often unable to select one of these alternatives. In this chapter, we show how we can extend the traditional decision theory to such realistic (interval) cases. 1
A Systematic Approach to
 SER Estimation and Solutions,” Proc. Intl. Reliability Physical Symp
, 2003
"... • To make a decision, we must: – find out the user’s preference, and – help the user select an alternative which is the best – according to these preferences. • Traditional approach is based on an assumption that for each two alternatives A ′ and A ′ ′ , a user can tell: – whether the first alternat ..."
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Cited by 1 (0 self)
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• To make a decision, we must: – find out the user’s preference, and – help the user select an alternative which is the best – according to these preferences. • Traditional approach is based on an assumption that for each two alternatives A ′ and A ′ ′ , a user can tell: – whether the first alternative is better for him/her; we will denote this by A ′ ′ < A ′; – or the second alternative is better; we will denote this by A ′ < A ′ ′; – or the two given alternatives are of equal value to the user; we will denote this by A ′ = A ′ ′.
Estimating Variance under Interval and Fuzzy Uncertainty: Case of Hierarchical Estimation
 Foundations of Fuzzy Logic and Soft Computing, Proc. World Congress of the Int’l Fuzzy Systems Association IFSA’2007, Cancun, Mexico, June 18–21, 2007, Springer Lecture Notes on Artificial Intelligence
"... Traditional data processing in science and engineering starts with computing the basic statistical characteristics such as the population mean E and population variance V. In computing these characteristics, it is usually assumed that the corresponding data values x1,..., xn are known exactly. In ma ..."
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Traditional data processing in science and engineering starts with computing the basic statistical characteristics such as the population mean E and population variance V. In computing these characteristics, it is usually assumed that the corresponding data values x1,..., xn are known exactly. In many practical situations, we only know intervals [x i, xi] that contain the actual (unknown) values of xi or, more generally, a fuzzy number that describes xi. In this case, different possible values of xi lead, in general, to different values of E and V. In such situations, we are interested in producing the intervals of possible values of E and V – or fuzzy numbers describing E and V. There exist algorithms for producing such interval and fuzzy estimates. However, these algorithms are more complex than the typical data processing formulas and thus, require a larger amount of computation time. If we have several processors, then, it is desirable to perform these algorithms in parallel on several processors, and thus, to speed up computations. In this paper, we show how the algorithms for estimating variance under interval and fuzzy uncertainty can be parallelized.
From pBoxes to pEllipsoids: Towards an Optimal Representation of Imprecise Probabilities
"... Abstract—One of the most widely used ways to represent a probability distribution is by describing its cumulative distribution function (cdf) F (x). In practice, we rarely know the exact values of F (x): for each x, we only know F (x) with uncertainty. In such situations, it is reasonable to describ ..."
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Abstract—One of the most widely used ways to represent a probability distribution is by describing its cumulative distribution function (cdf) F (x). In practice, we rarely know the exact values of F (x): for each x, we only know F (x) with uncertainty. In such situations, it is reasonable to describe, for each x, the interval [F (x), F (x)] of possible values of x. This representation of imprecise probabilities is known as a pbox; it is effectively used in many applications. Similar interval bounds are possible for probability density function, for moments, etc. The problem is that when we transform from one of such representations to another one, we lose information. It is therefore desirable to come up with a universal representation of imprecise probabilities in which we do not lose information when we move from one representation to another. We show that under reasonable objective functions, the optimal representation is an ellipsoid. In particular, ellipsoids lead to faster computations, to narrower bounds, etc. I. FORMULATION OF THE PROBLEM Probabilistic information is important. In describing and processing uncertainty, it is very important to take into account information about the probabilities of different possible values [23]. This is especially true in many engineering applications, when we have a long history of similar situations, and we can use this history to estimate the probabilities of different scenarios. For example, for measurement uncertainty, it is important to use the available information about the probabilities of different possible values of the measurement error; see, e.g., [19]. How probability distributions are usually represented. There are many different ways of representing information about the probability distribution of a random variable X; see, e.g., [23]: • we can use the cumulative distribution function F (x) def
RELIABLE SIMULATION WITH INPUT UNCERTAINTIES USING AN INTERVALBASED APPROACH
"... Uncertainty associated with input parameters and models in simulation has gained attentions in recent years. The sources of uncertainties include lack of data and lack of knowledge about physical systems. In this paper, we present a new reliable simulation mechanism to help improve simulation robust ..."
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Uncertainty associated with input parameters and models in simulation has gained attentions in recent years. The sources of uncertainties include lack of data and lack of knowledge about physical systems. In this paper, we present a new reliable simulation mechanism to help improve simulation robustness when significant uncertainties exist. The new mechanism incorporates variabilities and uncertainties based on imprecise probabilities, where the statistical distribution parameters in the simulation are intervals instead of precise real numbers. The mechanism generates random interval variates to model the inputs. Interval arithmetic is applied to simulate a set of scenarios simultaneously in each simulation run. To ensure that the interval results bound those from the traditional realvalued simulation, a generic approach is also proposed to specify the number of replications in order to achieve the desired robustness. This new reliable simulation mechanism can be applied to address input uncertainties to support robust decision making. 1
7th International Symposium on Imprecise Probability: Theories and Applications, Innsbruck, Austria, 2011 LikelihoodBased Naive Credal Classifier
"... The naive credal classifier extends the classical naive Bayes classifier to imprecise probabilities, substituting the imprecise Dirichlet model for the uniform prior. As an alternative to the naive credal classifier, we present a likelihoodbased approach, which extends in a novel way the naive Baye ..."
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The naive credal classifier extends the classical naive Bayes classifier to imprecise probabilities, substituting the imprecise Dirichlet model for the uniform prior. As an alternative to the naive credal classifier, we present a likelihoodbased approach, which extends in a novel way the naive Bayes towards imprecise probabilities, by considering any possible quantification (each one defining a naive Bayes classifier) apart from those assigning to the available data a probability below a given threshold level. Besides the available supervised data, in the likelihood evaluation we also consider the instance to be classified, for which the value of the class variable is assumed missingatrandom. We obtain a closed formula to compute the dominance according to the maximality criterion for any threshold level. As there are currently no wellestablished metrics for comparing credal classifiers which have considerably different determinacy, we compare the two classifiers when they have comparable determinacy, finding that in those cases they generate almost equivalent classifications.
Estimating Statistical Characteristics of Lognormal and DeltaLognormal Distributions under Interval Uncertainty: Algorithms and Computational Complexity
"... Traditional statistical estimates ̂S(x1,..., xn) for different statistical characteristics S (such as mean, variance, etc.) implicitly assume that we know the sample values x1,..., xn exactly. In practice, the sample values ˜xi come from measurements and are, therefore, in general, different from th ..."
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Traditional statistical estimates ̂S(x1,..., xn) for different statistical characteristics S (such as mean, variance, etc.) implicitly assume that we know the sample values x1,..., xn exactly. In practice, the sample values ˜xi come from measurements and are, therefore, in general, different from the actual (unknown) values xi of the corresponding quantities. Sometimes, we know the probabilities of different values of the measurement error ∆xi = ˜xi − xi, but often, the only information that we have about the measurement error is the upper bound ∆i on its absolute value – provided by the manufacturer of the corresponding measuring instrument. In this case, the only information that we have about the actual values xi is that they belong to the intervals [˜xi − ∆i, ˜xi + ∆i]. In general, different values xi ∈ [˜xi−∆i, ˜xi+∆i] lead to different values of the corresponding estimate ̂S(x1,..., xn). In this case, it is desirable to find the range of all possible values of this characteristic. In this paper, we consider the problem of computing the corresponding range for the cases of lognormal and deltalognormal distributions. Interestingly, it turns out that, in contrast to the case of normal distribution for which it is feasible to compute the range of the mean, for lognormal and deltalognormal distributions, computing the range of the mean is an NPhard problem. 1 1