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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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Cited by 11 (7 self)
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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 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.
Estimating Variance under Interval and Fuzzy Uncertainty: Parallel Algorithms
"... 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, x ..."
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Cited by 1 (0 self)
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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. I. COMPUTING STATISTICS IS IMPORTANT Traditional data processing in science and engineering starts with computing the basic statistical characteristics such as the population mean and population variance E = 1 n ·
Estimating Risk under Interval Uncertainty: Sequential and Parallel Algorithms
"... In traditional econometrics, the quality of an individual investment – and of the investment portfolio – is characterized by its expected return and its risk (variance). For an individual investment or portfolio, we can estimate the future expected return and a future risk by tracing the returns x1, ..."
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In traditional econometrics, the quality of an individual investment – and of the investment portfolio – is characterized by its expected return and its risk (variance). For an individual investment or portfolio, we can estimate the future expected return and a future risk by tracing the returns x1,..., xn of this investment (and/or similar investments) over the past years, and computing the statistical characteristics based on these returns. The return (per unit investment) is defined as the selling of the corresponding financial instrument at the ends of, e.g., a oneyear period, divided by the buying price of this instrument at the beginning of this period. It is usually assumed that we know the exact return values x1,..., xn. In practice, however, both the selling and the buying prices unpredictably fluctuate from day to day – and even within a single day. These minutebyminute fluctuations are rarely recorded; what we usually have recorded is the daily range of prices. As a result, we can only find the range [x i, xi] of possible values of the return xi. In this case, different 1 possible values of xi lead, in general, to different values of the expected return E and of the risk V. In such situations, we are interested in producing the intervals of possible values of E and V. In the paper, we describe algorithms for producing such interval estimates. The corresponding sequential algorithms, however, are reasonably complex and timeconsuming. In financial applications, it is often very important to produce the result as fast as possible. One way to speed up computations is 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 uncertainty can be parallelized. 1 Computing
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
NoFreeLunch Result for Interval and Fuzzy Computing: When Bounds Are Unusually Good, Their Computation is Unusually Slow
"... Abstract. On several examples from interval and fuzzy computations and from related areas, we show that when the results of data processing are unusually good, their computation is unusually complex. This makes us think that there should be an analog of Heisenberg’s uncertainty principle well known ..."
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Abstract. On several examples from interval and fuzzy computations and from related areas, we show that when the results of data processing are unusually good, their computation is unusually complex. This makes us think that there should be an analog of Heisenberg’s uncertainty principle well known in quantum mechanics: when we an unusually beneficial situation in terms of results, it is not as perfect in terms of computations leading to these results. In short, nothing is perfect.
Maximum Likelihood Approach to Pointwise Estimation in Statistical Data Processing under Interval Uncertainty
"... Traditional statistical estimates C(x1,..., xn) for different statistical characteristics (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 a ..."
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Traditional statistical estimates C(x1,..., xn) for different statistical characteristics (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 statistical characteristic C(x1,..., xn). In this case, it is desirable to find the set of all possible values of this characteristic.
Estimating Mean and Variance under Interval Uncertainty: Dynamic Case
"... In many practical situations, it is important to estimate the mean E and the variance V from the sample values x1,..., xn. Usually, in statistics, we consider the case when the parameters like E and V do not change with time and when the sample values xi are known exactly. In practice, the values xi ..."
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In many practical situations, it is important to estimate the mean E and the variance V from the sample values x1,..., xn. Usually, in statistics, we consider the case when the parameters like E and V do not change with time and when the sample values xi are known exactly. In practice, the values xi come from measurements, and measurements are never 100 % accurate. In many cases, we only know the upper bound ∆i on the measurement error. In this case, once we know the measured value ˜xi, we can conclude that the actual (unknown) value xi belongs to the interval [˜xi − ∆i, ˜xi + ∆i]. Different values xi from these intervals lead, in general, to different values of E and V. It is therefore desirable to find the ranges E and V of all possible values of E and V. While this problem is, in general, NPhard, in many practical situations, there exist efficient algorithms for computing such ranges. In practice, processes are dynamic. As a result, reasonable estimates for E and V assign more weight to more recent measurements and less weight to the past ones. In this paper, we extend known algorithms for computing the ranges E and V to such dynamic estimates. 1