## Exact bounds on finite populations of interval data (2001)

Venue: | Reliable Computing |

Citations: | 14 - 10 self |

### BibTeX

@ARTICLE{Ferson01exactbounds,

author = {Scott Ferson and Lev Ginzburg and Vladik Kreinovich and Luc Longpré and Monica Aviles},

title = {Exact bounds on finite populations of interval data},

journal = {Reliable Computing},

year = {2001},

volume = {11},

pages = {207--233}

}

### OpenURL

### Abstract

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

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Citation Context ...rd Our second result is that the general problem of computing σ 2 from given intervals xi is computationally difficult, or, in precise terms, NP-hard (for exact definitions of NP-hardness, see, e.g., =-=[5, 8, 13]-=-). Theorem 3.1. Computing σ 2 is NP-hard. Comment. This result was first announced in [3]. The very fact that computing the range of a quadratic function is NP-hard was first proven by Vavasis [15] (s... |

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Citation Context ... of the results of straightforward interval computations. Not surprisingly, we do get excess width when applying straightforward interval computations to the formula (1.1). For example, for x1 = x2 = =-=[0, 1]-=-, the actual σ 2 = (x1 − x2) 2 /4 and hence, the actual range σ 2 = [0, 0.25]. On the other hand, µ = [0, 1], hence (x1 − µ) 2 + (x2 − µ) 2 = [0, 1] ⊃ [0, 0.25]. 2 It is worth mentioning that there ar... |

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Citation Context ...rd Our second result is that the general problem of computing σ 2 from given intervals xi is computationally difficult, or, in precise terms, NP-hard (for exact definitions of NP-hardness, see, e.g., =-=[5, 8, 13]-=-). Theorem 3.1. Computing σ 2 is NP-hard. Comment. This result was first announced in [3]. The very fact that computing the range of a quadratic function is NP-hard was first proven by Vavasis [15] (s... |

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Citation Context ...rd Our second result is that the general problem of computing σ 2 from given intervals xi is computationally difficult, or, in precise terms, NP-hard (for exact definitions of NP-hardness, see, e.g., =-=[5, 8, 13]-=-). Theorem 3.1. Computing σ 2 is NP-hard. Comment. This result was first announced in [3]. The very fact that computing the range of a quadratic function is NP-hard was first proven by Vavasis [15] (s... |

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Citation Context ... 8, 13]). Theorem 3.1. Computing σ 2 is NP-hard. Comment. This result was first announced in [3]. The very fact that computing the range of a quadratic function is NP-hard was first proven by Vavasis =-=[15]-=- (see also [8]). We have shown that this difficulty happens even for the very simple quadratic functions (1.1) frequently used in data processing. A natural question is: maybe the difficulty comes fro... |

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Citation Context ...putationally difficult, or, in precise terms, NP-hard (for exact definitions of NP-hardness, see, e.g., [5, 8, 13]). Theorem 3.1. Computing σ 2 is NP-hard. Comment. This result was first announced in =-=[3]-=-. The very fact that computing the range of a quadratic function is NP-hard was first proven by Vavasis [15] (see also [8]). We have shown that this difficulty happens even for the very simple quadrat... |

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Citation Context ...ge and their finite population variance µ def = x1 + . . . + xn n σ 2 def = (x1 − µ) 2 + . . . + (xn − µ) 2 n (1.1) (or, equivalently, the finite population standard deviation σ = √ σ 2 ); see, e.g., =-=[14]-=-. 1sIn some practical situations, we only have intervals xi = [x i, xi] of possible values of xi. This happens, for example, if instead of observing the actual value xi of the random variable, we obse... |

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Citation Context ...ce σ 2 > 0. The question is (see, e.g., [18]): What is the total set of possible values of σ 2 when the above intersection is empty? The practical importance of this question was emphasized, e.g., in =-=[10, 11]-=- on the example of processing geophysical data. A similar question can (and will) be asked not only about the finite population variance, but also about other finite population parameters. 1.2 For thi... |

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Citation Context ...rval inputs [x i, xi], [y i , y i] is NP-hard. Theorem 5.2. The problem of computing C from the interval inputs [x i, xi], [y i , y i] is NP-hard. 8 i=1sComment. These results were first announced in =-=[12]-=-. 5.2 Finite Population Correlation As we have mentioned, finite population covariance C between the data sets x1, . . . , xn and y1, . . . , yn is often used to compute finite population correlation ... |

19 |
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Citation Context ...ce σ 2 > 0. The question is (see, e.g., [18]): What is the total set of possible values of σ 2 when the above intersection is empty? The practical importance of this question was emphasized, e.g., in =-=[10, 11]-=- on the example of processing geophysical data. A similar question can (and will) be asked not only about the finite population variance, but also about other finite population parameters. 1.2 For thi... |

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Citation Context ...ce is 0. In other words, if the intervals have a non-empty intersection, then σ 2 = 0. Conversely, if the intersection of xi is empty, then σ 2 cannot be 0, hence σ 2 > 0. The question is (see, e.g., =-=[18]-=-): What is the total set of possible values of σ 2 when the above intersection is empty? The practical importance of this question was emphasized, e.g., in [10, 11] on the example of processing geophy... |

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Citation Context ...e range of finite population average, as we have mentioned, straightforward interval computations lead to exact bounds. The reason: in the above formula for µ, each interval variable only occurs once =-=[6]-=-. For the problem of computing the range of finite population variance, the situation is somewhat more difficult, because in the expression (1.1), each variable xi occurs several times: explicitly, in... |

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