## Interval-Valued and Fuzzy-Valued Random Variables: From Computing Sample Variances to Computing Sample Covariances (2004)

Citations: | 13 - 8 self |

### BibTeX

@MISC{Beck04interval-valuedand,

author = {Jan B. Beck and Vladik Kreinovich and Berlin Wu},

title = {Interval-Valued and Fuzzy-Valued Random Variables: From Computing Sample Variances to Computing Sample Covariances},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper, we describe an algorithm that computes this range Cx;y for the case when the measurements are accurate enough -- so that the intervals corresponding to different measurements do not intersect much

### Citations

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Citation Context ...als corresponding to different degrees of confidence. Such a nested family of intervals is also called a fuzzy set, because it turns out to be equivalent to a more traditional definition of fuzzy set =-=[1, 7, 14, 15, 16]-=- (if a traditional fuzzy set is given, then different intervals from the nested family can be viewed as α-cuts corresponding to different levels of uncertainty α). To provide statistical values of fuz... |

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Citation Context ... ∈ xn}. For continuous functions f(x1, . . . , xn), this range is an interval. The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., =-=[4]-=-. Interval computations techniques: brief reminder. Historically the first method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval computati... |

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Citation Context ...more sophisticated techniques for producing a narrower enclosure, e.g., a centered form method. However, for each of these techniques, there are cases when we get an excess width. Reason: as shown in =-=[5]-=-, the problem of computing the exact range is known to be NP-hard even for polynomial functions f(x1, . . . , xn) (actually, even for quadratic functions f).sInterval-Valued Random Variables: How to C... |

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Citation Context ...ence. Such a nested family of intervals is also called a fuzzy set, because its8 Jan B. Beck, Vladik Kreinovich, and Berlin Wu turns out to be equivalent to a more traditional definition of fuzzy set =-=[7, 8]-=- (if a traditional fuzzy set is given, then different intervals from the nested family can be viewed as α-cuts corresponding to different levels of uncertainty α). To provide statistical values of fuz... |

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Citation Context ... of def these measurement errors ∆xi = �xi − xi, the result �y = f(�x1, . . . , �xn) of data processing is, in general, different from the actual value y = f(x1, . . . , xn) of the desired quantity y =-=[12]-=-. It is desirable to describe the error ∆y def = �y − y of the result of data processing. To do that, we must have some information about the errors of direct measurements. What do we know about the e... |

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Citation Context ...ent values of Ex, Ey, Vx, and Cx,y. The question is: what are the intervals Ex, Vx, and Cx,y of possible values of Ex, Vx, and Cx,y? The practical importance of this question was emphasized, e.g., in =-=[9, 10]-=- on the example of processing geophysical data. Comment: the problem reformulated in terms of set-valued random variables. Traditional statistical data processing means that we assume that the measure... |

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2001b. Propagating interval uncertainties in supervised pattern recognition for reservoir characterization
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Citation Context ...ent values of Ex, Ey, Vx, and Cx,y. The question is: what are the intervals Ex, Vx, and Cx,y of possible values of Ex, Vx, and Cx,y? The practical importance of this question was emphasized, e.g., in =-=[9, 10]-=- on the example of processing geophysical data. Comment: the problem reformulated in terms of set-valued random variables. Traditional statistical data processing means that we assume that the measure... |

14 |
Probabilities, intervals, what next? optimization problems related to extension of interval computations to situations with partial information about probabilities
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Citation Context ...hat the different measured values ˜xi are indeed different – e.g., the corresponding intervals xi do not intersect. In this case, there exists a quadratic-time algorithm for computing V x; see, e.g., =-=[2, 8, 10, 11]-=-. What about covariance? The only thing that we know is that in general, computing covariance Cx,y is NP-hard. A natural question is: can we find reasonable cases for which it is possible to compute t... |

10 |
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Citation Context ... ∈ xn}. For continuous functions f(x1, . . . , xn), this range is an interval. The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., =-=[4, 5, 6, 13]-=-. Interval computations techniques: brief reminder. Historically the first method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval computati... |

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Citation Context ... ∈ xn}. For continuous functions f(x1, . . . , xn), this range is an interval. The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., =-=[4, 5, 6, 13]-=-. Interval computations techniques: brief reminder. Historically the first method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval computati... |

6 |
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(Show Context)
Citation Context ...als corresponding to different degrees of confidence. Such a nested family of intervals is also called a fuzzy set, because it turns out to be equivalent to a more traditional definition of fuzzy set =-=[1, 7, 14, 15, 16]-=- (if a traditional fuzzy set is given, then different intervals from the nested family can be viewed as α-cuts corresponding to different levels of uncertainty α). To provide statistical values of fuz... |

5 |
Nonlinear Optimization: Complexity Issues
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Citation Context ...more sophisticated techniques for producing a narrower enclosure, e.g., a centered form method. However, for each of these techniques, there are cases when we get an excess width. Reason: as shown in =-=[9, 21]-=-, the problem of computing the exact range is known to be NP-hard even for polynomial functions f(x1, . . . , xn) (actually, even for quadratic functions f). Comment. NP-hard means, crudely speaking, ... |

3 |
Ginzburg L, Kreinovich V, Longpr L, Aviles M (2002) Computing variance for interval data is NP-hard. SIGACT News 33:108–118. and epistemic uncertainty : a critical discussion 37
- Ferson
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Citation Context ...dik Kreinovich, and Berlin Wu For variance, the problem is difficult. For Vx, all known algorithms lead to an excess width. Specifically, there exist feasible algorithms for computing V x (see, e.g., =-=[1]-=-), but in general, the problem of computing V x is NP-hard [1]. It is also known that in some practically important cases, feasible algorithms for computing V x are possible. One such practically usef... |

1 |
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Citation Context ...In real life, due to (inevitable) measurement uncertainty, often, what we actually observe is a set Si that contains the actual (unknown) value of xi. This phenomenon is called coarsening; see, e.g., =-=[3]-=-. Due to coarsening, instead of the actual values xi, all we know is the sets X1, . . . , Xn, . . . that are known the contain the actual (un-observable) values xi: xi ∈ Xi.s2 Jan B. Beck, Vladik Krei... |

1 |
Ogura Y, Kreinovich V (2002) Limit Theorems and Applications of Set Valued and Fuzzy Valued Random Variables
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Citation Context ...al research on set-valued random variables and corresponding statistics. In many cases, the corresponding statistics have been designed, and their asymptotical properties have been proven; see, e.g., =-=[2, 6]-=- and references therein. In many such situations, the main obstacle on the way of practically using these statistics is the fact that going from random numbers to random sets drastically increases the... |

1 |
Potluri L, Aló R (2002) Non-Destructive Testing of Aerospace Structures: Granularity and Data Mining Approach
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Citation Context ...t. In this case, there exists a quadratic-time algorithm for computing V x; see, e.g., [1]. What about covariance? The only thing that we know is that in general, computing covariance Cx,y is NP-hard =-=[11]-=-. In this paper, we show that (similarly to the case of variance), it is possible to compute the interval covariance when the measurement are accurate enough to enable us to distinguish between differ... |

1 |
Ferson S, Ginzburg L (2003) Outlier Detection Under Interval and Fuzzy Uncertainty: Algorithmic Solvability and Computational Complexity
- Kreinovich, Patangay, et al.
(Show Context)
Citation Context ...hat the different measured values ˜xi are indeed different – e.g., the corresponding intervals xi do not intersect. In this case, there exists a quadratic-time algorithm for computing V x; see, e.g., =-=[2, 8, 10, 11]-=-. What about covariance? The only thing that we know is that in general, computing covariance Cx,y is NP-hard. A natural question is: can we find reasonable cases for which it is possible to compute t... |

1 |
Kreinovich V (2003) Real-Time Algorithms for Statistical Analysis of Interval Data
- Wu, HT
- 2003
(Show Context)
Citation Context ...o Compute Sample Covariance 5 For variance, the problem is difficult. For Vx, all known algorithms lead to an excess width. Specifically, there exist feasible algorithms for computing V x (see, e.g., =-=[2, 22]-=-), but in general, the problem of computing V x is NP-hard [2]. It is also known that in some practically important cases, feasible algorithms for computing V x are possible. One such practically usef... |