## Application-Motivated Combinations of Fuzzy, Interval, and Probability Approaches, with Application to Geoinformatics, . . .

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@MISC{Kreinovich_application-motivatedcombinations,

author = {Vladik Kreinovich},

title = {Application-Motivated Combinations of Fuzzy, Interval, and Probability Approaches, with Application to Geoinformatics, . . . },

year = {}

}

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### Abstract

Since the 1960s, many algorithms have been designed to deal with interval uncertainty. In the last decade, there has been a lot of progress in extending these algorithms to the case when we have a combination of interval, probabilistic, and fuzzy uncertainty. We provide an overview of related algorithms, results, and remaining open problems.

### Citations

770 | Statistical Reasoning with Imprecise Probabilities - Walley - 1991 |

190 | Computational Complexity and Feasibility of Data Processing and Interval Computations
- Kreinovich, Lakayev, et al.
- 1998
(Show Context)
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 =-=[25]-=-, 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). Practical problem. In some practical situa... |

188 |
A first Course in Fuzzy Logic
- Nguyen, Walker
- 2000
(Show Context)
Citation Context ...the original fuzzy set. Vice versa, if we know α-cuts for every α, then, for each object x, we can determine the degree of possibility that x belongs to the original fuzzy set [11], [21], [39], [41], =-=[43]-=-. A fuzzy set can be thus viewed as a nested family of its α-cuts. If instead of a (crisp) interval xi of possible values of the measured quantity, we have a fuzzy set µi(x) of possible values, then w... |

176 |
Fuzzy Sets and Fuzzy Logic
- Klir, Yuan
- 1995
(Show Context)
Citation Context ...{x | µ(x) ≥ α} of the original fuzzy set. Vice versa, if we know α-cuts for every α, then, for each object x, we can determine the degree of possibility that x belongs to the original fuzzy set [11], =-=[21]-=-, [39], [41], [43]. A fuzzy set can be thus viewed as a nested family of its α-cuts. If instead of a (crisp) interval xi of possible values of the measured quantity, we have a fuzzy set µi(x) of possi... |

168 |
Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics
- Jaulin, Kieffer, et al.
- 2001
(Show Context)
Citation Context ...n) | x1 ∈ x1, . . . , xn ∈ xn}. x1 x2 · · · xn ✲ ✲ ✲ f y = f(x1, . . . , xn) ✲ The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., =-=[19]-=-, [20]. Interval computations techniques: brief reminder. Historically the rst method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval compu... |

122 |
Operations on fuzzy numbers
- Dubois, Prade
- 1978
(Show Context)
Citation Context ...oblem. In some practical situations, in addition to the lower and upper bounds on each random variable xi, we know the bounds Ei = [Ei, Ei] on its mean Ei. Indeed, in measurement practice (see, e.g., =-=[11]-=-), the overall measurement error ∆x is usually represented as a sum of two components: • a systematic error component ∆sx which is de ned as the expected value E[∆x], and • a random error component ∆r... |

81 |
Nested intervals and sets: concepts, relations to fuzzy sets, and applications
- Nguyen, Kreinovich
- 1996
(Show Context)
Citation Context ...α} of the original fuzzy set. Vice versa, if we know α-cuts for every α, then, for each object x, we can determine the degree of possibility that x belongs to the original fuzzy set [11], [21], [39], =-=[41]-=-, [43]. A fuzzy set can be thus viewed as a nested family of its α-cuts. If instead of a (crisp) interval xi of possible values of the measured quantity, we have a fuzzy set µi(x) of possible values, ... |

73 | Partial Identification of Probability Distributions - Manski - 2003 |

62 | Computing Variance for Interval Data is NP-Hard - Ferson, Ginzburg, et al. |

44 | Statool: A Tool for Distribution Envelope Determination (DEnv), an Interval-Based Algorithm for Arithmetic on Random Variables
- Berleant, Xie, et al.
- 2003
(Show Context)
Citation Context ... r1 · r2. The main idea behind straightforward interval computations is to perform the same operations, but with intervals instead of numbers: r1 := [1, 2] − [2, 2] = [−1, 0]; r2 := [1, 2] + [2, 2] = =-=[3, 4]-=-; r3 := [−1, 0] · [3, 4] = [−4, 0]. For this function, the actual range is f(x) = [−3, 0]. Comment: this is just a toy example, there are more ef cient ways of computing an enclosure Y ⊇ y. There exis... |

44 |
Kearfott and
- B
- 1996
(Show Context)
Citation Context ...1 ∈ x1, . . . , xn ∈ xn}. x1 x2 · · · xn ✲ ✲ ✲ f y = f(x1, . . . , xn) ✲ The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., [19], =-=[20]-=-. Interval computations techniques: brief reminder. Historically the rst method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval computation... |

41 | Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity
- Kreinovich, Xiang, et al.
- 2006
(Show Context)
Citation Context ...out that most such problems are computationally dif cult (to be more precise, NP-hard), and we provide feasible algorithms that compute these bounds under reasonable easily veri able conditions [15], =-=[31]-=-. Challenges. What is, in addition to intervals and rst moments, we also know second moments (this problem is important for design of computer chips): x1, E1, V1 x2, E2, V2 · · · xn, En, Vn ✲ ✲ ✲ f y,... |

34 |
Interval analysis and fuzzy set theory
- Moore, Lodwick
(Show Context)
Citation Context ...ble problems:• If we have complete information about the distributions of xi, then, to get validated estimates on uncertainty of y, we have to use Monte-Carlo-type techniques; see, e.g., [33], [34], =-=[39]-=-. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we ... |

31 | Representation and problem solving with the distribution envelope determination (DEnv) method, Reliability Engineering and System Safety
- Berleant, Zhang
(Show Context)
Citation Context ...e-Carlo-type techniques; see, e.g., [33], [34], [39]. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], =-=[6]-=-, [7], [48], [57]. • If we have moments, then we can use methods from [18], [22], [30], [44], [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practical a... |

30 | Error Estimations for Indirect Measurements: Randomized Vs. Deterministic Algorithms for 'Black-Box' Programs
- Trejo, Kreinovich
- 2001
(Show Context)
Citation Context ...ef ciency of these techniques are given, e.g., in [46]; in particular, examples related to estimating how the uncertainty of inputs leads to uncertainty of the results of data processing are given in =-=[50]-=-.Proof of Theorem 5. By de nition, y0 is the largest value of yp over all possible distributions p ∈ P. This means that for the given y0, for all possible distributions p ∈ P, we have Prob(D ≤ y0) ≥ ... |

26 |
Exact Bounds on Finite
- Ferson, Ginzburg, et al.
- 2005
(Show Context)
Citation Context ...turns out that most such problems are computationally dif cult (to be more precise, NP-hard), and we provide feasible algorithms that compute these bounds under reasonable easily veri able conditions =-=[15]-=-, [31]. Challenges. What is, in addition to intervals and rst moments, we also know second moments (this problem is important for design of computer chips): x1, E1, V1 x2, E2, V2 · · · xn, En, Vn ✲ ✲ ... |

26 | Novel approaches to numerical software with result verification. Pages 274-305 in Numerical Software with Result Verification
- Granvilliers, Kreinovich, et al.
- 2004
(Show Context)
Citation Context ...s, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we can use methods from =-=[18]-=-, [22], [30], [44], [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practical applications, like the appearance of histogram-type distributions in proble... |

26 | Equivalence of five methods for bounding uncertainty - Regan, Ferson, et al. - 2004 |

24 |
Using Pearson correlation to improve envelopes around the distributions of functions
- Berleant, Zhang
(Show Context)
Citation Context ... Monte-Carlo-type techniques; see, e.g., [33], [34], [39]. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], =-=[5]-=-, [6], [7], [48], [57]. • If we have moments, then we can use methods from [18], [22], [30], [44], [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practi... |

22 |
Estimating and validating the cumulative distribution of a function of random variables: toward the development of distribution arithmetic
- Lodwick, Jamison
- 2003
(Show Context)
Citation Context ...ion of possible problems:• If we have complete information about the distributions of xi, then, to get validated estimates on uncertainty of y, we have to use Monte-Carlo-type techniques; see, e.g., =-=[33]-=-, [34], [39]. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have momen... |

22 |
Probability bounds analysis
- Ferson, Donald
- 1998
(Show Context)
Citation Context ...n, to get validated estimates on uncertainty of y, we have to use Monte-Carlo-type techniques; see, e.g., [33], [34], [39]. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], =-=[51]-=-. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we can use methods from [18], [22], [30], [44], [45]. There are also additional issue... |

21 |
What Next? Optimization Problems Related to Extension of Interval Computations to Situations with Partial Information about Probabilities
- Kreinovich, Probabilities
(Show Context)
Citation Context ...can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we can use methods from [18], =-=[22]-=-, [30], [44], [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practical applications, like the appearance of histogram-type distributions in problems rel... |

21 | Automatic error analysis in digital computation - Moore - 1959 |

20 | Population Variance under Interval Uncertainty: A - Dantsin, Kreinovich, et al. - 2006 |

20 | Outlier Detection Under Interval Uncertainty: Algorithmic Solvability and Computational Complexity - Kreinovich, Longpré, et al. |

19 | RAMAS Risk Calc 4.0 - Ferson - 2002 |

18 | Closing the Gap Between ASIC & Custom - Chinnery, Keutzer - 2003 |

17 | Exact Upper Bound on the - Kreinovich, Ferson, et al. - 2004 |

13 | 2004. Interval-valued and fuzzy-valued random variables: from computing sample variances to computing sample covariances
- Beck, Kreinovich, et al.
(Show Context)
Citation Context ...ses (see examples below), the enclosure has excess width. Example. Let us illustrate the above idea on the example of estimating the range of the function f(x) = (x − 2) · (x + 2) on the interval x ∈ =-=[1, 2]-=-. We start with parsing the expression for the function, i.e., describing how a computer will compute this expression; it will implement the following sequence of elementary operation: r1 := x − 2; r2... |

13 | Probabilities, intervals, what next? Optimization problems related to extension of interval computations to situations with partial information about probabilities - Kreinovich |

12 | Risk analysis without Monte Carlo methods - Moore - 1984 |

12 | Fast algorithm for computing the upper endpoint of sample variance for interval data: case of sufficiently accurate measurements - Xiang - 2006 |

11 | New Algorithms for Statistical Analysis of Interval Data - Xiang, Starks, et al. - 2004 |

11 | Decision making under interval probabilities - Yager, Kreinovich - 1998 |

10 | Probabilities, intervals, what next? Extension of interval computations to situations with partial information about probabilities
- Kreinovich, Solopchenko, et al.
- 2004
(Show Context)
Citation Context ...e methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we can use methods from [18], [22], =-=[30]-=-, [44], [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practical applications, like the appearance of histogram-type distributions in problems related t... |

10 | Interval-Based Robust Statistical Techniques for NonNegative Convex Functions with Applications to Timing Analysis of Computer Chips, Reliable Engineering Computing
- Orshansky, Wang, et al.
- 2006
(Show Context)
Citation Context ...ods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, then we can use methods from [18], [22], [30], =-=[44]-=-, [45]. There are also additional issues, including: • how we get these bounds for xi? • speci c practical applications, like the appearance of histogram-type distributions in problems related to priv... |

10 | Computing Best-Possible Bounds for the Distribution of a Sum of Several Variables is NP-Hard - Kreinovich, Ferson - 2006 |

9 | 2006a) ‘Detecting Outliers under Interval Uncertainty: A New Algorithm Based on Constraint Satisfaction - Dantsin, Wolpert, et al. |

8 | On-Line Algorithms for Computing Mean and Variance - Kreinovich, Nguyen, et al. - 2007 |

8 | Dependable handling of uncertainty - Berleant, Cheong, et al. - 2003 |

7 |
Dependable Handling of Uncertainty, Reliable Computing
- Berleant, Cheong, et al.
- 2003
(Show Context)
Citation Context ... r1 · r2. The main idea behind straightforward interval computations is to perform the same operations, but with intervals instead of numbers: r1 := [1, 2] − [2, 2] = [−1, 0]; r2 := [1, 2] + [2, 2] = =-=[3, 4]-=-; r3 := [−1, 0] · [3, 4] = [−4, 0]. For this function, the actual range is f(x) = [−3, 0]. Comment: this is just a toy example, there are more ef cient ways of computing an enclosure Y ⊇ y. There exis... |

7 | Fast quantum algorithms for handling probabilistic and interval uncertainty - Kreinovich, Longpré - 2004 |

7 | Real-Time Algorithms for Statistical Analysis of Interval Data - Wu, Nguyen, et al. - 2003 |

6 |
Computing Variance under Interval Uncertainty: A New Algorithm and Its Potential Application to Privacy in Statistical Databases
- Aló, Beheshti, et al.
- 2006
(Show Context)
Citation Context ...ses (see examples below), the enclosure has excess width. Example. Let us illustrate the above idea on the example of estimating the range of the function f(x) = (x − 2) · (x + 2) on the interval x ∈ =-=[1, 2]-=-. We start with parsing the expression for the function, i.e., describing how a computer will compute this expression; it will implement the following sequence of elementary operation: r1 := x − 2; r2... |

5 |
Optimization under uncertainity: methods and applications
- Lodwick, Neumaier, et al.
- 2001
(Show Context)
Citation Context ... possible problems:• If we have complete information about the distributions of xi, then, to get validated estimates on uncertainty of y, we have to use Monte-Carlo-type techniques; see, e.g., [33], =-=[34]-=-, [39]. • If we have p-boxes, we can use methods from [13], [16], [17], [23], [47], [51]. • If we have histograms, we can use methods from [3], [4], [5], [6], [7], [48], [57]. • If we have moments, th... |

5 | Interval risk analysis of real estate investment: a non-MonteCarlo approach - Moore - 1985 |

5 | Fuzzy modeling in terms of surprise, Fuzzy Sets and Systems 135 - Neumaier - 2003 |

5 | Foundations of Statistical Processing of Set-Valued Data: Towards Efficient Algorithms - Nguyen, Kreinovich, et al. - 2004 |

4 | Interval Statistical Models (in Russian). Radio i Svyaz - Kuznetsov - 1991 |

4 | Towards combining probabilistic and interval uncertainty in engineering calculations - Starks, Kreinovich, et al. - 2004 |