## Interval Computations as an Important Part of Granular Computing: An Introduction

Citations: | 1 - 0 self |

### BibTeX

@MISC{Kreinovich_intervalcomputations,

author = {Vladik Kreinovich},

title = {Interval Computations as an Important Part of Granular Computing: An Introduction},

year = {}

}

### OpenURL

### Abstract

This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.

### Citations

11575 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...ost generality. This strategy was further clarified in the 1970s, when it turned out, crudely speaking, that some problem cannot be efficiently solved; such difficult problems are called NP-hard; see =-=[12, 28, 45]-=- for detailed description. If a problem is NPhard, then it is hopeless to search for a general efficient solution; we must look for efficient solutions to subclasses of this problem and/or approximate... |

2446 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...ost generality. This strategy was further clarified in the 1970s, when it turned out, crudely speaking, that some problem cannot be efficiently solved; such difficult problems are called NP-hard; see =-=[12, 28, 45]-=- for detailed description. If a problem is NPhard, then it is hopeless to search for a general efficient solution; we must look for efficient solutions to subclasses of this problem and/or approximate... |

1024 | Monte Carlo Statistical Methods
- Robert, Casella
- 1999
(Show Context)
Citation Context ... situations, the problem of 19sestimating the effect of these approximation errors ∆xi on the result of data processing is computationally easy. Namely, we can use Monte-Carlo simulations (see, e.g., =-=[49]-=-), when for several iterations k = 1, . . . , N, we do the following: • we simulate the inputs ∆x (k) i butions; • we substitute the resulting simulated values x (k) i according to the known probabili... |

1014 |
Interval Analysis
- Moore
- 1966
(Show Context)
Citation Context ...nterval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland [54, 55], by T. Sunaga in Japan [51], and by R. Moore in the USA =-=[33, 38, 34, 35, 36, 37, 39, 40]-=-. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enhanced by the fact that in addition to estimates for general numerical algorithms, Moore’s ... |

984 |
Fuzzy Sets and Fuzzy Logic Theory and Applications
- Klir, Yuan
- 1995
(Show Context)
Citation Context ...with 100% certainty), • an expert can also say that with 90% certainty, this inaccuracy is ≤ 4, and • with 70% certainty, this inaccuracy is ≤ 2. Thus, instead of a single interval [30 − 5, 30 + 5] = =-=[25, 35]-=- that is guaranteed to contain the (unknown) age with certainty 100%, the expert also produces a narrower interval [30−4, 30+4] = [26, 34] which contains this age with certainty 90%, and an even narro... |

593 |
Methods and applications of interval analysis
- Moore
- 1979
(Show Context)
Citation Context ...nterval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland [54, 55], by T. Sunaga in Japan [51], and by R. Moore in the USA =-=[33, 38, 34, 35, 36, 37, 39, 40]-=-. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enhanced by the fact that in addition to estimates for general numerical algorithms, Moore’s ... |

588 | Probability theory: The logic of science
- Jaynes
- 2003
(Show Context)
Citation Context ...x) · ln(ρ(x)) dx (here ρ(x) denotes the probability density). For details on this Maximum Entropy approach and its relation to interval uncertainty and Laplace’s principle of indifference, see, e.g., =-=[5, 21, 24]-=- Maximum entropy method for the case of interval uncertainty. One can easily check that for a single variable x1, among all distributions located on a given interval, the entropy is the largest when t... |

316 |
Rigorous Global Search: Continuous Problems
- Kearfott
- 1996
(Show Context)
Citation Context ...ch-and-bound techniques: e.g., they check for monotonicity to weed out subboxes where local maxima are only possible at the endpoints, they look for solutions to the equation ∂f = 0, etc.; see, e.g., =-=[17, 22]-=-. ∂xi 36sOptimization: granularity helps. In the above text, we assumed that we know the exact value of the objective function F (x) for each alternative x. In reality, we often only have approximate ... |

313 | Handbook of Parametric and Nonparametric Statistical Procedures, 3rd edn - Sheskin - 2004 |

199 |
A first course in fuzzy logic
- Nguyen, Walker
- 2000
(Show Context)
Citation Context ...25, 35]. In general, instead of single interval, we have a nested family of intervals corresponding to different degree of certainty. Such a nested family of intervals can be viewed as a fuzzy number =-=[25, 44]-=-: for every value x, we can define the degree µ(x) to which x is possible as 1 minus the largest degree of certainty α for which x belongs to the α-interval. Interval computations are needed to proces... |

197 |
Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions
- Mendel
- 2001
(Show Context)
Citation Context ...he (unknown) age with certainty 100%, the expert also produces a narrower interval [30−4, 30+4] = [26, 34] which contains this age with certainty 90%, and an even narrower interval [30 − 2, 30 + 2] = =-=[28, 32]-=- which contains the age with certainty 70%. So, we have three intervals which are nested in the sense that every interval corresponding to a smaller degree of certainty is contained in the interval co... |

191 | Computational Complexity and Feasibility of Data Processing and Interval Computations
- Kreinovich, Lakeyev, et al.
- 1998
(Show Context)
Citation Context ...of the given objective function is the largest. Such problems are called optimization problems. Both constraint satisfaction and optimization often requires a large number of computations; see, e.g., =-=[28]-=-. Comment. Our main objective is to describe interval computations. They were originally invented for the first two classes of problems – i.e., for data processing, but they turned out to be very usef... |

172 |
Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics
- Jaulin, Kieffer, et al.
- 2001
(Show Context)
Citation Context ...ll the uncertainties into account, the trajectory of a space flight is guaranteed to reach the Moon. Since then, interval computations have been actively used in many areas of science and engineering =-=[19, 20]-=-. Comment. An early history of interval computations is described in detail in [31] and in [41]; early papers on interval computations can be found on the interval computations website [19]. 9 Interva... |

128 |
Nonlinear Optimization: Complexity Issues
- Vavasis
- 1991
(Show Context)
Citation Context ...is: is the corresponding interval computations problem still feasible? Alas, it turns out that for quadratic functions, interval computations problem is, in general, NP-hard; this was first proved in =-=[52]-=-. Moreover, it turns out that it is NP-hard not just for some rarely-used exotic quadratic functions: it is known that the problem of computing the exact range V = [V , V ] for the variance V = 1 n · ... |

120 |
New Tools for Robustness of Linear Systems
- Barmish
- 1994
(Show Context)
Citation Context ...l. Robust control methods, i.e., methods which stabilize a system (known with interval uncertainty) for all possible values of the parameters from the corresponding intervals, are presented, e.g., in =-=[2, 4]-=-. Applications to optimization: practical need. As we have mentioned earlier, one of the main objectives of engineering s to find the alternative which is the best (in some reasonable sense). In many ... |

120 | A proof of the kepler conjecture
- Hales
(Show Context)
Citation Context ...m, etc., has the largest possible density. This hypothesis was proved in 1998 by T. Hales, who, in particular, used interval computations to prove that many other placements lead to a smaller density =-=[14]-=-. Beyond interval computations, towards general granular computing. In the previous text, we consider situations when we have either probabilistic, or interval, or fuzzy uncertainty. In practice, we o... |

109 |
Measurement Errors and Uncertainties: Theory and Practice
- Rabinovich
- 2005
(Show Context)
Citation Context ...abilities. In many practical situations, we not only know the interval [−∆i, ∆i] of possible values of the measurement error; we also know the probability of different values ∆xi within this interval =-=[47]-=-. In most practical applications, it is assumed that the corresponding measurement errors are normally distributed with 0 means and known standard deviation. Numerous engineering techniques are known ... |

98 |
Handbook of Parametric and Nonparametric
- Sheskin
- 2011
(Show Context)
Citation Context ...t in this case. According to this approach, we assume that ∆xi are independent random variables, each of which is uniformly distributed on the interval [−∆, ∆]. According to the Central Limit theorem =-=[50, 53]-=-, when n → ∞, the distribution of the sum of n independent identically distributed bounded random variables tends to Gaussian. This means that for large values n, the distribution of ∆y is approximate... |

88 |
Robust Control. The Parametric Approach
- Bhattacharyya, Chapellat, et al.
- 1995
(Show Context)
Citation Context ...l. Robust control methods, i.e., methods which stabilize a system (known with interval uncertainty) for all possible values of the parameters from the corresponding intervals, are presented, e.g., in =-=[2, 4]-=-. Applications to optimization: practical need. As we have mentioned earlier, one of the main objectives of engineering s to find the alternative which is the best (in some reasonable sense). In many ... |

63 |
C++ Toolbox for Verified Computing: Basic Numerical Problems
- Hammer, Hocks, et al.
- 1995
(Show Context)
Citation Context ...ications are described in the interval-related chapters of this handbook. Most of these applications use special software tools and packages specifically designed for interval computations; see, e.g, =-=[15]-=-; a reasonably current list of such tools is available from the interval website [19]. Applications to control. One of the areas where guaranteed bounds are important is the area of control. Robust co... |

62 | Computing Variance for Interval Data is NP-Hard
- Ferson, Ginzburg, et al.
(Show Context)
Citation Context ... of computing the exact range V = [V , V ] for the variance V = 1 n · n� i=1 (xi − E) 2 = 1 n · n� i=1 x 2 i − � 1 n · over interval data xi ∈ [�xi − ∆i, �xi + ∆i] is, in general, NP-hard; see, e.g., =-=[8, 9]-=-. To be more precise, there is a polynomial-time algorithm for computing V , but computing V is, in general, NP-hard. Historical comment. NP-hardness of interval computations was first proven in [10, ... |

56 |
A note on the extension principle for fuzzy sets
- Nguyen
- 1978
(Show Context)
Citation Context ...or y: y(α) = f(x1(α), . . . , xn(α)). It turns out that the resulting fuzzy number is exactly what we would get if we simply apply Zadeh’s extension principle to the fuzzy numbers corresponding to xi =-=[25, 43, 44]-=- So, in processing fuzzy expert opinions, we also need interval computations. 15s6 Interval Computations Are Sometimes Easy but In General, They Are Computationally Difficult (NP-Hard) Interval comput... |

52 |
Uncertainty and information. Foundations of generalized information theory
- Klir
- 2006
(Show Context)
Citation Context ...x) · ln(ρ(x)) dx (here ρ(x) denotes the probability density). For details on this Maximum Entropy approach and its relation to interval uncertainty and Laplace’s principle of indifference, see, e.g., =-=[5, 21, 24]-=- Maximum entropy method for the case of interval uncertainty. One can easily check that for a single variable x1, among all distributions located on a given interval, the entropy is the largest when t... |

47 |
Theory of interval algebra and its application to numerical analysis
- Sunaga
- 1958
(Show Context)
Citation Context ...igins of interval computations. Interval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland [54, 55], by T. Sunaga in Japan =-=[51]-=-, and by R. Moore in the USA [33, 38, 34, 35, 36, 37, 39, 40]. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enhanced by the fact that in add... |

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 ...now the p-boxes corresponding to the auxiliary quantities xi, we need to find the p-box corresponding to the desired quantity y = f(x1, . . . , xn); such methods are described, e.g., in [7] (see also =-=[29, 30]-=-). Similarly, in fuzzy logic, we considered the case when for every property A and for every value x, we know the exact value of the degree µA(x) to which x satisfies the property. In reality, as we h... |

40 |
Interval Arithmetic and Automatic Error Analysis in Digital Computing
- Moore
- 1962
(Show Context)
Citation Context ...nterval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland [54, 55], by T. Sunaga in Japan [51], and by R. Moore in the USA =-=[33, 38, 34, 35, 36, 37, 39, 40]-=-. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enhanced by the fact that in addition to estimates for general numerical algorithms, Moore’s ... |

37 | Computation and application of Taylor polynomials with interval remainder bounds
- Berz, Hoffstätter
- 1998
(Show Context)
Citation Context ... have mentioned, another way to get more accurate estimates is to use so-called Taylor techniques, i.e., to explicitly consider second-order and higher-order terms in the Taylor expansion; see, e.g., =-=[3, 42, 48]-=- and references therein. Let us illustrate the main ideas of Taylor analysis on the case when we allow second order terms. In this case, the formula with a remainder takes the form f(x1, . . . , xn) =... |

28 | A new Cauchy-Based black-box technique for uncertainty in risk analysis. Reliability Engineering and Systems Safety
- Kreinovich, Ferson
- 2004
(Show Context)
Citation Context ...le to estimate ∆ faster? The answer is “yes”, it is possible to have an algorithm which estimates ∆ by using only a constant number of calls to the data processing algorithm f; for details, see, e.g. =-=[26, 27]-=-. In some situations, we need a guaranteed enclosure. In many application areas, it is sufficient to have an approximate estimate of y. However, in some applications, it is important to guarantee that... |

28 | forms – use and limits - Taylor - 2003 |

23 |
Automatic error analysis in digital computation
- Moore
- 1959
(Show Context)
Citation Context |

22 |
Computational complexity of the range of the polynomial in several variables
- Gaganov
- 1985
(Show Context)
Citation Context ...8, 9]. To be more precise, there is a polynomial-time algorithm for computing V , but computing V is, in general, NP-hard. Historical comment. NP-hardness of interval computations was first proven in =-=[10, 11]-=-. A general overview of computational complexity of different problems of data processing and interval computations is given in [28]. 18 n� i=1 xi � 2s7 Maximum Entropy and Linearization: Useful Techn... |

20 | Linear Computations - Dwyer - 1951 |

20 | The Double Bubble Conjecture - Hass, Hutchings, et al. - 1995 |

14 | 2005a. Exact bounds on finite populations of interval data - Ferson, Ginzburg, et al. |

13 | Models and Floating-Point Arithmetic: Proof that Arithmetic Operations are Validated
- Revol, Makino, et al.
- 2005
(Show Context)
Citation Context ... have mentioned, another way to get more accurate estimates is to use so-called Taylor techniques, i.e., to explicitly consider second-order and higher-order terms in the Taylor expansion; see, e.g., =-=[3, 42, 48]-=- and references therein. Let us illustrate the main ideas of Taylor analysis on the case when we allow second order terms. In this case, the formula with a remainder takes the form f(x1, . . . , xn) =... |

12 | Monte-Carlo-type techniques for processing interval uncertainty, and their potential engineering applications
- Kreinovich, Beck, et al.
(Show Context)
Citation Context ...is ≤ 2. Thus, instead of a single interval [30 − 5, 30 + 5] = [25, 35] that is guaranteed to contain the (unknown) age with certainty 100%, the expert also produces a narrower interval [30−4, 30+4] = =-=[26, 34]-=- which contains this age with certainty 90%, and an even narrower interval [30 − 2, 30 + 2] = [28, 32] which contains the age with certainty 70%. So, we have three intervals which are nested in the se... |

12 |
Automatic local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. Error
- Moore
- 1965
(Show Context)
Citation Context ...with 100% certainty), • an expert can also say that with 90% certainty, this inaccuracy is ≤ 4, and • with 70% certainty, this inaccuracy is ≤ 2. Thus, instead of a single interval [30 − 5, 30 + 5] = =-=[25, 35]-=- that is guaranteed to contain the (unknown) age with certainty 100%, the expert also produces a narrower interval [30−4, 30+4] = [26, 34] which contains this age with certainty 90%, and an even narro... |

12 |
Calculus of Approximations. Bulletin de l’Académie Polonaise de Sciences
- Warmus
- 1956
(Show Context)
Citation Context ...erval computations are about. Origins of interval computations. Interval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland =-=[54, 55]-=-, by T. Sunaga in Japan [51], and by R. Moore in the USA [33, 38, 34, 35, 36, 37, 39, 40]. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enha... |

11 |
forms – use and limits, Reliable Computing 9
- Neumaier, Taylor
- 2003
(Show Context)
Citation Context ... have mentioned, another way to get more accurate estimates is to use so-called Taylor techniques, i.e., to explicitly consider second-order and higher-order terms in the Taylor expansion; see, e.g., =-=[3, 42, 48]-=- and references therein. Let us illustrate the main ideas of Taylor analysis on the case when we allow second order terms. In this case, the formula with a remainder takes the form f(x1, . . . , xn) =... |

11 | Calculus of approximations - Warmus - 1956 |

10 | How far are we from complete knowledge: complexity of knowledge acquisition in Dempster-Shafer approach
- Chokr, Kreinovich
- 1994
(Show Context)
Citation Context ...x) · ln(ρ(x)) dx (here ρ(x) denotes the probability density). For details on this Maximum Entropy approach and its relation to interval uncertainty and Laplace’s principle of indifference, see, e.g., =-=[5, 21, 24]-=- Maximum entropy method for the case of interval uncertainty. One can easily check that for a single variable x1, among all distributions located on a given interval, the entropy is the largest when t... |

10 |
The automatic analysis and control of error in digital computation based on the use of interval numbers
- Moore
- 1965
(Show Context)
Citation Context ...is ≤ 2. Thus, instead of a single interval [30 − 5, 30 + 5] = [25, 35] that is guaranteed to contain the (unknown) age with certainty 100%, the expert also produces a narrower interval [30−4, 30+4] = =-=[26, 34]-=- which contains this age with certainty 90%, and an even narrower interval [30 − 2, 30 + 2] = [28, 32] which contains the age with certainty 70%. So, we have three intervals which are nested in the se... |

7 | Interval versions of statistical techniques, with applications to environmental analysis, bioinformatics, and privacy in statistical databases - Kreinovich, Longpré, et al. - 2007 |

6 |
K: The Contribution of T. Sunaga to Interval Analysis and Reliable Computing
- Markov, Okumura
- 1999
(Show Context)
Citation Context ...ach the Moon. Since then, interval computations have been actively used in many areas of science and engineering [19, 20]. Comment. An early history of interval computations is described in detail in =-=[31]-=- and in [41]; early papers on interval computations can be found on the interval computations website [19]. 9 Interval Computations: Main Techniques General comment about algorithms and parsing. Our g... |

6 | The algebra of multi-valued quantities - Young |

6 | eds.: Applications of Interval Computations - Kearfott, Kreinovich - 1996 |

5 |
of the Circle
- Archimedes
- 1980
(Show Context)
Citation Context ...matics. The notion of interval computations is reasonably recent, it dates from the 1950s, but the main problem is known since Archimedes who used guaranteed two-sided bounds to compute π; see, e.g., =-=[1]-=-. Since then, many useful guaranteed bounds have been developed for different numerical methods. There have also been several general descriptions of such bounds, often formulated in terms similar to ... |

5 |
Approximations and inequalities in the calculus of approximations. classification of approximate numbers
- Warmus
- 1961
(Show Context)
Citation Context ...erval computations are about. Origins of interval computations. Interval computations were independently invented by three researchers in three different parts of the world: by 24sM. Warmus in Poland =-=[54, 55]-=-, by T. Sunaga in Japan [51], and by R. Moore in the USA [33, 38, 34, 35, 36, 37, 39, 40]. The active interest in interval computations started with Moore’s 1966 monograph [39]. This interest was enha... |

4 |
The double bubble conjecture, Electron
- Hass, Hutchings, et al.
- 1995
(Show Context)
Citation Context ...y between them in the disk). The actual proof required proving that for this configuration, the area is indeed larger than for all possible other configurations. This proof was done by Haas et al. in =-=[13]-=- who computed an interval enclosure [Y , Y ] for all other configurations, and showed that Y is smaller than the area Y0 corresponding to the double bubble. Another well-known example is the Kepler’s ... |

4 |
An Improved Algorithm for Computing the Product of Two
- Heindl
- 1993
(Show Context)
Citation Context ...imes increase in computation time. Computational comment: interval multiplication can be performed faster. It is known that we can compute the interval product faster, by using only 3 multiplications =-=[18, 16]-=-. Namely: • if x 1 ≥ 0 and x 2 ≥ 0, then x1 · x2 = [x 1 · x 2, x1 · x2]; • if x 1 ≥ 0 and x 2 ≤ 0 ≤ x2, then x1 · x2 = [x1 · x 2, x1 · x2]; • if x 1 ≥ 0 and x2 ≤ 0, then x1 · x2 = [x1 · x 2 , x 1 · x2... |