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Volumn , Issue , 2008, Pages 1-31

Interval Computation as an Important Part of Granular Computing: An Introduction

Author keywords

Constraint satisfaction and optimization; Data processing notion; Handling measurement uncertainty; Interval computation; Interval computation problem; Interval to fuzzy uncertainty; Measuring difficult to measure quantities; Real life computations

Indexed keywords


EID: 84889374621     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470724163.ch1     Document Type: Chapter
Times cited : (26)

References (57)
  • 1
    • 0003553371 scopus 로고    scopus 로고
    • Computational Complexity and Feasibility of Data Processing and Interval Computations
    • Kluwer, Dordrecht
    • V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl. Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht, 1997.
    • (1997)
    • Kreinovich, V.1    Lakeyev, A.2    Rohn, J.3    Kahl, P.4
  • 2
    • 84892299362 scopus 로고    scopus 로고
    • Measurement Errors and Uncertainty. Theory and Practice
    • Springer-Verlag, Berlin
    • S.G. Rabinovich. Measurement Errors and Uncertainty. Theory and Practice. Springer-Verlag, Berlin, 2005.
    • (2005)
    • Rabinovich, S.G.1
  • 3
    • 0003471143 scopus 로고
    • Fuzzy Sets and Fuzzy Logic: Theory and Applications
    • Prentice Hall, Upper Saddle River, NJ
    • G. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, NJ, 1995.
    • (1995)
    • Klir, G.1    Yuan, B.2
  • 4
    • 0003766367 scopus 로고    scopus 로고
    • A First Course in Fuzzy Logic
    • CRC Press, Boca Raton, FL
    • H.T. Nguyen and E.A. Walker. A First Course in Fuzzy Logic. CRC Press, Boca Raton, FL, 2005.
    • (2005)
    • Nguyen, H.T.1    Walker, E.A.2
  • 5
    • 49349119632 scopus 로고
    • A note on the extension principle for fuzzy sets
    • H.T. Nguyen. A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64 (1978) 359-380.
    • (1978) J. Math. Anal. Appl. , vol.64 , pp. 359-380
    • Nguyen, H.T.1
  • 6
    • 0003603813 scopus 로고
    • Computers and Intractability: A Guide to the Theory of NP-Completeness
    • W.F. Freeman, San Francisco, CA
    • M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.F. Freeman, San Francisco, CA, 1979.
    • (1979)
    • Garey, M.R.1    Johnson, D.S.2
  • 7
    • 0003897694 scopus 로고
    • Computational Complexity
    • Addison-Wesley, Reading, MA
    • C.H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, MA, 1994.
    • (1994)
    • Papadimitriou, C.H.1
  • 8
    • 0003523947 scopus 로고
    • Nonlinear Optimization: Complexity Issues
    • Oxford University Press, New York
    • S. Vavasis. Nonlinear Optimization: Complexity Issues. Oxford University Press, New York, 1991.
    • (1991)
    • Vavasis, S.1
  • 11
    • 0011844006 scopus 로고
    • Computational Complexity of the Range of the Polynomial in Several Variables
    • M.S. Thesis. Mathematics Department, Leningrad University, Leningrad, USSR
    • A.A. Gaganov. Computational Complexity of the Range of the Polynomial in Several Variables. M.S. Thesis. Mathematics Department, Leningrad University, Leningrad, USSR, 1981.
    • (1981)
    • Gaganov, A.A.1
  • 12
    • 0022091645 scopus 로고
    • Computational complexity of the range of the polynomial in several variables
    • A.A. Gaganov, Computational complexity of the range of the polynomial in several variables. Cybernetics, Vol. 21, (1985) 418-421.
    • (1985) Cybernetics , vol.21 , pp. 418-421
    • Gaganov, A.A.1
  • 13
    • 0003919677 scopus 로고    scopus 로고
    • Monte Carlo Statistical Methods
    • Springer-Verlag, New York
    • C.P. Robert and G. Casella. Monte Carlo Statistical Methods. Springer-Verlag, New York, 2004.
    • (2004)
    • Robert, C.P.1    Casella, G.2
  • 14
    • 27144484205 scopus 로고
    • How far are we from the complete knowledge: complexity of knowledge acquisition in Dempster-Shafer approach
    • In: R.R. Yager, J. Kacprzyk, and M. Pedrizzi (eds) , Wiley, New York
    • B. Chokr and V. Kreinovich. How far are we from the complete knowledge: complexity of knowledge acquisition in Dempster-Shafer approach. In: R.R. Yager, J. Kacprzyk, and M. Pedrizzi (eds) . Advances in the Dempster-Shafer Theory of Evidence. Wiley, New York, 1994, pp. 555-576.
    • (1994) Advances in the Dempster-Shafer Theory of Evidence , pp. 555-576
    • Chokr, B.1    Kreinovich, V.2
  • 15
    • 0003998828 scopus 로고    scopus 로고
    • Probability Theory: The Logic of Science
    • Cambridge University Press, Cambridge, UK
    • E. T. Jaynes and G.L. Bretthorst. Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, UK, 2003.
    • (2003)
    • Jaynes, E.T.1    Bretthorst, G.L.2
  • 16
    • 84992631062 scopus 로고    scopus 로고
    • Uncertainty and Information: Foundations of Generalized Information Theory
    • Wiley, Hoboken, NJ
    • G.J. Klir. Uncertainty and Information: Foundations of Generalized Information Theory. Wiley, Hoboken, NJ, 2005.
    • (2005)
    • Klir, G.J.1
  • 17
    • 0003683294 scopus 로고    scopus 로고
    • Handbook of Parametric and Nonparametric Statistical Procedures
    • Chapman & Hall/CRC, Boca Raton, FL
    • D.J. Sheskin. Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC, Boca Raton, FL, 2004.
    • (2004)
    • Sheskin, D.J.1
  • 18
    • 0003928308 scopus 로고
    • Handbook of Statistical Methods for Engineers and Scientists
    • (ed.), McGraw-Hill, New York
    • H.M. Wadswort (ed.) . Handbook of Statistical Methods for Engineers and Scientists. McGraw-Hill, New York, 1990.
    • (1990)
    • Wadswort, H.M.1
  • 19
    • 33845335842 scopus 로고    scopus 로고
    • Monte-Carlo-type techniques for processing interval uncertainty, and their potential engineering applications
    • V. Kreinovich, J. Beck, C. Ferregut, A. Sanchez, G.R. Keller, M. Averill, and S.A. Starks. Monte-Carlo-type techniques for processing interval uncertainty, and their potential engineering applications. Reliab. Comput. 13(1) (2007) 25-69.
    • (2007) Reliab. Comput. , vol.13 , Issue.1 , pp. 25-69
    • Kreinovich, V.1    Beck, J.2    Ferregut, C.3    Sanchez, A.4    Keller, G.R.5    Averill, M.6    Starks, S.A.7
  • 20
    • 2642529400 scopus 로고    scopus 로고
    • A new Cauchy-based black-box technique for uncertainty in risk analysis
    • V. Kreinovich and S. Ferson. A new Cauchy-based black-box technique for uncertainty in risk analysis. Reliab. Eng. Syst. Saf. 85(1-3) (2004) 267-279.
    • (2004) Reliab. Eng. Syst. Saf. , vol.85 , Issue.1-3 , pp. 267-279
    • Kreinovich, V.1    Ferson, S.2
  • 21
    • 84889362548 scopus 로고
    • The Works of Archimedes
    • Archimedes. On the measurement of the circle. In: T.L. Heath (ed.), Cambridge University Press, Cambridge, 1897; Dover edition
    • Archimedes. On the measurement of the circle. In: T.L. Heath (ed.) , The Works of Archimedes. Cambridge University Press, Cambridge, 1897; Dover edition, 1953, pp. 91-98.
    • (1953) , pp. 91-98
  • 22
    • 0347484652 scopus 로고
    • Sull due funzioni a piu valori constituite dai limiti d'una funzione di variable reale a destra ed a sinistra di ciascun punto
    • W.H. Young. Sull due funzioni a piu valori constituite dai limiti d'una funzione di variable reale a destra ed a sinistra di ciascun punto. Rend. Acad. Lincei CI. Sei. Fes. 17(5) (1908) 582-587.
    • (1908) Rend. Acad. Lincei CI. Sei. Fes. , vol.17 , Issue.5 , pp. 582-587
    • Young, W.H.1
  • 23
    • 0001927944 scopus 로고
    • The algebra of multi-valued quantities
    • R.C. Young. The algebra of multi-valued quantities. Math. Ann. 104 (1931) 260-290.
    • (1931) Math. Ann. , vol.104 , pp. 260-290
    • Young, R.C.1
  • 24
    • 0004246193 scopus 로고
    • Linear Computations
    • Wiley, New York
    • P.S. Dwyer. Linear Computations. Wiley, New York, 1951.
    • (1951)
    • Dwyer, P.S.1
  • 25
    • 0013414995 scopus 로고
    • Calculus of approximations
    • M. Warmus. Calculus of approximations. Bull. Acad. Pol. sei. 4(5) (1956) 253-257.
    • (1956) Bull. Acad. Pol. sei. , vol.4 , Issue.5 , pp. 253-257
    • Warmus, M.1
  • 26
    • 0342879201 scopus 로고
    • Approximations and inequalities in the calculus of approximations. Classification of approximate numbers
    • M. Warmus. Approximations and inequalities in the calculus of approximations. Classification of approximate numbers. Bull. Acad. Polon. Sei., Ser. Sei. Math. Astron. Phys. 9 (1961) 241-245.
    • (1961) Bull. Acad. Polon. Sei., Ser. Sei. Math. Astron. Phys. , vol.9 , pp. 241-245
    • Warmus, M.1
  • 27
    • 0002038715 scopus 로고
    • Theory of interval algebra and its application to numerical analysis
    • In:, Research Association of Applied Geometry (RAAG) , Tokyo, Japan, 1958
    • T. Sunaga. Theory of interval algebra and its application to numerical analysis. In: RAAG Memoirs, Ggujutsu Bunken Fukuy-kai. Research Association of Applied Geometry (RAAG) , Tokyo, Japan, 1958, Vol. 2, 1958, pp. 29-46 (547-564) .
    • (1958) RAAG Memoirs, Ggujutsu Bunken Fukuy-kai , vol.2
    • Sunaga, T.1
  • 28
    • 0003642009 scopus 로고
    • Automatic Error Analysis in Digital Computation
    • Technical Report, Space Div. Report LMSD84821. Lockheed Missiles and Space Co., Sunnyvale, California
    • R.E. Moore. Automatic Error Analysis in Digital Computation. Technical Report, Space Div. Report LMSD84821. Lockheed Missiles and Space Co., Sunnyvale, California, 1959.
    • (1959)
    • Moore, R.E.1
  • 29
    • 0003406326 scopus 로고
    • Interval Arithmetic and Automatic Error Analysis in Digital Computing
    • Ph.D. Dissertation. Department of Mathematics, Stanford University, Stanford, CA,, Published as Applied Mathematics and Statistics Laboratories Technical Report No. 25.
    • R.E. Moore. Interval Arithmetic and Automatic Error Analysis in Digital Computing. Ph.D. Dissertation. Department of Mathematics, Stanford University, Stanford, CA, 1962. Published as Applied Mathematics and Statistics Laboratories Technical Report No. 25.
    • (1962)
    • Moore, R.E.1
  • 30
    • 0041061758 scopus 로고
    • The automatic analysis and control of error in digital computing based on the use of interval numbers
    • In: L.B. Rail (ed.) ,, Wiley, New York
    • R.E. Moore. The automatic analysis and control of error in digital computing based on the use of interval numbers. In: L.B. Rail (ed.) , Error in Digital Computation. Wiley, New York, 1965, Vol. I, pp. 61-130.
    • (1965) Error in Digital Computation , vol.I , pp. 61-130
    • Moore, R.E.1
  • 31
    • 0041061754 scopus 로고
    • Automatic local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations
    • In: L.B. Rail (ed.) ,, Wiley, New York
    • R.E. Moore. Automatic local coordinate transformations to reduce the growth of error bounds in interval computation of solutions of ordinary differential equations. In: L.B. Rail (ed.) , Error in Digital Computation. Wiley, New York, 1965, Vol. II, pp. 103-140.
    • (1965) Error in Digital Computation , vol.II , pp. 103-140
    • Moore, R.E.1
  • 32
    • 78651262985 scopus 로고
    • Interval Integrals
    • Technical Report, Space Div. Report LMSD703073, Lockheed Missiles and Space Co.
    • R.E. Moore, W. Strother, and C.T. Yang. Interval Integrals. Technical Report, Space Div. Report LMSD703073, Lockheed Missiles and Space Co., 1960.
    • (1960)
    • Moore, R.E.1    Strother, W.2    Yang, C.T.3
  • 33
    • 18944392689 scopus 로고
    • Interval Analysis I
    • Technical Report, Space Div. Report LMSD285875. Lockheed Missiles and Space Co.
    • R.E. Moore and C.T. Yang. Interval Analysis I. Technical Report, Space Div. Report LMSD285875. Lockheed Missiles and Space Co., 1959.
    • (1959)
    • Moore, R.E.1    Yang, C.T.2
  • 34
    • 0004293209 scopus 로고
    • Interval Analysis
    • Prentice Hall, Englewood Cliffs, NJ
    • R.E. Moore. Interval Analysis. Prentice Hall, Englewood Cliffs, NJ, 1966.
    • (1966)
    • Moore, R.E.1
  • 35
    • 0003717416 scopus 로고
    • Methods and Applications of Interval Analysis
    • SIAM, Philadelphia
    • R.E. Moore. Methods and Applications of Interval Analysis. SIAM, Philadelphia, 1979.
    • (1979)
    • Moore, R.E.1
  • 36
    • 84889397443 scopus 로고    scopus 로고
    • Interval computations Web site, Helvetica
    • Interval computations Web site, Helveticahttp://www.cs.utep.edu/interval-comp, 2008.
    • (2008)
  • 37
    • 0003440056 scopus 로고    scopus 로고
    • Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics
    • Springer-Verlag, London
    • L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics. Springer-Verlag, London, 2001.
    • (2001)
    • Jaulin, L.1    Kieffer, M.2    Didrit, O.3    Walter, E.4
  • 38
    • 0013364106 scopus 로고    scopus 로고
    • The contribution of T. Sunaga to interval analysis and reliable computing
    • In: T. Csendes (ed.) ,, Kluwer, Dordrecht
    • S. Markov and K. Okumura. The contribution of T. Sunaga to interval analysis and reliable computing. In: T. Csendes (ed.) , Developments in Reliable Computing. Kluwer, Dordrecht, 1999, pp. 167-188.
    • (1999) Developments in Reliable Computing , pp. 167-188
    • Markov, S.1    Okumura, K.2
  • 39
    • 34547686767 scopus 로고    scopus 로고
    • The dawning
    • R.E. Moore, The dawning. Reliab. Comput. 5 (1999) 423-424.
    • (1999) Reliab. Comput. , vol.5 , pp. 423-424
    • Moore, R.E.1
  • 40
    • 84889280793 scopus 로고
    • An Improved Algorithm for Computing the Product of Two Machine Intervals
    • Interner Bericht IAGMPI-9304. Fachbereich Mathematik, Gesamthochschule Wuppertal
    • G. Heindl. An Improved Algorithm for Computing the Product of Two Machine Intervals. Interner Bericht IAGMPI-9304. Fachbereich Mathematik, Gesamthochschule Wuppertal, 1993.
    • (1993)
    • Heindl, G.1
  • 42
    • 0003088011 scopus 로고    scopus 로고
    • Computation and application of Taylor polynomials with interval remainder bounds
    • M. Berz and G. Hoffstätter. Computation and application of Taylor polynomials with interval remainder bounds. Reliab. Comput. 4(1) (1998) 83-97.
    • (1998) Reliab. Comput. , vol.4 , Issue.1 , pp. 83-97
    • Berz, M.1    Hoffstätter, G.2
  • 43
    • 0037292825 scopus 로고    scopus 로고
    • Taylor forms-use and limits
    • A. Neumaier. Taylor forms-use and limits. Reliab. Comput. 9 (2002) 43-79.
    • (2002) Reliab. Comput. , vol.9 , pp. 43-79
    • Neumaier, A.1
  • 44
    • 17644369306 scopus 로고    scopus 로고
    • Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY
    • N. Revol, K. Makino, and M. Berz. Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY. J. Log. Algebr. Program. 64(1) (2005) 135-154.
    • (2005) J. Log. Algebr. Program. , vol.64 , Issue.1 , pp. 135-154
    • Revol, N.1    Makino, K.2    Berz, M.3
  • 45
    • 0004217246 scopus 로고    scopus 로고
    • Applications of Interval Computations
    • (eds), Kluwer, Dordrecht
    • R.B. Kearfott and V. Kreinovich (eds) . Applications of Interval Computations. Kluwer, Dordrecht, 1996.
    • (1996)
    • Kearfott, R.B.1    Kreinovich, V.2
  • 46
    • 0003560312 scopus 로고
    • Numerical Toolbox for Verified Computing. I. Basic Numerical Problems
    • Springer-Verlag, Heidelberg, New York
    • R. Hammer, M. Hocks, U. Kulisch, and D. Ratz. Numerical Toolbox for Verified Computing. I. Basic Numerical Problems. Springer-Verlag, Heidelberg, New York, 1993.
    • (1993)
    • Hammer, R.1    Hocks, M.2    Kulisch, U.3    Ratz, D.4
  • 47
    • 0004121716 scopus 로고
    • New Tools for Robustness of Linear Systems
    • McMillan, New York
    • B.R. Barmish. New Tools for Robustness of Linear Systems. McMillan, New York, 1994.
    • (1994)
    • Barmish, B.R.1
  • 48
    • 0003717399 scopus 로고
    • Robust Control: The Parametric Approach
    • Prentice-Hall, Engle-wood Cliffs, NJ
    • S.R Bhattacharyya, H. Chapellat, and L. Keel. Robust Control: The Parametric Approach. Prentice-Hall, Engle-wood Cliffs, NJ, 1995.
    • (1995)
    • Bhattacharyya, S.R.1    Chapellat, H.2    Keel, L.3
  • 49
    • 0003571608 scopus 로고    scopus 로고
    • Global Optimization Using Internal Analysis
    • MIT Press, Cambridge, MA
    • E.R. Hansen and G.W. Walster. Global Optimization Using Internal Analysis. MIT Press, Cambridge, MA, 2004.
    • (2004)
    • Hansen, E.R.1    Walster, G.W.2
  • 50
    • 0004044405 scopus 로고    scopus 로고
    • Rigorous Global Search: Continuous Problems
    • Kluwer, Dordrecht
    • R.B. Kearfott. Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht, 1996.
    • (1996)
    • Kearfott, R.B.1
  • 52
    • 22044441063 scopus 로고    scopus 로고
    • A proof of the Kepler conjecture
    • T.C. Hales. A proof of the Kepler conjecture. Ann. Math. 162 (2005) 1065-1185.
    • (2005) Ann. Math. , vol.162 , pp. 1065-1185
    • Hales, T.C.1
  • 53
    • 84889451030 scopus 로고    scopus 로고
    • Risk Assessment with Uncertainty Numbers: Risk Calc
    • CRC Press, Boca Raton, FL
    • S. Ferson. Risk Assessment with Uncertainty Numbers: Risk Calc. CRC Press, Boca Raton, FL, 2002.
    • (2002)
    • Ferson, S.1
  • 54
    • 33749650825 scopus 로고    scopus 로고
    • Interval versions of statistical techniques, with applications to environmental analysis, bioinformatics, and privacy in statistical databases
    • V. Kreinovich, L. Longpré, S.A. Starks, G. Xiang, J. Beck, R. Kandathi, A. Nayak, S. Ferson, and J. Hajagos. Interval versions of statistical techniques, with applications to environmental analysis, bioinformatics, and privacy in statistical databases. J. Comput. Appl. Math. 199(2) (2007) 418-123.
    • (2007) J. Comput. Appl. Math. , vol.199 , Issue.2 , pp. 418-123
    • Kreinovich, V.1    Longpré, L.2    Starks, S.A.3    Xiang, G.4    Beck, J.5    Kandathi, R.6    Nayak, A.7    Ferson, S.8    Hajagos, J.9
  • 55
    • 33750717212 scopus 로고    scopus 로고
    • Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity
    • V. Kreinovich, G. Xiang, S.A. Starks, L. Longpré, M. Ceberio, R. Araiza, J. Beck, R. Kandathi, A. Nayak, R. Torres, and J. Hajagos. Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity. Reliab. Comput. 12(6) (2006) 471-501.
    • (2006) Reliab. Comput. , vol.12 , Issue.6 , pp. 471-501
    • Kreinovich, V.1    Xiang, G.2    Starks, S.A.3    Longpré, L.4    Ceberio, M.5    Araiza, R.6    Beck, J.7    Kandathi, R.8    Nayak, A.9    Torres, R.10    Hajagos, J.11
  • 56
    • 0003698514 scopus 로고    scopus 로고
    • Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions
    • Prentice Hall, Englewood Cliffs, NJ
    • J.M. Mendel. Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice Hall, Englewood Cliffs, NJ, 2001.
    • (2001)
    • Mendel, J.M.1
  • 57
    • 0012016870 scopus 로고    scopus 로고
    • Granular Computing: An Emerging Paradigm
    • (ed.), Springer-Verlag. New York
    • W. Pedrycz (ed.) . Granular Computing: An Emerging Paradigm. Springer-Verlag. New York, 2001.
    • (2001)
    • Pedrycz, W.1


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