Random Sets pp 129-164 | Cite as

Random Sets in Information Fusion an Overview

  • Ronald P. S. Mahler
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 97)


“Information fusion” refers to a range of military applications requiring the effective pooling of data concerning multiple targets derived from a broad range of evidence types generated by diverse sensors/sources. Methodologically speaking, information fusion has two major aspects: multisource, multitarget estimation and inference using ambiguous observations (expert systems theory). In recent years some researchers have begun to investigate random set theory as a foundation for information fusion. This paper offers a brief history of the application of random sets to information fusion, especially the work of Mori et. al., Washburn, Goodman, and Nguyen. It also summarizes the author’s recent work suggesting that random set theory provides a systematic foundation for both multisource, multitarget estimation and expert-systems theory. The basic tool is a statistical theory of random finite sets which directly generalizes standard single-sensor, single-target statistics: density functions, a theory of differential and integral calculus for set functions, etc.

Key words

Data Fusion Random Sets Nonadditive Measure Expert Systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R.T. AntonyPrinciples of Data Fusion AutomationArtech House, Dedham, Massachusetts, 1995.Google Scholar
  2. [2]
    C.A. Barlow, L.D. Stone, and M.V. FinnUnified data fusionProceedings of the 9th National Symposium on Sensor Fusion, vol. I (Unclassified), Naval Postgraduate School, Monterey CA, March 11–13, 1996.Google Scholar
  3. [3]
    Y. Bar-Shalom and T.E. FortmannTracking and Data AssociationAcademic Press, New York City, New York, 1988.zbMATHGoogle Scholar
  4. [4]
    Y. Bar-Shalom and X.-R. LiEstimation and Tracking: Principles Techniques and SoftwareArtech House, Dedham, Massachusetts, 1993.Google Scholar
  5. [5]
    S.S. BlackmanMultiple-Target Tracking with Radar ApplicationsArtech House, Dedham MA, 1986.Google Scholar
  6. [6]
    P.P. Bonissone and N.C. WoodT-norm based reasoning in situation assessment applicationsUncertainty in Artificial Intelligence (L.N. Kanal, T.S. Levitt, and J.F. Lemmer, eds.), vol. 3, New York City, New York: Elsevier Publishers, 1989, pp. 241–256.Google Scholar
  7. [7]
    D.J. Daley and D. Vere-Jones, AnIntroduction to the Theory of Point Pro-cessesSpringer-Verlag, New York City, New York, 1988.Google Scholar
  8. [8]
    I.R. GoodmanFuzzy sets as equivalence classes of random setsFuzzy Sets and Possibility Theory (R. Yager, ed.), Pergamon Press, 1982, pp. 327–343.Google Scholar
  9. [9]
    I.R. GoodmanA new characterization of fuzzy logic operators producing homomorphic-like relations with one-point coverages of random setsAd-vances in Fuzzy Theory and Technology, vol. II, (P.P. Wang, ed.), Duke Uni-versity, Durham, NC, 1994, pp. 133–159.Google Scholar
  10. [10]
    I.R. GoodmanPact: An approach to combining linguistic-based and probabilistic information in correlation and trackingTech. Rep. 878, Naval Ocean Com-mand and Control Ocean Systems Center, RDT&E Division, San Diego, Cali-fornia, March 1986; andA revised approach to combining linguistic and prob-abilistic information in correlationTech. Rep. 1386, Naval Ocean Command and Control Ocean Systems Center, RDT&E Division, San Diego, California, July 1992.Google Scholar
  11. [11]
    I.R. GoodmanToward a comprehensive theory of linguistic and probabilistic ev-idence: Two new approaches to conditional event algebraIEEE Transactions on Systems, Man and Cybernetics, 24 (1994), pp. 1685–1698.CrossRefGoogle Scholar
  12. [12]
    I.R. Goodman, Aunified approach to modeling and combining of evidence through random set theoryProceedings of the 6thMIT/ONR Workshop on C3 Systems, Massachusetts Institute of Technology, Cambridge, MA, pp. 42–47.Google Scholar
  13. [13]
    I.R. Goodman and H.T. NguyenUncertainty Models for Knowledge Based Sys-temsNorth-Holland, Amsterdam, The Netherlands, 1985.Google Scholar
  14. [14]
    I.R. Goodman, H.T. Nguyen, and E.A. WalkerConditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free ConditioningNorth-Holland, Amsterdam, The Netherlands, 1991.Google Scholar
  15. [15]
    M. Grabisch, H.T. Nguyen, and E.A. WalkerFundamentals of Uncertainty Calculi With Applications to Fuzzy InferenceKluwer Academic Publishers, Dordrecht, The Netherlands, 1995.Google Scholar
  16. [16]
    S. Graf, ARadon-Nikodÿm theorem for capacitiesJournal für Reine and Ange-wandte Mathematik, 320 (1980), pp. 192–214.MathSciNetzbMATHGoogle Scholar
  17. [17]
    D.L. HallMathematical Techniques in Multisensor Data FusionArtech House, Dedham, Massachusetts, 1992.Google Scholar
  18. [18]
    K. Hestir, H.T. Nguyen, and G.S. RogersA random set formalism for evi-dential reasoningConditional Logic in Expert Systems (I.R. Goodman, M.M. Gupta, H.T. Nguyen and G.S. Rogers, eds.), Amsterdam, The Netherlands: North-Holland, 1991, pp. 309–344.Google Scholar
  19. [19]
    T.L. HillStatistical Mechanics: Principles and Selected ApplicationsDover Pub-lications, New York City, New York, 1956.zbMATHGoogle Scholar
  20. [20]
    P.B. KantorOrbit space and closely spaced targetsProceedings of the SDI Panels on Tracking, no. 2, 1991.Google Scholar
  21. [21]
    P.J. Huber and V. StrassenMinimax tests and the Neyman-Pearson lemma for capacitiesAnnals of Statistics, 1 (1973), pp. 251–263.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    K. Kastella and C. LutesCoherent maximum likelihood estimation and mean—field theory in multi—target trackingProceedings of the 6t hJoint Service Data Fusion Symposium, vol. I (Part 2), Johns Hopkins Applied Physics Laboratory, Laurel, MD, June 14–18, 1993, pp. 971–982.Google Scholar
  23. [23]
    R. Kruse and K.D. MeyerStatistics with Vague DataD. Reidel/Kluwer Academic Publishers, Dordrecht, The Netherlands, 1987.CrossRefGoogle Scholar
  24. [24]
    R. Kruse, E. Schwencke, and J. HeinsohnUncertainty and Vagueness in Knowledge—Based SystemsSpringer—Verlag, New York City, New York, 1991.CrossRefGoogle Scholar
  25. [25]
    D. LewisProbabilities of conditionals and conditional probabilitiesPhilosophical Review, 85 (1976), pp. 297–315.CrossRefGoogle Scholar
  26. [26]
    Y. LiProbabilistic Interpretations of Fuzzy Sets and SystemsPh.D. Thesis, De-partment of Electrical Engineering and Computer Science, Massachusetts In-stitute of Technology, Cambridge, MA, 1994.Google Scholar
  27. [27]
    N.N. LyshenkoStatistics of random compact sets in Euclidean spaceJournal of Soviet Mathematics, 21 (1983), pp. 76–92.CrossRefGoogle Scholar
  28. [28]
    R.P.S. MahlerCombining ambiguous evidence with respect to ambiguous a priori knowledge. Part II: Fuzzy logicFuzzy Sets and Systems, 75 (1995), pp. 319–354.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    R.P.S. MahlerGlobal integrated data fusionProceedings of the 7th National Symposium on Sensor Fusion, vol. I (Unclassified), March 16–18, 1994, Sandia National Laboratories, Albuquerque, NM, pp. 187–199.Google Scholar
  30. [30]
    R.P.S. MahlerGlobal optimal sensor allocationProceedings of the 9th National Symposium on Sensor Fusion, vol. I (Unclassified), Naval Postgraduate School, Monterey CA, March 11–13, 1996.Google Scholar
  31. [31]
    R.P.S. MahlerInformation theory and data fusionProceedings of the 8th National Symposium on Sensor Fusion, vol. I (Unclassified), Texas Instruments, Dallas, TX, March 17–19, 1995, pp. 279–292.Google Scholar
  32. [32]
    R.P.S. MahlerNonadditive probability finite—set statistics and information fusionProceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, December 1995, pp. 1947–1952.Google Scholar
  33. [33]
    R.P.S. MahlerThe random—set approach to data fusionSPIE Proceedings, vol. 2234, 1994, pp. 287–295.CrossRefGoogle Scholar
  34. [34]
    R.P.S. MahlerRepresenting rules as random sets. I: Statistical correlations between rulesInformation Sciences, 88 (1996), pp. 47–68.MathSciNetCrossRefGoogle Scholar
  35. [35]
    R.P.S. MahlerRepresenting rules as random sets. II: Iterated rulesInternational Journal of Intelligent Systems, 11 (1996), pp. 583–610.zbMATHCrossRefGoogle Scholar
  36. [36]
    R.P.S. MahlerUnified data fusion: Fuzzy logic evidence and rulesSPIE Proceedings, 2755 (1996), pp. 226–237.CrossRefGoogle Scholar
  37. [37]
    R.P.S. MahlerUnified nonparametric data fusionSPIE Proceedings, vol. 2484, 1995, pp. 66–74.CrossRefGoogle Scholar
  38. [38]
    G. MathüronRandom Sets and Integral GeometryJohn Wiley, New York City, New York, 1975.Google Scholar
  39. [39]
    I.S. MolchanovLimit Theorems for Unions of Random Closed SetsSpringer—Verlag Lecture Notes in Mathematics, vol. 1561, Springer—Verlag, Berlin, Germany, 1993.Google Scholar
  40. [40]
    S. Mori, C.-Y. Chong, E. TSE, and R.P. WishnerMultitarget multisensor tracking problems. Part I: A general solution and a unified view on Bayesian approachesRevised Version, Tech. Rep. TR-1048–01 Advanced Information and Decision Systems, Inc. Mountain View CAAugust1984. My thanks to Dr. Mori for making this report available to me (Dr. Shozo Mori, Personal Communication, February 28, 1995).Google Scholar
  41. [41]
    S. Mori, C.-Y. Chong, E. TSE, and R.P. WishnerTracking and classifying multiple targets without a priori identificationIEEE Transactions on Auto-matic Control, 31 (1986), pp. 401–409.zbMATHCrossRefGoogle Scholar
  42. [42]
    R.E. Neapolitan, Asurvey of uncertain and approximate inferenceFuzzy Logic for the Management of Uncertainty (L. Zadeh andJ.Kocprzyk, eds.), New York City, New York: John Wiley, 1992.Google Scholar
  43. [43]
    H.T. NguyenOn random sets and belief functionsJournal of Mathematical Analysis and Applications, 65 (1978), pp. 531–542.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    A.I. OrlovRelationships between fuzzy and random sets: Fuzzy tolerancesIssledovania po Veroyatnostnostatishesk. Modelirovaniu Realnikh System, 1977, Moscow, Union of Soviet Socialist Republics.Google Scholar
  45. [45]
    A.I. OrlovFuzzy and random setsPrikladnoi Mnogomerni Statisticheskii Analys, 1978, Moscow, Union of Soviet Socialist Republics.Google Scholar
  46. [46]
    P. Quinio and T. MatsuyamaRandom closed sets: A unified approach to the representation of imprecision anduncertainty, Symbolic and Quantitative Ap-proaches to Uncertainty (R. Kruse and P. Siegel, eds.), New York City, New York: Springer—Verlag, 1991, pp. 282–286.CrossRefGoogle Scholar
  47. [47]
    D.B. ReidAn algorithm for tracking multiple targetsIEEE Transactions on Au-tomatic Control, 24 (1979), pp. 843–854.CrossRefGoogle Scholar
  48. [48]
    B. Schweizer and A.SklarProbabilistic Metric SpacesNorth—Holland, Ams-terdam, The Netherlands, 1983.zbMATHGoogle Scholar
  49. [49]
    G.E. Shilov and B.L. GurevichIntegral Measure and Derivative: A Unified ApproachPrentice—Hall, New York City, New York, 1966.Google Scholar
  50. [50]
    P. Smets The transferable belief model and random setsInternational Journal of Intelligent Systems, 7 (1992), pp. 37–46.zbMATHCrossRefGoogle Scholar
  51. [51]
    L.D. Stone, M.V. Finn, and C.A. BarlowUnified data fusionTech. Rep., Metron Corp., January 26, 1996.Google Scholar
  52. [52]
    J.K. UhlmannAlgorithms for multiple—target trackingAmerican Scientist, 80 (1992), pp. 128–141.Google Scholar
  53. [53]
    B.C. Van FraassenProbabilities of conditionalsFoundations of Probability Theory, Statistical Inference, and Statistical Theories of Science (W.L. Harper and E.A. Hooker, eds.), vol. I, Dordrecht, The Netherlands: D. Reidel, 1976, pp. 261–308.CrossRefGoogle Scholar
  54. [54]
    E. Waltz and J. LianasMultisensor Data FusionArtech House, Dedham, Mas-sachusetts, 1990.Google Scholar
  55. [55]
    R.B. Washburn, Arandom point process approach to multiobject trackingPro-ceedings of the American Control Conference, vol. 3, June 10–12, 1987, Min-neapolis, Minnesota, pp. 1846–1852.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ronald P. S. Mahler
    • 1
  1. 1.Lockheed Martin Tactical Defense SystemsEaganUSA

Personalised recommendations