Skip to main content

Introduction to Scan Statistics

  • Chapter
Scan Statistics and Applications

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

In this chapter, we define discrete and continuous scan statistics in one-dimensional as well as multidimensional cases. We then mention some related applications and open problems. We also present a brief account of order statistics which naturally arise in the study of scan statistics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ajne, B. (1968). A simple test for uniformity of a circular distributionBiometrika 55343–354.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alm, S. E. (1983). On the distribution of the scan statistic of a Poisson processProbability and Mathematical Statistics Essays in Honour of Carl-Gustav Esseen1–10.

    Google Scholar 

  3. Alm, S. E. (1997). On the distribution of scan statistics of a two-dimensional Poisson processAdvances in Applied Probability 291–18.

    Article  MathSciNet  MATH  Google Scholar 

  4. Alm, S. E. (1998). Approximation and simulation of the distributions of scan statistics for Poisson process in higher dimensionsExtremes 1111–126.

    Article  MathSciNet  MATH  Google Scholar 

  5. Altschul, S. F. and Erickson, B. W. (1988). Significance levels for biological sequence comparisons using non-linear similarity functionsBulletin of Mathematical Biology 5077–92.

    MATH  Google Scholar 

  6. Arnold, B. C. and Balakrishnan, N. (1989).Relations Bounds and Approximations for Order StatisticsLecture Notes in Statistics53Springer-Verlag, New York.

    Google Scholar 

  7. Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). AFirst Course in Order StatisticsJohn Wiley&Sons, New York.

    Google Scholar 

  8. Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen-Stein methodAnnals of Probability 179–25.

    Article  MathSciNet  MATH  Google Scholar 

  9. Balakrishnan, N., Balasubramanian, K. and Viveros, R. (1993). On sampling inspection plans based on the theory of runsThe Mathematical Scientist 18113–126.

    MathSciNet  MATH  Google Scholar 

  10. Balakrishnan, N. and Cohen, A. C. (1991).Order Statistics and Inference: Estimation MethodsAcademic Press, San Diego, California.

    Google Scholar 

  11. Balakrishnan, N. and Rao, C. R. (Eds.) (1998a).Handbook of Statistics -16: Order Statistics: Theory and MethodsNorth-Holland, Amsterdam, The Netherlands.

    Google Scholar 

  12. Balakrishnan, N. and Rao, C. R. (Eds.) (1998b).Handbook of Statistics - 17: Order Statistics: ApplicationsNorth-Holland, Amsterdam, The Netherlands.

    Google Scholar 

  13. Barbour, A. D., Chryssaphinou, O. and Roos, M. (1996). Compound Poisson approximation in system reliabilityNaval Research Logistics 43251–264.

    Article  MathSciNet  MATH  Google Scholar 

  14. Berman, M. and Eagleson, G. K. (1985). A useful upper bound for the tail probabilities of the scan statistic when the sample size is largeJournal of the American Statistical Association 80886–889.

    Article  MathSciNet  Google Scholar 

  15. Bogush, Jr., A. J. (1972). Correlated clutter and resultant properties of binary signalsIEEE Transactions on Aerospace and Electronic Systems 9208–213.

    Article  Google Scholar 

  16. Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of reliability studies of consecutive-k-out-of-n: F and related systemsIEEE Transactions on Reliability 44120–127.

    Article  Google Scholar 

  17. Chen, J. (1998). Approximations and Inequalities for Discrete Scan StatisticsPh.D. DissertationUniversity of Connecticut, Storrs, CT.

    Google Scholar 

  18. Chen, J. and Glaz, J. (1996). Two-dimensional discrete scan statisticsStatistics Probability Letters 3159–68.

    Article  MathSciNet  MATH  Google Scholar 

  19. .Chen, J. and Glaz, J. (1998). Approximations for discrete scan statistics on the circleSubmitted for publication.

    Google Scholar 

  20. .Chen, J., Glaz, J., Naus, J. and Wallenstein, S. (1998). Bonferroni-type inequalities for conditional scan statisticsUnder preparation.

    Google Scholar 

  21. Conover, W. J., Bement, T. R. and Iman, R. L. (1979). On a method for detecting clusters of possible uranium depositsTechnometrics 21277–282.

    Article  Google Scholar 

  22. Cressie, N. (1977). On some properties of the scan statistic on the circle and the lineAnnals of Probability 14272–283.

    MathSciNet  MATH  Google Scholar 

  23. Cressie, N. (1978). Power results for tests based on higher order gapsBiometrika 65214–218.

    Article  MATH  Google Scholar 

  24. Cressie, N. (1979). An optimal statistic based on higher order gapsBiometrika 66619–627.

    Article  MathSciNet  MATH  Google Scholar 

  25. Cressie, N. (1980). The asymptotic distribution of the scan statistic under uniformityAnnals of Probability 8828–840.

    Article  MathSciNet  MATH  Google Scholar 

  26. Cressie, N. (1984). Using the scan statistic to test uniformityColloquia Mathematica Societatis János Bolyai 45pp. 87–100, Debrecen, Hungary.

    MathSciNet  Google Scholar 

  27. Cressie, N. (1991).Statistics for Spatial DataJohn Wiley & Sons, New York.

    MATH  Google Scholar 

  28. Darling, R. W. R. and Waterman, M. S. (1986). Extreme value distribution for the largest cube in random latticeSIAM Journal on Applied Mathematics 46118–132.

    Article  MathSciNet  MATH  Google Scholar 

  29. David, H. A. (1981).Order StatisticsSecond edition, John Wiley&Sons, New York.

    Google Scholar 

  30. Ederer, F., Myers, M. H. and Mantel, N. (1964). A statistical problem in space and time: Do leukemia cases come in clusters?Biometrics 20626–638.

    Article  Google Scholar 

  31. Eggleton, P. and Kermack, W. O. (1944). A problem in the random distribution of particlesProceedings of the Royal Society of Edinburgh Section A 62103–115.

    MathSciNet  MATH  Google Scholar 

  32. Fu, J. and Koutras, M. (1994). Distribution theory of runs: A Markov chain approachJournal of the American Statistical Association 891050–1058.

    Article  MathSciNet  MATH  Google Scholar 

  33. Fu, Y. X. and Curnow, R. N. (1990). Locating a changed segment in a sequence of Bernoulli variablesBiometrika 77295–304.

    Article  MathSciNet  MATH  Google Scholar 

  34. Galambos, J. and Simonelli, I. (1996).Bonferroni-type Inequalities with ApplicationsSpringer-Verlag, New York.

    MATH  Google Scholar 

  35. Gates, D. J. and Westcott, M. (1984). On the distributions of scan statisticsJournal of the American Statistical Association 79423–429.

    Article  MathSciNet  MATH  Google Scholar 

  36. Glaz, J. (1979). Expected waiting time for a visual responseBiological Cybernetics 3539–41

    Article  MathSciNet  MATH  Google Scholar 

  37. Glaz, J. (1981). Clustering of events in a stochastic processJournal of Applied Probability 18268–275.

    Article  MathSciNet  MATH  Google Scholar 

  38. Glaz, J. (1983). Moving window detection for discrete dataIEEE Transactions on Information Theory IT-29457–462.

    Article  Google Scholar 

  39. Glaz, J. (1989). Approximations and bounds for the distribution of the scan statisticJournal of the American Statistical Association 84560–566.

    Article  MathSciNet  MATH  Google Scholar 

  40. Glaz, J. (1992). Approximations for tail probabilities and moments of the scan statisticComputational Statistics and Data Analysis 14213–227.

    Article  MathSciNet  MATH  Google Scholar 

  41. Glaz, J. (1996). Discrete scan statistics with applications to minefield detectionProceedings of the Conference of SPIEOrlando, FL2765420–429.

    Article  Google Scholar 

  42. Glaz, J. and Naus, J. (1991). Tight bounds and approximations for scan statistic probabilities for discrete dataAnnals of Applied Probability 1306–318.

    Article  MathSciNet  MATH  Google Scholar 

  43. Glaz, J., Naus, J., Roos, M. and Wallenstein, S. (1984). Poisson approximations for the distribution and moments of ordered m-spacingsJournal of Applied Probability 31271–281.

    Article  MathSciNet  Google Scholar 

  44. Goldstein, L. and Waterman, M.S. (1992). Poisson, compound Poisson and process approximations for testing statistical significance in sequence comparisonsBulletin of Mathematical Biology 54785–812.

    MATH  Google Scholar 

  45. Greenberg, I. (1970). The first occurrence ofnsuccesses inNtrialsTechnometrics 12 627–634.

    Article  Google Scholar 

  46. Hamilton, J. F., Lawton, W. H. and Trabka, E. A. (1972). Some spatial and temporal point processes in photographic scienceStochastic Processes: Statistical Analysis Theory and Applications(Ed., P. A. W. Lewis), pp. 817–867, New York: Wiley Interscience.

    Google Scholar 

  47. Harter, H. L. and Balakrishnan, N. (1996).CRC Handbook of Tables for the Use of Order Statistics in EstimationCRC Press, Boca Raton, FL.

    Google Scholar 

  48. Huffer, F. and Lin, C. T. (1997). Approximating the distribution of the scan statistic using moments of the number of clumpsJournal of the American Statistical Association 921466–1475.

    Article  MathSciNet  MATH  Google Scholar 

  49. Huntington, R. and Naus, J. (1975). A simpler expression for Kth nearest neighbor coincidence probabilitiesAnnals of Probability 3894–896.

    Article  MathSciNet  MATH  Google Scholar 

  50. Hüsler, J. (1982). Random coverage of the circle and asymptotic distributionsJournal of Applied Probability 19578–587.

    Article  MathSciNet  MATH  Google Scholar 

  51. Ikeda, S. (1965). On Bouman-Velden-Yamamoto’s asymptotic evaluation formula for the probability of visual response in a certain experimental research in quantum biophysics of visionAnnals of the Institute of Statistical Mathematics17, 295–310.

    Article  MATH  Google Scholar 

  52. Janson, S. (1984). Bounds on the distributions of extremal values of a scanning processStochastic Processes and Their Applications 18313–328.

    Article  MathSciNet  MATH  Google Scholar 

  53. Karlin, S. and Brendel, V. (1992). Chance and statistical significance in Protein and DNA sequence analysisScience 25739–49.

    Article  Google Scholar 

  54. Karlin, S., Ghandour, G., Ost, F., Tavare, S. and Korn, L. J. (1983). New approaches for computer analysis of nucleic acid sequencesProceedings of the National Academy of Sciences 805660–5664.

    Article  MATH  Google Scholar 

  55. Koen, C. (1991). A computer program package for the statistical analysis of spatial point processes in a squareBiometrical Journal 33493–503.

    Article  MATH  Google Scholar 

  56. Kokic, P. N. (1987). On tests of uniformity for randomly distributed arcs on the circleThe Australian Journal of Statistics 29179–187.

    Article  MathSciNet  MATH  Google Scholar 

  57. Koutras, M. V. and Alexandrou, V. A. (1995). Runs, scans and urn model distributions: A unified Markov chain approachAnnals of the Institute of Statistical Mathematics 47743–766.

    Article  MathSciNet  MATH  Google Scholar 

  58. Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1993). Reliability of 2-dimensional consecutive-k-out-of-n: F systemsIEEE Transactions on Reliability R-42658–661.

    Article  Google Scholar 

  59. Kounias, E. G. (1968). Bounds for the probability of a union of events, with applicationsAnnals of Mathematical Statistics 392154–2158.

    Article  MathSciNet  MATH  Google Scholar 

  60. Krauth, J. (1992a). Bounds for the upper-tail probabilities of the circular ratchet scan statisticBiometrics 481177–1185.

    Article  MathSciNet  Google Scholar 

  61. Krauth, J. (1992b). Bounds for the upper-tail probabilities of the linear ratchet scan statistic, InAnalyzing and Modeling Data and Knowledge(Ed., M. Schader), pp. 51–61, Springer-Verlag, Berlin.

    Google Scholar 

  62. Leslie, R. T. (1969). Recurrence times of clusters of Poisson pointsJournal of Applied Probability 6372–388.

    Article  MathSciNet  MATH  Google Scholar 

  63. Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statisticsAdvances in Applied Probability 23751–771.

    Article  MathSciNet  MATH  Google Scholar 

  64. Mack, C. (1949). The expected number of aggregates in a random distribution of n pointsProceedings of the Cambridge Philosophical Society 46285–292.

    Article  MathSciNet  Google Scholar 

  65. Moye, L. A., Kapadia, A. S., Cech, I. M. and Hardy, R. J. (1988). The theory of runs with applications to drought predictionJournal of Hydrology 103127–137.

    Article  Google Scholar 

  66. Muises, R. R. and Smith, C. M. (1992). Nonparametric minefield detection and localizationTechnical Report CSS-TM 591–91Coastal Systems Station, Naval Surface Warfare Center.

    Google Scholar 

  67. Naus, J. (1965). The distributions of the size of the maximum cluster of points on a lineJournal of the American Statistical Association 60532–538.

    Article  MathSciNet  Google Scholar 

  68. Naus, J. (1966a). Some probabilities, expectations, and variances for the size of the largest clusters and smallest intervalsJournal of the American Statistical Association 611191–1199.

    Article  MathSciNet  MATH  Google Scholar 

  69. Naus, J. (1966b). A power comparison of two tests of non-random clustersTechnometrics 8493–517.

    MathSciNet  MATH  Google Scholar 

  70. Naus, J. (1974). Probabilities for a generalized birthday problemJournal of the American Statistical Association 69810–815.

    Article  MathSciNet  MATH  Google Scholar 

  71. Naus, J. (1982). Approximations for distributions of scan statisticsJournal of the American Statistical Association 77377–385.

    Article  MathSciNet  Google Scholar 

  72. Naus, J. and Sheng, K. N. (1997). Matching among multiple random sequencesBulletin of Mathematical Biology 59483–496.

    Article  MATH  Google Scholar 

  73. Nelson, J. B. (1978). Minimal order models for false alarm calculations on sliding windowsIEEE Transactions on Aerospace and Electronic Systems 15352–363.

    Google Scholar 

  74. Newell, G. F. (1963). Distribution for the smallest distance between any pair of Kth nearest-neighbor random points on a lineTime Series AnalysisProceedings of the Conference, Brown University (Ed., M. Rosenblatt), New York: Academic Press.

    Google Scholar 

  75. Orear, J. and Cassel, D. (1971). Applications of statistical inference to physics, InFoundation of Statistical Inference(Eds., V. Godambe and D. Sprott), pp. 280–289, Toronto: Holt, Rinehart and Winston.

    Google Scholar 

  76. Panayirci, E. and Dubes, R. C. (1983). A test for multidimensional clustering tendencyPattern Recognition 16433–444.

    Article  MATH  Google Scholar 

  77. Pfaltz, J. L. (1983). Convex clusters in discrete m-dimensional spaceJournal of Computation 12746–750.

    Google Scholar 

  78. Roos, M. (1993). Stein-Chen method for compound Poisson approximationPh.D. DissertationUniversity of Zurich, Zurich, Switzerland.

    Google Scholar 

  79. Rosenfeld, (1978). Clusters in digital picturesInformation Control 3919–34.

    Article  MathSciNet  MATH  Google Scholar 

  80. Salvia, A. A. and Lasher, W. C. (1990). 2-dimensional consecutive-k-outof-n: F modelsIEEE Transactions on Reliability R-39382–385.

    Article  Google Scholar 

  81. Samuel-Cahn, E. (1983). Simple approximations to the expected waiting time for a cluster of any given size for point processesAdvances in Applied Probability 1521–38.

    Article  MathSciNet  MATH  Google Scholar 

  82. Saperstein, B. (1972). The generalized birthday problemJournal of the American Statistical Association 67425–428.

    Article  MathSciNet  MATH  Google Scholar 

  83. Saperstein, B. (1973). On the occurrence of n successes withinNBernoulli trialsTechnometrics 15 809–818.

    MathSciNet  MATH  Google Scholar 

  84. Sarhan, A. E. and Greenberg, B. G. (Eds.) (1962).Contributions to Order StatisticsJohn Wiley&Sons, New York.

    MATH  Google Scholar 

  85. Sheng, K. N. and Naus, J. I. (1996). Matching rectangles in 2-dimensionsStatistics & Probability Letters 2683–90.

    Article  MathSciNet  MATH  Google Scholar 

  86. Shepard, J., Creasey, J. W. and Fisher, N. I. (1981). Statistical analysis of spacings between geological discontinuities in coal mines, with appli-cations to short-range forecasting of mining conditionsAustralian Coal Geol. 371–80.

    Google Scholar 

  87. Smith, C. M. (1991). Two-dimensional minefield simulationTechnical ReportNCSM-TM-558–91, Coastal Systems Center, Naval Surface Warfare Center.

    Google Scholar 

  88. Takács, L. (1996). On a test for uniformity of a circular distributionMathematical Methods of Statistics 577–98.

    MATH  Google Scholar 

  89. Trusov, A. G. (1970). Estimation of the optimal signal arrival time under conditions of photon counting in free spaceProceedings of the IEEE Radio-Optics 19137–139.

    Google Scholar 

  90. Van de Grind, W. A., Koenderink, J. J., Van der Heyde, G. L., Landman, H. and Bowman, M. A. (1971). Adapting coincidence scalars and neural modeling studies of visionKybernetik 885–105.

    Article  Google Scholar 

  91. Wallenstein, S. (1980). A test for detection of clustering over timeAmerican Journal of Epidemiology 11367–372.

    Google Scholar 

  92. Wallenstein, S. and Naus, J. (1974). Probabilities for the size of the largest clusters and smallest intervalsJournal of the American Statistical Association 69690–697.

    Article  MathSciNet  MATH  Google Scholar 

  93. Wallenstein, S., Naus, J. and Glaz, J. (1994). Power of the scan statistic in detecting a changed segment in a Bernoulli sequenceBiometrika 81595–601.

    Article  MathSciNet  MATH  Google Scholar 

  94. Wallenstein, S., Naus, J. and Glaz, J. (1995). Power of the scan statisticsProceedings Section EpidemiologyAnnual ASA Meeting Toronto, Canada, pp. 70–75.

    Google Scholar 

  95. Wallenstein, S. and Neff, N. (1987). An approximation for the distribution of the scan statisticStatistics in Medicine 121–15.

    Google Scholar 

  96. Wallenstein, S., Weinberg, C. R. and Gould, M. (1989). Testing for a pulse in seasonal event dataBiometrics 45817–830.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Glaz, J., Balakrishnan, N. (1999). Introduction to Scan Statistics. In: Glaz, J., Balakrishnan, N. (eds) Scan Statistics and Applications. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1578-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1578-3_1

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7201-4

  • Online ISBN: 978-1-4612-1578-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics