• Bernoulli
  • Volume 12, Number 5 (2006), 821-839.

Locating lines among scattered points

Peter Hall, Nader Tajvidi, and P.E. Malin

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Consider a process of events on a line L, where, for the most part, the events occur randomly in both time and location. A scatterplot of the pair that represents position on the line, and occurrence time, will resemble a bivariate stochastic point process in a plane, P say. If, however, some of the points on L arise through a more regular phenomenon which travels along the line at an approximately constant speed, creating new points as it goes, then the corresponding points in P will occur roughly in a straight line. It is of interest to locate such lines, and thereby identify, as nearly as possible, the points on L which are associated with the (approximately) constant-velocity process. Such a problem arises in connection with the study of seismic data, where L represents a fault-line and the constant-velocity process there results from the steady diffusion of stress. We suggest methodology for solving this needle-in-a-haystack problem, and discuss its properties. The technique is applied to both simulated and real data. In the latter case it draws particular attention to events occurring along the San Andreas fault, in the vicinity of Parkville, California, on 5 April 1995.

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Bernoulli, Volume 12, Number 5 (2006), 821-839.

First available in Project Euclid: 23 October 2006

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earthquake hypothesis test large-deviation probability ley-line point process Poisson process San Andreas fault spatial process


Hall, Peter; Tajvidi, Nader; Malin, P.E. Locating lines among scattered points. Bernoulli 12 (2006), no. 5, 821--839. doi:10.3150/bj/1161614948.

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  • [1] Bakun, W.H. and Lindh, A.G. (1985) The Parkfield, California, earthquake prediction experiment. Science, 229, 619-624.
  • [2] Bakun, W.H. and McEvilly, T.V. (1979) Earthquakes near Parkfield, California: Comparing the 1934 and 1966 sequences. Science, 205, 1375-1377.
  • [3] Bakun, W.H. and McEvilly T.V. (1984) Recurrence models and Parkfield, California earthquakes. J. Geophys. Res., 89, 3051-3058.
  • [4] Broadbent, S. (1980) Simulating the ley hunter. J. Roy. Statist. Soc. Ser. A, 143, 109-140.
  • [5] Cressie, N. (1977) On some properties of the scan statistic on the circle and the line. J. Appl. Probab., 14, 272-283.
  • [6] Danuser, G. and Stricker, M. (1998) Parametric model fitting: From inlier characterization to outlier detection. IEEE Trans. Pattern Anal. Machine Intelligence, 20, 263-280.
  • [7] Desolneux, A., Moisan, L. and Morel, J.M. (2003a) Maximal meaningful events and applications to image analysis. Ann. Statist., 31, 1822-1851.
  • [8] Desolneux, A., Moisan, L. and Morel, J.M. (2003b) Computational gestalts and perception thresholds. J. Physiol. Paris, 97, 311-324.
  • [9] Fletcher, M. and Lock, G. (1981) Computerised pattern perception within posthole distributions. Sci. Archaeol., 22, 1520.
  • [10] Fletcher, M. and Lock, G. (1984) Post built structures at Danebury Hilfort: An analytical search method with statistical discussion. Oxford J. Archaeol., 3, 175-196.
  • [11] Fletcher, M. and Lock, G. (1991) Digging Numbers: Elementary Statistics for Archaelogists. Oxford: Oxbow.
  • [12] Frigui, H. and Krishnapuram, R. (1999) A robust competitive clustering algorithm with applications to computer vision. IEEE Trans. Pattern Anal. Machine Intelligence, 21, 450-465.
  • [13] Gates, J. (1986) Measures and tests of alignment. Biometrika, 73, 731-734.
  • [14] Goodall, C. (1991) Procrustes methods in the statistical-analysis of shape. J. Roy. Statist. Soc. Ser. B, 53, 285-339.
  • [15] Hall, P., Tajvidi, N. and Malin, P. (2005) Locating lines among scattered points.
  • [16] Kendall, D.G. (1984) Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc., 16, 81-121.
  • [17] Kendall, D.G. (1985) Exact distributions for shapes of random triangles in convex sets. Adv. in Appl. Probab., 17, 308-329.
  • [18] Kendall, D.G. (1986) Further developments and applications of the statistical theory of shape. Teor. Veroyatnost. i Primenen., 31, 467-473.
  • [19] Kendall, D.G. and Kendall, W.S. (1980) Alignments in two-dimensional random sets of points. Adv. in Appl. Probab., 12, 380-424.
  • [20] Kent, J.T., Briden, J.C. and Mardia, K.V. (1983) Linear and planar structure in ordered multivariate data, as applied to progressive demagnetization remanence. Geophys. J. Roy. Astronom. Soc., 75, 593-621.
  • [21] Kulldorff, M., Athas, W.F., Feurer, E.J., Miller, B.A. and Key, C.R. (1998) Evaluating cluster alarms: a space-time scan statistic and brain cancer in Los Alamos, New Mexico. Amer. J. Public Health, 88, 1377-1380.
  • [22] Lindh, A.G. and Malin, P.E. (1987) Introduction to Parkfield Earthquake Studies: Instrumentation and Geology. El Cerrito, CA: Seismological Society of America.
  • [23] Mack, C. (1950) The expected number of aggregates in a random distribution of n points. Proc. Cambridge Philos. Soc., 46, 285-292.
  • [24] Malin, P.E. and Alvarez, M.G. (1992) Stress diffusion along the San Andreas fault at Parkfield, CA. Science, 256, 1005-1007.
  • [25] Malin, P.E., Oancea, V.G., Shalev, E. and Tang, C. (2002) A friction-feedback model for microearthquake sequences at Parkfield, California. Manuscript.
  • [26] McEvilly, T.V., Bakun W.H. and Casady, K.B. (1967) The Parkfield, California earthquakes of 1966. Bull. Seismol. Soc. Amer., 57, 1221-1244.
  • [27] Meer, P., Stewart, C.V. and Tyler, D.E. (2000) Robust computer vision: An interdisciplinary challenge. Comput. Vision Image Understanding, 78, 1-7.
  • [28] Priebe, C.E., Olson, T.H. and Dennis, M. Jr. (1997) A spatial scan statistic for stochastic scan partitions. J. Amer. Statist. Assoc., 92, 1476-1484.
  • [29] Saunders, I.W. (1978) Locating bright spots in a point process. Adv. in Appl. Probab., 10, 587-612.
  • [30] Segall, P. and Harris, R. (1987) Earthquake deformation cycle on the San Andreas fault near Parkfield, California. J. Geophys. Res., 92, 10 511-10 525.
  • [31] Silverman, B. and Brown, T. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Probab., 15, 815-825.
  • [32] Small, C.G. (1982) Random uniform triangles and the alignment problem. Math. Proc. Cambridge Philos. Soc., 91, 315-322.
  • [33] Small, C.G. (1984) A classification theorem for planar distributions based on the shape statistics of independent tetrads. Math. Proc. Cambridge Philos. Soc., 96, 543-547.
  • [34] Small, C.G. (1988) Techniques of shape analysis on sets of points. Internat. Statist. Rev., 56, 243-257.
  • [35] Small, C.G. (1996) The Statistical Theory of Shape. New York: Springer-Verlag.
  • [36] Stewart, C.V. (1995a) MINIPRAN - new robust estimator for computer vision. IEEE Trans. Pattern Anal. Machine Intelligence, 17, 925-938.
  • [37] Stewart, C.V. (1995b) Robust parameter estimation in computer vision. Comput. Vision Image Understanding, 76, 54-69.
  • [38] Susaki, J., Hara, K., Kajiwara, K. and Honda, Y. (2004) Robust estimation of BRDF model parameters. Remote Sensing of Environment, 89, 63-71.
  • [39] Stuart, W.D., Archuleta, R.J. and Lindh, A.G. (1985) Forecast model for moderate earthquakes near Parkfield, California. J. Geophys. Res., 90, 592-604.
  • [40] Weinstock, M.A. (1981) A generalised scan statistic for the detection of clusters. Internat. J. Epidemiol., 10, 289-293.