A J-Function for Marked Point Patterns

We propose a new summary statistic for marked point patterns. The underlying principle is to compare the distance from a marked point to the nearest other marked point in the pattern to the same distance seen from an arbitrary point in space. Information about the range of interaction can be inferred, and the statistic is well-behaved under random mark allocation. We develop a range of Hanisch style kernel estimators to tackle the problems of exploding tail variance earlier associated with J-function plug-in estimators, and carry out an exploratory analysis of a forestry data set.

[1]  Dietrich Stoyan,et al.  Improving Ratio Estimators of Second Order Point Process Characteristics , 2000 .

[2]  D. Stoyan.  On Correlations of Marked Point Processes , 1984 .

[3]  Adrian Baddeley,et al.  The empty space hazard of a spatial pattern , 1994 .

[4]  Regularity in the distribution of superclusters , 1997, astro-ph/9805088.

[5]  M. J. Paulo,et al.  Statistical sampling and modelling for cork oak and eucalyptus stands , 2002 .

[6]  A. Stein,et al.  Spatial interaction of methylene blue stained soil pores , 2001 .

[7]  Adrian Baddeley,et al.  Nonparametric measures of association between a spatial point process and a random set, with geological applications , 2002 .

[8]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[9]  Simon D. M. White,et al.  The hierarchy of correlation functions and its relation to other measures of galaxy clustering , 1979 .

[10]  D. Vere-Jones,et al.  Elementary theory and methods , 2003 .

[11]  R. Gill,et al.  Kaplan-Meier estimators of distance distributions for spatial point processes , 1997 .

[12]  Y. Ogata,et al.  Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns , 1989 .

[13]  Adrian Baddeley,et al.  Kaplan-Meier type estimators for linear contact distributions , 1996 .

[14]  van Marie-Colette Lieshout,et al.  Indices of Dependence Between Types in Multivariate Point Patterns , 1999 .

[15]  Peter J. Diggle,et al.  Simple Monte Carlo Tests for Spatial Pattern , 1977 .

[16]  Nguyen Xuan Xanh,et al.  Integral and differential characterizations of the Gibbs process , 1977, Advances in Applied Probability.

[17]  Martin Kerscher,et al.  Fluctuations in the IRAS 1.2 Jy catalogue , 1997, astro-ph/9704028.

[18]  B. Hambly.  Fractals, random shapes, and point fields , 1994 .

[19]  P. A. W. Lewis,et al.  Multivariate point processes , 2018, Point Processes.

[20]  A. Baddeley,et al.  A cautionary example on the use of second-order methods for analyzing point patterns , 1984 .

[21]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[22]  Jürgen Symanzik,et al.  Statistical Analysis of Spatial Point Patterns , 2005, Technometrics.

[23]  B. Ripley,et al.  Markov Point Processes , 1977 .

[24]  Dietrich Stoyan,et al.  Statistical analysis for a class of line segment processes , 1989 .

[25]  A. Baddeley,et al.  Estimating the J function without edge correction , 1999, math/9910011.

[26]  Günter Last,et al.  On the empty space function of some germ-grain models , 1999, Pattern Recognit..

[27]  E. Thönnes,et al.  A comparative study on the power of van Lieshout and Baddeley's J-function , 1998 .

[28]  T. Bedford,et al.  A Remark on the Van Lieshout and Baddeley J-Function for Point Processes , 1997, Advances in Applied Probability.

[29]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

[30]  Hans Zessin,et al.  Integral and Differential Characterizations of the GIBBS Process , 1979 .

[31]  A. Baddeley,et al.  Nearest-Neighbour Markov Point Processes and Random Sets , 1989 .

[32]  Martin Kerscher,et al.  A Global Descriptor of Spatial Pattern Interaction in the Galaxy Distribution , 1999 .

[33]  K. Hanisch,et al.  Some remarks on estimators of the distribution function of nearest neighbour distance in stationary spatial point processes , 1984 .

[34]  D. Stoyan,et al.  Estimators of distance distributions for spatial patterns , 1998 .

[35]  A. Baddeley,et al.  A non-parametric measure of spatial interaction in point patterns , 1996, Advances in Applied Probability.

[36]  B. Ripley.  Statistical inference for spatial processes , 1990 .

[37]  Jeffrey D. Scargle,et al.  An Introduction to the Theory of Point Processes, Vol. I: Elementary Theory and Methods , 2004, Technometrics.