AISM (2006) 58: 235–259 DOI 10.1007/s10463-005-0015-7

M.N.M. van Lieshout A J - for marked patterns

Received: 5 July 2004 / Revised: 7 February 2005 / Published online: 30 May 2006 © The Institute of Statistical , Tokyo 2006

Abstract We propose a new summary statistic for marked point patterns. The underlying principle is to compare the distance from a marked point to the near- est other marked point in the pattern to the same distance seen from an arbitrary point in . 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 earlier associated with J -function plug-in estimators, and carry out an exploratory analysis of a forestry data . Keywords Empty space function · J -function · Marked point pattern · Mark correlation function · Nearest neighbour distance distribution function · Product density · Random labelling · Reduced second · Spatial interaction · Spatial

1 Introduction

Marked point patterns are spatial point configurations with a mark attached to each point (Stoyan and Stoyan, 1994). The points could represent the locations in (Euclidean) space of objects, while the marks capture additional information. The latter could be a type label, in which case we also speak of a multivariate point pattern (Cox and Lewis, 1972), a continuous measurement or shape descriptor, or a combination of these. The statistical analysis of such a pattern in general begins by plotting a few summary statistics. Which statistic is used depends on taste and the type of mark.

M.N.M. van Lieshout Department PNA4, Centrum voor Wiskunde en Informatica, P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands E-mail: [email protected]