A J-Function for Marked Point Patterns

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsA J-function for marked point patterns 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 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. AbstractWe 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.

Tests for Independence between Marks and Points of a Marked Point Process

“…Define the mark K function and the mark G function Note that K m ( r ) is a special case of the K f ( r ) summary of Penttinen and Stoyan (1989) with f ( m 1 , m 2 ) = m 1 . G m ( r ) is different from the G B ( r ) function used in van Lieshout (2004) where the latter is the cumulative distribution function of the distance from a typical point of the process with mark in B to its nearest neighbor in N , where B is a given Borel set from the mark space. When the marks take only the value 1, the marked point process Ψ reduces to the spatial point process N .…”

“…To detect dependence between marks and points, several graphical approaches have been proposed. These include the use of the mark correlation/variogram function (Wälder and Stoyan, 1996), the J function (van Lieshout, 2004), and the E and V functions (Schlather, Ribeiro, and Diggle, 2004). Schalther et al (2004) also proposed a formal Monte Carlo test for independence between marks and points for a class of stationary (transformed) Gaussian marks.…”

Spatial statistics in ecological analysis: from indices to functions

“…The J-function is another distance-dependent function for analysis of the spatial point pattern (Van Lieshout and Baddeley 2006). The idea of this function is to compare distances from an arbitrary point to the nearest neighbor (empty space F-function) and distances from typical point of the pattern measured by the nearest-neighbor distance G-function.…”

“…Then, if J(r) ≡ 1 the distance distribution follows the Poisson process. Deviation J(r) > 1or J(r) < 1 indicates spatial regularity and clustering, respectively (Van Lieshout and Baddeley 1999;Paolo et al 2002;Fortin and Dale 2005;Van Lieshout and Baddeley 2006). Because practically observation of points is restricted to some bounded area (measurement plot) the estimate of the J-function is hampered also by the edge effect.…”

A J–function for inhomogeneous point processes

“…Similarly, provided that the series is absolutely convergent. Thence ( van L ieshout , 2006), for all t ≥0 for which F ( t )<1, where J n ( t )=∫ B (0, t ) ⋯∫ B (0, t ) ξ n +1 (0, x 1 ,…, x n )d x 1 ⋯d x n . If product densities of all orders do not exist, one may truncate the series.…”