Tests for correlation between points
Examining the correlation between points in space is a critical step in determining which point processes may have shaped a point-pattern and at what spatial scale. In a point-pattern with a completely random distribution of points, the average number of neighbors at a given distance from any random point follows a Poisson distribution (Baddeleyet al. 2015). Departures from this distribution indicate points may be either more clustered or overdispersed than if the pattern were truly random. The K-function is a commonly used method to evaluate correlation between points and determine whether patterns reflect clustering, overdispersion, or randomness (Ripley 1977). Each homogeneous point-pattern in the Midwestern and Oklahoma datasets was evaluated for departures from CSR using the standard K-function. Inhomogeneous data were tested using a modified K-function designed for non-homogeneous point-patterns. We used an isotropic-edge correction for all K-function tests to account for unobserved trees on the margins of the sampling window (Horvitz and Thompson 1952). Two methods were used to assess differences between a theoretical Poisson distribution for a random set of n points and the empirical K-function for then points within each point-pattern: 1) A Monte Carlo test (1999 simulations; rank = 50; α = 0.05) for each of the n points in each point-pattern was conducted to construct simulation acceptance envelopes of a theoretical random distribution. These acceptance envelopes contain the range of values that are used to test the hypothesis of CSR. Acceptance envelopes overlaid with the empirical K-functions for each point-pattern were visually inspected for areas where the empirical K-function diverged from the envelope, indicating a departure from CSR. Empirical K-function lines above the envelope indicate neighboring trees are more closely spaced than in a random distribution. Conversely, if the empirical K-function line is below the envelope, then neighboring trees are more regularly-spaced than in a random distribution. 2) The Diggle-Cressie-Loosmore-Ford (DCLF) test was applied to detect departures from the homogeneous Poisson process for each point-pattern (Diggle, 1986; Cressie, 1991; Loosmore and Ford, 2006). The DCLF test allows an upper range of interaction between points to be specified, and a quantitative assessment that indicates a deviation from CSR.