Kfunctie

The K-function, or Kfunctie in Dutch, is a central concept in spatial statistics used to describe the spacing of events in a region. Introduced by Brian D. Ripley in the 1970s, it summarizes how many other events lie within a given distance of a typical event, after accounting for overall density.

Formally, for a stationary point process with intensity λ, K(r) = (1/λ) E[number of additional points within distance r of a typical point]. For a homogeneous Poisson process, K(r) = π r^2 in two dimensions; in general it relates to the pair correlation function g by K(r) = ∫_0^r 2πu g(u) du.

To estimate from observed data in a study area A with n points, a common estimator is K_hat(r) = (A / (n(n−1))) ∑_{i≠j} w_ij^{-1} I(d_ij ≤ r), where d_ij is the distance between points i and j, I is the indicator function, and w_ij are edge-correction weights. Edge effects are important in finite regions; different corrections exist. A related transformation, L(r) = sqrt(K(r)/π), stabilizes variance.

Interpretation: comparing K_hat to the CSR (complete spatial randomness) expectation reveals clustering or regularity at multiple scales. If L(r)−r > 0, points cluster at distance r; if L(r)−r < 0, the pattern is more regular than random. Variants include the bivariate K-function for two types of points and the cross K-function for interaction between types.

Applications span ecology, epidemiology, criminology, and materials science, where researchers assess whether events such as tree locations, disease cases, or defects are clustered beyond random chance. Software implementations include the Kest function in R's spatstat package and similar tools in Python's PySAL.