Ripley’s Cross-L Function

Answers the question: At what distances does group B accumulate around or avoid group A?

The tool uses group $A$ as the reference and measures the distances from its cells to cells from group $B$. It gradually increases a radius $r$ around every $A$-cell and counts how many $B$-cells fall within that radius.

More specifically, let $C(A,B)(r)$ be the number of observed $A$-to-$B$ pairs separated by at most $r$. The cross-K function adjusts this count for the overall density $\rho (B)$ of group $B$ and for $m(A)(r)$, the number of $A$-cells included at radius $r$:

Under complete spatial randomness, $K(A,B)(r)=\pi r^2$. The tool transforms this comparison into a curve centered around zero:

Positive values indicate more $B$-cells around $A$ than expected, while negative values indicate fewer.

Plot. The x-axis shows the radius $r$, and the y-axis shows $L(A,B)(r)-r$. The solid line is the observed curve; its peak indicates the spatial scale at which $B$ is most strongly accumulated around $A$. The grey band comes from random redistributions of the $B$ labels while keeping $A$ fixed. Values above the band indicate enrichment of $B$ around $A$, while values below indicate spatial avoidance.

Table. Each row represents one tested radius $r$. L(r) − r is the observed curve value, while Envelope low and Envelope high are the lowest and highest values from the random relabellings at that radius. Outside envelope indicates whether the observed value lies above or below this range; a blank entry means that it lies inside.