Meaning
Mathematical adjustments resolve the bias introduced when points near the boundary of a sample window have fewer neighbors than those in the center. Implementing isotropic edge correction involves weighting the contribution of each pair of points by the proportion of the circle centered at one point that lies within the study region. This method ensures that the estimated density of neighbors remains accurate even when part of the search radius extends outside the observable area.
Weighting Factor
The correction factor increases the weight of points close to the edge. This weighting factor compensates for the unobserved neighbors that likely exist just beyond the border.
Boundary Interaction
Calculation of the weight depends on the geometry of the sampling window. For a given distance, the algorithm calculates the circumference of a circle and determines what fraction of that circumference is inside the boundary. If only half of the circle is inside, the count for that pair is doubled in the final sum.
This technique is more sophisticated than simple border methods and it uses all the available data and avoids discarding points near the edges. It is particularly effective for rectangular or circular windows where the geometry is well defined.
Estimation Accuracy
Large distances relative to the window size increase the variance of the estimate. An estimation accuracy decreases as the search radius approaches the dimensions of the sample area.
