Distance and Nearest Neighbours

A fundamental concept in the analysis of the point pattern data is distance.  The (Euclidean) distance, d(i,j), between points i and j in 3D space with locations (or coordinates) (x_i , y_i , z_i) and (x_j , y_j , z_j) respectively is

d(i,j) = √{( x_i - x_j )² + ( y_i - y_j )² + ( z_i - z_j )²}

that is, the square-root of the squared differences between the x, y and z coordinates.

For a set of n data points corresponding to object locations, we have a total of N=n(n-1)/2 distances between objects taken pairwise.  We can arrange the distances into a symmetric n x n matrix, D, with (i,j)^th element d(i,j) and zeros on the diagonal.

The i^th row of D contains the distances from object i to all other objects; the nearest neighbour of i is the object whose distance from i is smallest.  That is, the nearest neighbour of i is denoted η_i and defined mathematically by

η_i  = argmin_j d(i,j)

and the nearest neighbour distance is denoted δ_i and defined by

δ_i  =  d(i,η_i) = min_j d(i,j)

that is, the smallest observed distance from object i.

Thus, for any data set of size n, we can extract a set of N inter-point distances, and a set of n nearest neighbour differences.  These sets of distances can be used to test our hypotheses of interest.

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