Pattern Recognition, 4th Edition

Tests Based on Nearest Neighbor Distances

Two tests of this kind are the Hopkins test [Hopk 54] and the Cox-Lewis test [Cox 76, Pana 83]. The tests rely on the distances between the vectors of X and a number of vectors which are randomly placed in the sampling window.

The Hopkins Test

Let X′ = {yⁱ, i = 1,…,M}, ≪ N,^[9] be a set of vectors that are randomly distributed in the sampling window, following the uniform distribution. Also let X¹ ⊂ X be a set of M randomly chosen vectors of X. Let d^j be the distance from y^j ∈ X′ to its closest vector in X¹, denoted by x^j, and δ^j be the distance from x^j to its closest vector in X¹ − {x^j}. Then the Hopkins statistic involves the lth powers of d^j and δ^j and it is defined as [Jain 88](16.61)This statistic ...