k (ϵ x , ϵ y ) represents the probability distribution that there are k -1 data points within the rectangle x i ± ϵ x (i)/2 and y i ± ϵ y (i)/2, a rectangle defined by the kth nearest neighbor within in the x subspace, N -k -1 points that are outside a different rectangle defined by x i ± (ϵ x (i) + dϵ x )/2 and y i ± (ϵ y (i) + dϵ y )/2, and one data point in the space between the two rectangles. The probability distribution P (c) k (ϵ x , ϵ y ) is similar, though the rectangles are defined by the kth nearest neighbor within the y subspace, which may be the same, or a different, point used to define P (b) k (ϵ x , ϵ y ). These quantities are then