where we choose to use the a similar max norm for ||x -x ′ || and ||y -y ′ ||, although this is not required -a Euclidean norm could be used, for instance. For each data point z i , let ϵ x (i)/2 and ϵ y (i)/2 represent the distances from z i to its kth nearest neighbor projected onto the X and Y subspaces, respectively. The value p i is the integrated density within a distance ϵ/2 of the point x i defined as: