TABLE I.

Partitioning algorithm for identifying maximally separated subsets For each class (multiset) X, partition X into two subsets A and B such that NCD₁(AB) − NCD₁(A) − NCD₁(B) is a maximum.

(Initialize) Pick two elements (seeds) of X at random, assigning one element to each A and B. For each remaining element x, assign x to the closer one of A or B using pairwise NCD to the random seeds For each element x, compute the distance from x to class A and B using (IV.1) and assign to whichever class achieves the smaller distance. Choose the single element that wants to change subsets, e.g. from A to B or vice versa and whose change maximizes NCD1(AB) − NCD1(A) − NCD1(B) and swap that element from A to B or vice versa. Repeat steps 2 and 3 until no more elements want to change subsets or until we exceed e.g. 100 iterations.

Repeat the whole process some fixed number of times (here we use 5) for each X and choose the subsets that achieve the maximum of NCD1(AB) − NCD1(A) − NCD1(B). If that value exceeds the minimum inter-class separation and the subsets are not smaller than the specified minimum size then divide X into A and B and repeat the process for A and B. If the value does not exceed the minimum inter-class separation of our training data or the subsets exceed the specified minimum size, then accept X as approximately monotonic and go on to the next class.