A Kernel Method for the Two Sample Problem
by Arthur Gretton, Karsten Borgwardt, Malte Rasch,
We propose to test whether distributions P and Q are different on the basis of samples drawn from each of them, by finding a smooth function (the witness function) which is large on the points drawn from P, and small (as negative as possible) on the points from Q. We use as our test statistic the difference between the mean function values on the two samples, or maximum mean discrepancy (MMD): when this is large, the samples are likely from different distributions. Smoothness is enforced by restricting the witness function to a unit ball in a reproducing kernel Hilbert space.
Three strategies are used to calculate the test threshold:
Note that an earlier version of this test was proposed in BorEtAl06, however the current test more accurately estimates the null distribution, and should be used in preference to the earlier algorithm.
The archive contains two files: mmd.m is the main code, and U4thmoment.c
contains additional optimised c-code for one of the test options. While
the algorithm runs in standalone form, it is also possible to use it
Spider machine learning toolbox. Code is written by Malte Rasch.
|[GreEtAl07a]||Gretton, A., K. Borgwardt, M. Rasch, B. Schoelkopf and A. Smola: A Kernel Method for the Two-Sample-Problem. NIPS 2006. download|
|[GreEtAl07b]||Gretton, A., K. Borgwardt, M. Rasch, B. Schoelkopf and A. Smola: A Kernel Method for the Two-Sample-Problem. MPI Technical Report 157, 2007.|
|[BorEtAl06]||Borgwardt, K., A. Gretton, M. Rasch, H.-P. Kriegel, B. Schoelkopf and A. Smola: Integrating structured biological data by Kernel Maximum Mean Discrepancy. Bioinformatics 22(14), 1-9 (2006) download|