mmdval = mmdtest (X, Y) returns the squared maximum mean discrepancy between the samples X and Y, whose rows are observations and whose columns are variables. It measures how far apart the two distributions lie, 0 where the two samples are alike. With m rows in X, n in Y and k the kernel, it is the biased estimate $$\mathrm{MMD}^2 = {1 \over m^2} \sum_{i,j=1}^{m} k(x_i, x_j) - {2 \over m n} \sum_{i=1}^{m} \sum_{j=1}^{n} k(x_i, y_j) + {1 \over n^2} \sum_{i,j=1}^{n} k(y_i, y_j).$$

X and Y are both numeric matrices with the same number of columns, or both tables, read over the variables they share, which must be all the variables of one of them. An observation holding a missing value in a variable used is left out.

The kernel is Gaussian, k(u, v) = exp (-d^2 / s). Each continuous variable is standardised by the mean and standard deviation of X, a standard deviation of 0 being taken as 1, and contributes its squared difference to d^2; a variable holding levels contributes 2 where the two observations differ, as one column per level would. The scale s is the median of d^2 over the pairs of observations in X, taken as 1 where that median is 0.

[mmdval, p, h] = mmdtest (…) also returns the p-value of the permutation test of the null hypothesis that X and Y come from the same distribution, and h, which is 1 where the null hypothesis is rejected at the significance level 'Alpha' and 0 otherwise. Each permutation deals the pooled observations out again as X and Y, standardises and scales them afresh, and computes the statistic; p is the share of the permutations whose statistic is at least the observed one. The permutations are drawn with randperm, so the state of rand decides them. They are drawn only where p or h is asked for.

mmdval = mmdtest (…, Name, Value) takes the following options.

Reference: A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schoelkopf and A. Smola (2012). A kernel two-sample test. Journal of Machine Learning Research, 13, 723-773.

See also: knntest, kstest2

Source Code: mmdtest