ummd 0.2.0

Efficient Maximum Mean Discrepancy two-sample testing for data with duplicate observations, scaling with unique values rather than sample size.

Unique Maximum Mean Discrepancy (uMMD)

An efficient implementation of the Maximum Mean Discrepancy two-sample test for datasets with duplicate observations via count-weighting of unique values. This implementation scales with unique data values rather than sample size.

Installation

pip install ummd

Quick start

import numpy as np
from ummd import MMD

rng = np.random.default_rng(0)
x = rng.integers(0, 10, size=500)   # sample from one distribution
y = rng.integers(2, 12, size=500)   # sample from a shifted distribution

result = MMD(x, y, unique=True, bandwidths=10, n_permutations=999)

print(result["biased_MMD"])   # MMD statistic per bandwidth
# [ 0.04408069 0.053788   0.06124013 0.06328209 0.06290089 0.0602459 0.04713144 0.02831863 0.01431563 0.0066321 ]

print(result["p-value"])    # combined p-value across bandwidths
# 0.001

Interpreting the result

MMD returns a dictionary with:

• biased_MMD: the MMD statistic for each tested bandwidth
• p-values_per_bandwidth: permutation p-value for each bandwidth
• p-value: a single Cauchy-combined p-value across the bandwidths
• bandwidths: the kernel bandwidths actually used

Custom kernels

By default MMD uses a Gaussian (RBF) kernel. You can supply your own via kernel_fn. A custom kernel should build the entire stacked kernel matrix. This is to allow control of how it is vectorised/optimised:

import numpy as np
from scipy.spatial.distance import cdist
from ummd import MMD

def my_rbf(x, y, bandwidths):
    gammas = 1.0 / (2.0 * bandwidths**2)
    D = cdist(x, y, metric="sqeuclidean")
    return np.exp(-gammas[:, None, None] * D[None, :, :])  # (len(bandwidths), m, n)

result = MMD(x, y, kernel_fn=my_rbf, bandwidths=10, n_permutations=999)

The contract is kernel_fn(x, y, bandwidths) -> ndarray of shape (len(bandwidths), m, n), one kernel matrix per bandwidth. bandwidths is the 1-D array of sigma length-scales that MMD resolves from the bandwidths= argument (median heuristic, geometric grid, or an explicit array); your kernel decides how to interpret them, and may ignore them entirely and return a single (1, m, n) matrix.

Why uMMD

A standard MMD test builds an N x N kernel matrix, so cost grows with sample size. When your data has many repeated values (counts, categories, discretised measurements), uMMD instead works over the u unique values, where u << n, giving the same test at a fraction of the cost.