eummd

Description

Computes maximum mean discrepancy two-sample test for univariate data using the Laplacian kernel. It is also possible to compute the p-value using permutations. Also includes implementation for computing the robust median difference statistic 'Q_n' from Croux and Rousseeuw (1992) doi:10.1007/978-3-662-26811-7_58 based on Johnson and Mizoguchi (1978) doi:10.1137/0207013.

Installation instructions

Install using pip:

python3 -m pip install eummd

Methods are implemented in C++, and then called using Cython.

Functions

There are various functions for computing the maximum mean discrepancy (MMD).

• eummd: efficient univariate MMD with the Laplacian kernel, O(n log n) for a total of n samples.
• mmd: MMD (univariate/multivariate).
• meammd: multivariate efficient approximate MMD, using either random projections or interpoint distances.
• mediandiff: computes the median difference. If univariate, can be computed in O(n log n), otherwise is O(n^2).
• medianheuristic: the inverse of the median difference.

eummd and mmd return a dictionary with the items pval for p-value (if specified), stat for the statistic and beta for the kernel parameter value that was used. If beta is specified as a non-positive number, then the median heuristic will be calculated.

Examples:

from eummd import eummd
from eummd import mediandiff

X = np.array([7.1, 1.2, 4.3, 0.4]).astype(np.float64)
Y = np.array([5.5, 2.6, 8.7]).astype(np.float64)
beta = 0.1

# only computes statistic, 
d = eummd(X, Y, beta, pval=False)
print(d['stat'])

# computes statistic and p-value, and used median heuristic for beta
d = eummd(X, Y, beta=-0.1, pval=True)
print("p-value for univariate data: ", d['pval'])
print("statistic for univariate data: ", d['stat'])
print("beta for univariate data: ", d['beta'])

# show median heuristic was used
print(1.0 / mediandiff(X, Y, fast=True))

# a multivariate example, using median heuristic for beta
X = np.array([[1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12]]).reshape((6, 2))
Y = np.array([[13, 14, 15, 16], [17, 18, 19, 20]]).reshape((4, 2))

d = mmd(X=X, Y=Y, beta=-0.5, pval=True, kernel="Laplacian")
print("p-value for multivariate data: ", d['pval'])
print("statistic for multivariate data: ", d['stat'])
print("beta for multivariate data: ", d['beta'])