# K-Sample¶

## Non-parametric K-Sample Test¶

class mgc.ksample.KSample(indep_test, compute_distance=<function euclidean>)[source]

Class for calculating the k-sample test statistic and p-value.

A k-sample test tests equality in distribution among groups. Groups can be of different sizes, but generally have the same dimensionality. There are not many non-parametric k-sample tests, but this version cleverly leverages the power of some of the implemented independence tests to test this equality of distribution.

Parameters: indep_test : {"CCA", "Dcorr", "HHG", "RV", "Hsic", "MGC", "MGCRF"} A string corresponding to the desired independence test from mgc.independence. compute_distance : callable(), optional (default: euclidean) A function that computes the distance among the samples within each data matrix. Set to None if x and y are already distance matrices. To call a custom function, either create the distance matrix before-hand or create a function of the form compute_distance(x) where x is the data matrix for which pairwise distances are calculated.

Notes

The ideas behind this can be found in an upcoming paper:

The k-sample testing problem can be thought of as a generalization of the two sample testing problem. Define $$\{ u_i \stackrel{iid}{\sim} F_U,\ i = 1, ..., n \}$$ and $$\{ v_j \stackrel{iid}{\sim} F_V,\ j = 1, ..., m \}$$ as two groups of samples deriving from different distributions with the same dimensionality. Then, problem that we are testing is thus,

$\begin{split}H_0: F_U &= F_V \\ H_A: F_U &\neq F_V\end{split}$

The closely related independence testing problem can be generalized similarly: Given a set of paired data $$\{\left(x_i, y_i \right) \stackrel{iid}{\sim} F_{XY}, \ i = 1, ..., N\}$$, the problem that we are testing is,

$\begin{split}H_0: F_{XY} &= F_X F_Y \\ H_A: F_{XY} &\neq F_X F_Y\end{split}$

By manipulating the inputs of the k-sample test, we can create concatenated versions of the inputs and another label matrix which are necessarily paired. Then, any nonparametric test can be performed on this data.

test(*args, reps=1000, workers=1, random_state=None)[source]

Calculates the k-sample test statistic and p-value.

Parameters: *args : ndarrays Variable length input data matrices. All inputs must have the same number of samples. That is, the shapes must be (n, p) and (m, p) where n and m are the number of samples and p are the number of dimensions. Alternatively, inputs can be distance matrices, where the shapes must all be (n, n). reps : int, optional (default: 1000) The number of replications used to estimate the null distribution when using the permutation test used to calculate the p-value. workers : int, optional (default: 1) The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process. random_state : int or np.random.RandomState instance, optional If already a RandomState instance, use it. If seed is an int, return a new RandomState instance seeded with seed. If None, use np.random.RandomState. Default is None. stat : float The computed k-Sample statistic. pvalue : float The computed k-Sample p-value.

Examples

>>> import numpy as np
>>> from mgc.ksample import KSample
>>> x = np.arange(7)
>>> y = x
>>> z = np.arange(10)
>>> stat, pvalue = KSample("Dcorr").test(x, y)
>>> '%.3f, %.1f' % (stat, pvalue)
'-0.136, 1.0'


The number of replications can give p-values with higher confidence (greater alpha levels).

>>> import numpy as np
>>> from mgc.ksample import KSample
>>> x = np.arange(7)
>>> y = x
>>> z = np.ones(7)
>>> stat, pvalue = KSample("Dcorr").test(x, y, z, reps=10000)
>>> '%.3f, %.1f' % (stat, pvalue)
'0.172, 0.0'