SmoothCFTest

class hyppo.ksample.SmoothCFTest(num_randfreq=5)

Smooth Characteristic Function test statistic and p-value

The Smooth Characteristic Function test is a two-sample test that uses differences in the smoothed (analytic) characteristic function of two data distributions in order to determine how different the two data are 1.

Parameters

num_randfreq (int) -- Used to construct random array with size (p, q) where p is the number of dimensions of the data and q is the random frequency at which the test is performed. These are the random test points at which test occurs (see notes).

Notes

The test statistic takes on the following form:

$$n{W}_n{\mathrm{\Sigma }}_n^{-1}{W}_n$$

As seen in the above formulation, this test-statistic takes the same form as the Hotelling $T^2$ statistic. However, the components are defined differently in this case. Given data sets X and Y, define the following as $Z_i$, the vector of differences:

$$Z_i=(k(X_i,T_1)-k(Y_i,T_1),\dots ,k(X_i,T_J)-k(Y_i,T_J))\in {\mathbb{R}}^J$$

The above is the vector of differences between kernels at test points, $T_j$. This same formulation is used in the Mean Embedding Test. Moving forward, $W_n$ can be defined:

$$W_n=\frac{1}{n}\sum _{i=1}^n{Z}_i$$

This leaves ${\mathrm{\Sigma }}_n$, the covariance matrix as:

$${\mathrm{\Sigma }}_n=\frac{1}{n}Z{Z}^T$$

In the specific case of the Smooth Characteristic function test, the vector of differences can be defined as follows:

$$Z_i=(f(X_i)\sin (X_i{T}_1)-f(Y_i)\sin (Y_i{T}_1),f(X_i)\cos (X_i{T}_1)-f(Y_i)\cos (Y_i{T}_1),\cdots )\in {\mathbb{R}}^{2J}$$

Once $S_n$ is calculated, a threshold $r_{\alpha }$ corresponding to the $1-\alpha$ quantile of a Chi-squared distribution w/ J degrees of freedom is chosen. Null is rejected if $S_n$ is larger than this threshold.

References

1

Kacper P Chwialkowski, Aaditya Ramdas, Dino Sejdinovic, and Arthur Gretton. Fast two-sample testing with analytic representations of probability measures. Advances in Neural Information Processing Systems, 2015.

Methods Summary

SmoothCFTest.statistic(x, y, random_state)	Calculates the smooth CF test statistic.
SmoothCFTest.test(x, y[, random_state])	Calculates the smooth CF test statistic and p-value.

---

SmoothCFTest.statistic(x, y, random_state)

Calculates the smooth CF test statistic.

Parameters

• x,y (ndarray of float) -- Input data matrices. x and y must have the same number of dimensions. That is, the shapes must be (n, p) and (m, p) where n is the number of samples and p and q are the number of dimensions.

• random_state (int) -- Set random seed for generation of test points

Returns

stat (float) -- The computed Smooth CF statistic.

SmoothCFTest.test(x, y, random_state=None)

Calculates the smooth CF test statistic and p-value.

Parameters

• x,y (ndarray of float) -- Input data matrices. x and y must have the same number of dimensions. That is, the shapes must be (n, p) and (m, p) where n is the number of samples and p and q are the number of dimensions.

• random_state (int) -- Set random seed for generation of test points

Returns

• stat (float) -- The computed Smooth CF statistic.

• pvalue (float) -- The computed smooth CF p-value.

Examples

>>>

>>> import numpy as np
>>> from hyppo.ksample import SmoothCFTest
>>> np.random.seed(1234)
>>> x = np.random.randn(500, 10)
>>> y = np.random.randn(500, 10)
>>> stat, pvalue = SmoothCFTest().test(x, y, random_state=1234)
>>> '%.2f, %.3f' % (stat, pvalue)
'4.70, 0.910'

Examples using hyppo.ksample.SmoothCFTest

K-Sample Testing

previous

Hotelling

next

MeanEmbeddingTest