linear_time_mmd.rst

Linear Time MMD

The linear time MMD implements a nonparametric statistical hypothesis test to reject the null hypothesis that to distributions p and q, each only observed via n samples, are the same, i.e. H₀ : p = q.

The (unbiased) statistic is given by

$$\frac{2}{n}\sum_{i=1}^n k(x_{2i},x_{2i}) + k(x_{2i+1}, x_{2i+1}) - 2k(x_{2i},x_{2i+1}).$$

See gretton2012kernel for a detailed introduction.

Example

Imagine we have samples from p and q. As the linear time MMD is a streaming statistic, we need to pass it CStreamingFeatures. Here, we use synthetic data generators, but it is possible to construct CStreamingFeatures from (large) files.

linear_time_mmd.sg:create_features

We create an instance of CLinearTimeMMD, passing it data and the kernel to use,

linear_time_mmd.sg:create_instance

An important parameter for controlling the efficiency of the linear time MMD is block size of the number of samples that is processed at once. As a guideline, set as large as memory allows.

linear_time_mmd.sg:set_burst

Computing the statistic is done as

linear_time_mmd.sg:estimate_mmd

We can perform the hypothesis test via computing the rejection threshold

linear_time_mmd.sg:perform_test_threshold

Alternatively, we can compute the p-value for the above value of the statistic

linear_time_mmd.sg:perform_test_p_value

References

../../references.bib