Multivariate Hypothesis Testing

Developing and implementing multivariate nonparametric hypothesis tests with high finite testing power in Python

MGC Map

Summary

Hypothesis testing is a fundamental problem in science and provides a framework for testing hypotheses. Two popular ones are independence and $k$-sample testing. Generally, independence testing asks whether the distributions of inputs $X$ and $Y$ are sampled independently or not from a joint distribution. In other words, if $F_X$ is the distribution of $X$, $F_Y$ is the distribution of $Y$, and $F_{XY}$ is the joint distribution of $X$ and $Y$, independence testing performs this test: $$ \begin{split} H_0 : F_{XY} &= F_X F_Y \\ H_A : F_{XY} &\neq F_X F_Y \end{split} $$ $k$-sample testing is similar. Here, we are asking if we had $k$ inputs, are the distributions of all these inputs the same or is at least one different? Specifically, for $k$ inputs $U_1, \ldots, U_k$, with distributions $F_{u_1}, \ldots F_{u_k}$, $k$-sample testing asks: $$ \begin{split} H_0 &: F_{U_1} = F_{U_2} = \ldots = F_{U_k} \\ H_A &: \exists\ i \neq j \text{ s.t. } F_{U_i} \neq F_{U_j} \end{split} $$ Many test have been designed to answer this question, however many do not work well for high-dimensional and highly nonlinear data. Our lab has designed multiscale graph correlation (MGC), which works well with this kind of data specifically, and I ported MGC into SciPy. We built upon these tests, whether it be ore accurate ones such as one based on Random Forest or faster implementations of existing ones. We also implemented a framework for running $k$-sample tests that reduces the $k$-sample testing problem to the indpendence testing problem. This allows us to use powerful independence tests that work well with nonlinear and multivariate data and opens up a whole new avenue of new powerful $k$-sample tests.

All of these algorithms and more have been implemented in the hyppo package. The package contains many state-of-the art hypothesis tests and was based on an old package that I had written with a team of students earlier. I currently maintain the package and wrote much of it, including the documentation. This work was what I did during my M.S.E. and the beginning part of my Ph.D. at Johns Hopkins.


📚 The Chi-Square Test of Distance Correlation

Cencheng Shen, Sambit Panda, Joshua T. Vogelstein

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📚 Nonpar MANOVA via Independence Testing

Sambit Panda, Cencheng Shen, Ronan Perry, Jelle Zorn, Antoine Lutz, Carey E. Priebe, Joshua T. Vogelstein

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📚 hyppo: A Multivariate Hypothesis Testing Python Package

Sambit Panda, Satish Palaniappan, Junhao Xiong, Eric W. Bridgeford, Ronak Mehta, Cencheng Shen, Joshua T. Vogelstein

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📚 Learning Interpretable Characteristic Kernels via Decision Forests

Cencheng Shen, Sambit Panda, Joshua T. Vogelstein

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Posted on:
August 25, 2018
Length:
3 minute read, 427 words
See Also: