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Two-sample statistics based on anisotropic kernels.

Xiuyuan Cheng1, Alexander Cloninger2, Ronald R Coifman3

  • 1Department of Mathematics, Duke University, Durham, NC, USA 27708.

Information and Inference : a Journal of the IMA
|September 15, 2020
PubMed
Summary
This summary is machine-generated.

Researchers developed a novel kernel-based Maximum Mean Discrepancy (MMD) statistic to measure distribution differences using sample data. This new method enhances statistical power for detecting distribution variations, particularly in low-dimensional settings.

Keywords:
anisotropic kernelmaximum mean discrepancytwo-sample statistics

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Area of Science:

  • Statistics
  • Machine Learning
  • Computational Biology

Background:

  • Comparing probability distributions is crucial in various scientific fields.
  • Existing methods like Maximum Mean Discrepancy (MMD) have limitations in detecting subtle differences, especially with limited data or complex distributions.
  • The need for more powerful and flexible statistical tests for distribution comparison is evident.

Purpose of the Study:

  • To introduce a novel kernel-based Maximum Mean Discrepancy (MMD) statistic for comparing multivariate distributions from finite samples.
  • To enhance the statistical power of distribution comparison tests by incorporating local geometric information (covariance matrices) and anisotropic kernels.
  • To establish theoretical guarantees for the proposed test, including consistency and finite-sample power bounds.

Main Methods:

  • Development of a new kernel-based MMD statistic utilizing local covariance matrices to construct an anisotropic kernel.
  • The kernel computes affinity between data points and a potentially smaller set of reference points.
  • Theoretical analysis to prove the consistency of the proposed test under mild kernel assumptions and derive finite-sample power bounds.

Main Results:

  • The proposed kernel-based MMD statistic provides a powerful method for distinguishing between distributions, especially when they are locally low-dimensional.
  • The test demonstrates consistency, ensuring reliable results as sample size increases.
  • A finite-sample lower bound for the testing power was obtained, quantifying the test's effectiveness with limited data.

Conclusions:

  • The novel anisotropic kernel-based MMD statistic offers a statistically sound and powerful approach for comparing distributions.
  • The method is particularly effective in scenarios with locally low-dimensional distributions and can handle asymmetric kernel computations.
  • Demonstrated applications in flow cytometry and diffusion MRI highlight the practical utility of the proposed method for distribution comparison.