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In statistics, the term independence means that one can directly obtain the probability of any event involving both variables by multiplying their individual probabilities. Tests of independence are chi-square tests involving the use of a contingency table of observed (data) values.
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A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Measuring Statistical Dependence via Characteristic Function IPM.

Povilas Daniušis1,2, Shubham Juneja3, Lukas Kuzma3

  • 1Neurotechnology, Laisvės av. 125A, 06118 Vilnius, Lithuania.

Entropy (Basel, Switzerland)
|December 24, 2025
PubMed
Summary

We introduce the uniform Fourier dependence measure (UFDM) for analyzing statistical dependence in the frequency domain. UFDM effectively detects complex dependencies and integrates into machine learning, outperforming other methods in feature extraction tasks.

Keywords:
IPMcharacteristic functionsindependence testingstatistical dependencesupervised feature extractionuniform norm

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Area of Science:

  • Statistics
  • Machine Learning
  • Frequency Domain Analysis

Background:

  • Statistical dependence is crucial for data analysis.
  • Existing measures may not capture all types of dependencies.
  • Frequency domain analysis offers a unique perspective.

Purpose of the Study:

  • To propose a novel measure for statistical dependence in the frequency domain.
  • To introduce the uniform Fourier dependence measure (UFDM).
  • To evaluate UFDM's theoretical properties and empirical performance.

Main Methods:

  • Defined UFDM using characteristic functions within the integral probability metric (IPM) framework.
  • Developed a gradient-based estimation algorithm with singular value decomposition (SVD) warm-up.
  • Compared UFDM against distance correlation (DCOR), HSIC, and MEF using independence testing and feature extraction.

Main Results:

  • UFDM exhibits desirable properties like invariances and monotonicity.
  • The SVD warm-up is critical for stable UFDM estimation.
  • UFDM demonstrated effectiveness in detecting sparse geometric dependencies.
  • UFDM outperformed baselines in 20 out of 160 feature extraction comparisons.

Conclusions:

  • UFDM is a powerful new tool for statistical dependence analysis.
  • Its differentiability allows seamless integration into machine learning pipelines.
  • UFDM shows promise for both independence testing and feature extraction.