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Categorical distance correlation under general encodings and its application to high-dimensional feature screening

Qingyang Zhang

TL;DR

This work extends distance correlation to categorical data by introducing general encodings (e.g., ordinal and semicircle) beyond one-hot, enabling $\mathrm{dCov}^{2}$ and $\mathrm{dVar}^{2}$ calculations that incorporate category spacings via $d^{X}_{ik}$ and $d^{Y}_{jl}$. It provides two sample estimators (maximum likelihood and a bias-corrected $U$-statistic) and derives their asymptotic null and alternative distributions using eigenstructures of encoding-specific matrices, facilitating p-value calculation and power analysis. The framework supports high-dimensional feature screening with sure screening guarantees under mild conditions and includes a data-driven change-point method for threshold selection. Simulations reveal that ordinal and semicircle encodings often outperform one-hot in monotone relationships, while one-hot can be advantageous for nonmonotone patterns; the method is demonstrated on the 2018 General Social Survey, identifying 32 relevant factors for subjective socioeconomic classification. Overall, the paper broadens the applicability of categorical distance correlation, offering principled encoding choices, rigorous theory, and practical screening tools for mixed nominal/ordinal data.

Abstract

In this paper, we extend distance correlation to categorical data with general encodings, such as one-hot encoding for nominal variables and semicircle encoding for ordinal variables. Unlike existing methods, our approach leverages the spacing information between categories, which enhances the performance of distance correlation. Two estimates including the maximum likelihood estimate and a bias-corrected estimate are given, together with their limiting distributions under the null and alternative hypotheses. Furthermore, we establish the sure screening property for high-dimensional categorical data under mild conditions. We conduct a simulation study to compare the performance of different encodings, and illustrate their practical utility using the 2018 General Social Survey data.

Categorical distance correlation under general encodings and its application to high-dimensional feature screening

TL;DR

This work extends distance correlation to categorical data by introducing general encodings (e.g., ordinal and semicircle) beyond one-hot, enabling and calculations that incorporate category spacings via and . It provides two sample estimators (maximum likelihood and a bias-corrected -statistic) and derives their asymptotic null and alternative distributions using eigenstructures of encoding-specific matrices, facilitating p-value calculation and power analysis. The framework supports high-dimensional feature screening with sure screening guarantees under mild conditions and includes a data-driven change-point method for threshold selection. Simulations reveal that ordinal and semicircle encodings often outperform one-hot in monotone relationships, while one-hot can be advantageous for nonmonotone patterns; the method is demonstrated on the 2018 General Social Survey, identifying 32 relevant factors for subjective socioeconomic classification. Overall, the paper broadens the applicability of categorical distance correlation, offering principled encoding choices, rigorous theory, and practical screening tools for mixed nominal/ordinal data.

Abstract

In this paper, we extend distance correlation to categorical data with general encodings, such as one-hot encoding for nominal variables and semicircle encoding for ordinal variables. Unlike existing methods, our approach leverages the spacing information between categories, which enhances the performance of distance correlation. Two estimates including the maximum likelihood estimate and a bias-corrected estimate are given, together with their limiting distributions under the null and alternative hypotheses. Furthermore, we establish the sure screening property for high-dimensional categorical data under mild conditions. We conduct a simulation study to compare the performance of different encodings, and illustrate their practical utility using the 2018 General Social Survey data.
Paper Structure (17 sections, 8 theorems, 70 equations, 7 figures, 3 tables)

This paper contains 17 sections, 8 theorems, 70 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Different encodings for categorical data
  4. Categorical distance correlation under general encodings
  5. Asymptotic distributions
  6. Application to high-dimensional feature screening
  7. Simulation study
  8. A real application
  9. Discussion and conclusions
  10. Appendix
  11. Proof of Lemma \ref{['dcov']}
  12. Proof of Lemma \ref{['dvar']}
  13. Proof of Lemma \ref{['diff']}
  14. Proof of Theorem \ref{['null']}
  15. Proof of Theorem \ref{['alter']}
  16. ...and 2 more sections

Key Result

Lemma 1

Under general encodings and distance matrices $D^{X}$ and $D^{Y}$, the squared distance covariance between categorical variables $X$ and $Y$ can be expressed as An equivalent matrix form is where $D^{X}\otimes D^{Y}$ represents the Kronecker product of $D^{X}$ and $D^{Y}$, and $\Delta$ is the column vector $(\pi_{11}-\pi_{1+}\pi_{+1}, ..., \pi_{1J}-\pi_{1+}\pi_{+J}, ~...,~\pi_{IJ}-\pi_{I+}\pi_{+

Figures (7)

  • Figure 1: Dependence patterns in the six simulation settings, with darker shades indicating a greater departure from independence.
  • Figure 2: ROC curves for distance correlation screening under three encodings (one-hot, ordinal, and semicircle) in Setting 1.
  • Figure 3: ROC curves for distance correlation screening under three encodings (one-hot, ordinal, and semicircle) in Setting 2.
  • Figure 4: ROC curves for distance correlation screening under three encodings (one-hot, ordinal, and semicircle) in Setting 3.
  • Figure 5: ROC curves for distance correlation screening under three encodings (one-hot, ordinal, and semicircle) in Setting 4.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Remark 2
  • Remark 3
  • Lemma 3
  • Remark 4
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • ...and 2 more