Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal Learning for Heterogeneous Subgroups Based on Nonlinear Causal Kernel Clustering

Lu Liu, Yang Tang, Kexuan Zhang, Qiyu Sun

TL;DR

This work tackles heterogeneity in causal relations across subgroups within multi-source observational data. It introduces a nonlinear Causal Kernel Clustering (CKC) framework that uses a $u$-centered sample mapping function to map samples into a high-dimensional space isomorphic to the causal graph space, enabling unbiased estimation and subgroup discovery. A nonlinear causal kernel built on the mapped representations clusters samples by their causal structure, and the authors establish space isomorphism and causal identifiability to justify the approach. Empirical results on synthetic data and real-world IOD and Boston Housing data demonstrate CKC’s ability to identify heterogeneous subgroups and enhance downstream causal learning, including early causal-warning signals. The framework provides a flexible, plug-in module for improving causal inference in the presence of distribution shifts and diverse environments.

Abstract

Due to the challenge posed by multi-source and heterogeneous data collected from diverse environments, causal relationships among features can exhibit variations influenced by different time spans, regions, or strategies. This diversity makes a single causal model inadequate for accurately representing complex causal relationships in all observational data, a crucial consideration in causal learning. To address this challenge, the nonlinear Causal Kernel Clustering method is introduced for heterogeneous subgroup causal learning, highlighting variations in causal relationships across diverse subgroups. The main component for clustering heterogeneous subgroups lies in the construction of the $u$-centered sample mapping function with the property of unbiased estimation, which assesses the differences in potential nonlinear causal relationships in various samples and supported by causal identifiability theory. Experimental results indicate that the method performs well in identifying heterogeneous subgroups and enhancing causal learning, leading to a reduction in prediction error.

Causal Learning for Heterogeneous Subgroups Based on Nonlinear Causal Kernel Clustering

TL;DR

This work tackles heterogeneity in causal relations across subgroups within multi-source observational data. It introduces a nonlinear Causal Kernel Clustering (CKC) framework that uses a -centered sample mapping function to map samples into a high-dimensional space isomorphic to the causal graph space, enabling unbiased estimation and subgroup discovery. A nonlinear causal kernel built on the mapped representations clusters samples by their causal structure, and the authors establish space isomorphism and causal identifiability to justify the approach. Empirical results on synthetic data and real-world IOD and Boston Housing data demonstrate CKC’s ability to identify heterogeneous subgroups and enhance downstream causal learning, including early causal-warning signals. The framework provides a flexible, plug-in module for improving causal inference in the presence of distribution shifts and diverse environments.

Abstract

Due to the challenge posed by multi-source and heterogeneous data collected from diverse environments, causal relationships among features can exhibit variations influenced by different time spans, regions, or strategies. This diversity makes a single causal model inadequate for accurately representing complex causal relationships in all observational data, a crucial consideration in causal learning. To address this challenge, the nonlinear Causal Kernel Clustering method is introduced for heterogeneous subgroup causal learning, highlighting variations in causal relationships across diverse subgroups. The main component for clustering heterogeneous subgroups lies in the construction of the -centered sample mapping function with the property of unbiased estimation, which assesses the differences in potential nonlinear causal relationships in various samples and supported by causal identifiability theory. Experimental results indicate that the method performs well in identifying heterogeneous subgroups and enhancing causal learning, leading to a reduction in prediction error.
Paper Structure (16 sections, 5 theorems, 21 equations, 4 figures, 3 tables)

This paper contains 16 sections, 5 theorems, 21 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Description
  4. Method
  5. Marginal Distance Covariance
  6. Sample Mapping Function Based on Hypothesis Testing
  7. Nonlinear Causal Kernel
  8. Causal Identifiability Theory
  9. Space Isomorphism
  10. Causal Identifiability
  11. Experiments
  12. Synthetic Data
  13. Real-world Data
  14. Experiments on Indian Ocean Dipole Data
  15. Experiments on Boston Housing Data
  16. ...and 1 more sections

Key Result

Lemma 4.1

For samples $\textbf{S}=[X_{1},\dots,X_{m}]\in \Bbb{R}^{n\times m}$, the sample distance matrix is denoted as $\textbf{H} \in \Bbb{R}^{n\times n\times m}$, with $\textbf{H}_{i,i',j}=|\textbf{S}_{i,j}-\textbf{S}_{i',j}|$. $\textbf{H}_{\cdot,\cdot,j}$ denotes the sample distance matrix for feature $X_

Figures (4)

  • Figure 1: An illustration of the process of generating multi-source and heterogeneous data from various environments and the approach to heterogeneous subgroup causal learning, taking four features ($m=4$) as an example. $E_{1},\dots$ denote the multiple environments, $X_{1},\dots$ denote the features, $CR_{1},\dots$ denote the diverse causal relationships between features, and $\textbf{l}^{s} (s\in \{1,2,\dots\})$ denote the data generated by the corresponding causal relationships. The objective is to obtain heterogeneous subgroups through clustering, identifying various causal relationships.
  • Figure 2: The operated framework of the method (CKC) which is based on samples $\textbf{S} \in \Bbb{R}^{n\times 2}$ with two features ($m=2$).
  • Figure 3: (a) The evolution of the standardized $\text{YC}(y)$ time series. When it undergoes transitions (passing through red line) and reaches an extreme value (marked by black boxes), there is a possibility of an IOD event in the upcoming year (pointed with the green line). (b) The years predicted by the method are displayed in the circles. Both circles of blue and red represent IOD events among the correctly predicted. The black circles represent the wrongly predicted.
  • Figure 4: The coefficients $\beta^{\text{g}_{i}}$ of features across heterogeneous subgroups $G=\{\text{g}_{1},\text{g}_{2},\dots,\text{g}_{6}\}$ as learned by the method with $\text{K}=6$. The features with lower variances of coefficients are more likely causal factors that the models need to prioritize.

Theorems & Definitions (11)

  • Definition 3.1: Subgroup Invariance
  • Lemma 4.1: Sample Distance Matrix ker2lemma1lemma2discorr2
  • Definition 4.1: Marginal Distance Covariance lemma1r28discorr2
  • Lemma 4.2: Hypothesis Testing r28lemma2tj2
  • Definition 4.2: Sample Mapping Function ker2r17lemma1r28tj2
  • Theorem 4.1
  • Definition 4.3: Nonlinear Causal Kernel ker2r17lemma1s2
  • Theorem 4.2
  • Definition 5.1: Causal Graph Space others5r6ker2others1
  • Definition 5.2: Causal Matrix Space ker2s2others1matrix1
  • ...and 1 more