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Copula Adapted Directed Acyclic Graph for Cluster Representation of Biomedical Data

Heranga K. Rathnasekara, Norou Diawara, Manar D. Samad

arXiv:2609.16240v1stat.MLcs.LGstat.CO

TL;DR

Biomedical labels can be unreliable, and raw-data clustering may miss complex feature dependencies. CopDAG combines copula modeling with DAG-based feature relationships to create cluster-friendly representations; with K-means, it achieved the highest overall average ACC and ARI among the evaluated representation and clustering combinations.

  • Problem

    Biomedical labels can be unreliable, and traditional clustering may overlook complex feature dependencies when applied directly to raw observations.

    Source: §1 · pp. 1–2

  • Method

    CopDAG combines multivariate copula modeling with DAG-based causal and directional feature dependencies to form cluster-friendly data representations.

    Source: §1 · p. 2

  • Results

    CopDAG with K-means achieved the highest overall average ACC and ARI among the feature-representation and clustering combinations.

    Source: §5 · p. 12

  • Takeaways & Limitations

    CopDAG offers a flexible, interpretable framework for dependence-guided representations of heterogeneous biomedical data.

    Source: §6 · p. 14

    Performance may be sensitive to choices such as bootstrap size, stability thresholds, DAG algorithms, candidate copula families, and clustering procedure.

    Source: §5.1 · p. 14

Abstract

from arXiv · show

Diagnostic errors and mislabeling are common in biomedicine, which compromise the reliability of predictive models and data-driven outcomes. Stratifying unlabeled biomedical data based on complex relationships between features eliminates the need for data labels and overcomes the limitations of supervised learning. Traditional clustering methods assume restrictive data distributions, making them suboptimal for capturing complex dependencies in high-dimensional biomedical data. This paper introduces a novel cluster-friendly data presentation framework that integrates the non-Gaussian and non-linear feature dependence of copula models with an ensemble of causal structure discovery (CSD) methods based on Directed Acyclic Graphs (DAGs). While copulas model flexible multivariate distributions by relaxing assumptions related to multivariate normality, linear dependence, and symmetric relationships, an ensemble of DAG-based CSD methods identifies stable causal relationships between features. When clustered using K-means, the new data representation obtained by the proposed copula-adapted DAG (CopDAG) ranks first among the 12 methods in normalized clustering accuracy and adjusted Rand index across 16 biomedical datasets. Our CopDAG method predicts ground-truth class labels directly from feature relationships without data annotations and supervised learning, while also providing cluster visualizations and explainable causal structures of the biomedical data features.

1 Introduction

Unreliable biomedical labels make supervised analyses difficult, while clustering can identify groups from observed feature relationships without using response features. The paper addresses gaps in translating complex dependencies into cluster-friendly representations, combining copula models with DAGs.

1 Introduction

  • Unreliable or costly outcome labels can compromise biomedical modeling, motivating methods that do not depend on those labels.Diagnostic errors, bias, unequal class distributions, and sampling distortions are identified as sources of inaccurate labels.Source: §1 · p. 1
  • Clustering identifies latent groups from relationships among observed features without using response features, and may reveal subgroups beyond known clinical stratifications.Traditional clustering can overlook nonlinear, asymmetric, and conditional dependencies when applied directly to raw observations.Source: §1 · pp. 1–2
  • CopDAG combines copula models of nonlinear and conditional dependencies with DAGs that learn causal and directional feature relationships.The framework transforms raw data into representations intended to recover ground-truth labels while providing interpretable feature relationships and cluster visualizations.Source: §1 · p. 2

1.1 Related work

  • Existing approaches leave a gap between dependence modeling and cluster-friendly representations: copula applications often interpret dependence, while some DAG methods omit directions or assume Gaussian linear structure.The cited work includes a DAG-skeleton approach that does not estimate edge directions and a phenotype-network method with a Gaussian linear structural assumption.Source: §1.1 · p. 2
  • Prior copula-based graphical models mainly addressed network inference and interpretation, leaving unclear how conditional feature relationships could facilitate clustering.Existing copula clustering approaches embed dependence in a specific clustering model rather than providing a general representation for different algorithms.Source: §1.1 · p. 2

2 Preliminaries

The preliminaries establish copulas as a way to separate marginal distributions from feature dependence, then describe vine and DAG structures for representing that dependence. They set up CopDAG’s use of copula densities for relationships between DAG-connected features.

2 Preliminaries

  • The preliminaries introduce copulas and directed acyclic graphs as theoretical foundations for modeling causal relationships.The section sets up the mathematical formulation used to explain the proposed CopDAG framework.Source: §2 · p. 2, §2.5 · p. 4

2.1 Copula Theory and Sklar’s Theorem

  • A copula separates dependence from marginal behavior; uniform probability-integral transforms and flexible copula families accommodate features with differing distributions and skewness.Gaussian, Student-t, Clayton, Gumbel, and Frank families represent different dependence patterns, including tail dependence.Source: §2.1 · p. 3

2.2 Vine Copulas

  • Vine copulas decompose multivariate dependence into conditional bivariate pair-copulas arranged in linked trees, allowing different families for different feature pairs.The R-vine construction does not impose the specific structural forms of C-vines or D-vines; the study selects among five candidate families using AIC.Source: §2.2 · p. 3

2.3 DAG-Based Causal Structure

  • A DAG represents features as nodes connected by directed edges without cycles; DAG learning seeks edges that reproduce the data’s conditional-independence structure.DAGs encode parent sets and Markov blankets and can provide sparse structures for dependence modeling.Source: §2.3 · pp. 3–4

2.4 R-vine Graphical Representation

  • An R-vine is represented as linked undirected trees, with later-tree nodes formed from edges in the preceding tree.The first tree is defined over the feature nodes and its edges.Source: §2.4 · p. 4

2.5 Copula-DAG Formulation

  • CopDAG models each feature’s dependence on its DAG parents with a copula density and defines the corresponding conditional distribution using parent sets.The framework considers both parent- and child-based joint formulations, but expresses DAG factorization in terms of parent sets.Source: §2.5 · p. 4

3 Proposed Framework

CopDAG combines Gaussian-copula scores with DAG-derived dependence structure to create a weighted feature representation, alongside raw-DAG, R-vine, and raw-data baselines.

3 Proposed Framework

  • CopDAG is a Gaussian copula-adapted DAG framework compared with raw-DAG, R-vine, and unweighted raw-data feature representations.The R-vine baseline aggregates estimated feature-dependence information across individual models.Source: §3 · p. 5

3.1 Preliminary Data Transformation

  • Standardized features are bootstrap-sampled and transformed into pseudo-observations; CopDAG further transforms these into Gaussian copula scores.The pseudo-observations are used directly for R-vine.Source: §3.1 · p. 5

3.2 Dependence Structure Learning and Ensemble Aggregation

  • Seven DAG learners are majority-voted within bootstrap samples; arcs meeting stability threshold γ form consensus graphs whose node connectivity yields normalized feature weights.The resulting weighted representations make features with larger connectivity scores contribute more strongly while retaining the original predictors’ standardized scale.Source: §3.2 · pp. 6–7

3.3 Clustering Methods and Evaluation Metrics

  • The framework is evaluated with K-means, Ward’s D2 hierarchical clustering, and Gaussian Mixture Models using ACC, ARI, and NMI.These represent distance-based, hierarchical agglomerative, and model-based clustering paradigms, respectively.Source: §3.3 · p. 8

3.4 Experiment: Application on Biomedical Datasets

  • The application covers 16 public biomedical datasets across disease domains, sample sizes, feature dimensions, and class structures; dataset-specific γ choices of 0.3, 0.4, or 0.6 changed clustering results only marginally.Missing continuous and binary values were imputed using the mean and mode, respectively, and observations with missing response variables were removed.Source: §3.4 · p. 9

4 Results

Across 16 biomedical datasets, results varied by representation, clustering method, and metric: CopDAG with K-means led in average ACC and ARI, while dependence- and DAG-guided representations led on 11 datasets. K-means improvements over raw data were not statistically significant after Holm correction, and feature-dependence graphs showed class-specific structures.

4.1 Best method for individual datasets

  • Dependence- and DAG-guided representations led on 11/16 datasets, uniquely on nine and tied with raw data on two; raw data led on the remaining five.R-vine led on four datasets, while CopDAG and raw-DAG each led on three.Source: §4.1 · p. 10, Table 2 · p. 10

4.2 Overall clustering performance

  • CopDAG with K-means attained the highest average ACC and ARI, while raw-DAG with GMM achieved the best overall average rank.Table 3 reports normalized scores averaged across datasets; lower rank indicates better performance.Source: Table 3 · p. 10, §4.2 · p. 11

4.3 Effects of clustering methods

  • GMM consistently ranked among the top-performing approaches, K-means varied more widely by representation, and every HWD2 combination ranked in the bottom half.Large performance standard deviations across datasets supported using average rank as a more reliable summary than mean performance alone.Source: §4.3 · p. 11

4.2 Overall clustering performance

  • Figure 2 compares t-SNE views for two datasets where K-means performed best; differences among representations suggest they alter observation distributions in the embedded space.Open circles mark predicted labels and crosses mark true labels.Source: §4.3 · p. 11, Figure 2 · p. 11

4.3 Effects of clustering methods

  • With K-means, R-vine improved ACC over raw data on 11/16 datasets (68.8%) and CopDAG on 10/16 (62.5%), but no comparison remained significant after Holm correction.The positive median change was offset by larger drops on a smaller subset of datasets.Source: §4.3 · pp. 11–12, Table 4 · p. 12

4.4 Feature dependence structures

  • Class-specific DAG and R-vine structures differed between diabetic and non-diabetic patients, suggesting class separation may reflect dependence patterns as well as marginal feature values.For non-diabetic patients, most raw-data clinical features depended only on age; the diabetic raw-data structure was more complex.Source: §4.4 · p. 12

5 Discussion

CopDAG with K-means achieved the highest overall average ACC and ARI among the tested representation–clustering combinations, while performance varied by dataset and clustering method. The discussion motivates dependence-guided representations and identifies both integration and parameter-selection limitations.

5 Discussion

  • CopDAG with K-means achieved the highest overall average ACC and ARI among all feature-representation and clustering combinations.The discussion also reports that effectiveness depends on the dataset and clustering technique, with the largest gains over raw data observed using K-means.Source: §5 · p. 12
  • The discussion describes DAG structures and R-vine structures as revealing distinct feature relationships across clusters or patient cohorts.For the Pima Indians Diabetes dataset, Figure 3 presents class-specific dependence structures.Source: §5 · p. 12, Figure 3 · p. 13
  • DAG-based representations may help separate overlapping clusters by encoding conditional feature dependencies, whose weights are assumed to vary across cluster observations.The discussion links this motivation to clustering without available class labels and to complex, potentially high-dimensional feature spaces.Source: §5 · p. 13
  • The authors state that no model is optimal for every data problem and that clustering performance varies across datasets and clustering methods.They note that accuracy improvement is not statistically significant, while describing a 5% increase as equivalent to 50 of 1,000 observations with practical value in science.Source: §5 · p. 13

5.1 Limitations

  • The proposed integration of DAGs and copulas may not be optimal, and translating DAG causal structures into cluster-friendly representations remains an open problem.The authors call for deeper analysis of why dependence weighting improves performance more on some datasets than others.Source: §5.1 · p. 14
  • CopDAG requires informed choices about bootstrap size, voting cutoff, stability threshold, DAG algorithms, candidate copula families, and clustering procedure.The authors state that performance may be sensitive to these choices, which can yield varying dependence structures and feature weights.Source: §5.1 · p. 14

6 Conclusion

The conclusion presents a representation shaped by DAG-learned feature dependencies and multivariate copula models, reporting strong recovery of true cluster structures across clustering settings. The accompanying declarations specify AI-tool use, funding, data availability, conflicts, and author contributions.

6 Conclusion

  • The proposed data representation combines feature dependencies learned through DAG-based causal structure discovery and multivariate copula models.The conclusion reports strong recovery of true cluster structures across different clustering settings and says multivariate copula modeling could further improve performance.Source: §6 · p. 14

Declaration of Generative AI and AI-Assisted Technologies

Supplementary Materials

The supplementary materials include clustering-performance summaries for all 16 biomedical datasets, dataset-specific preprocessing details, and additional t-SNE visualizations. The supplied passages identify the scope of the tables and figures but do not provide their performance values or visual outcomes.

S1 Clustering Performance for Biomedical Datasets

S2 Dataset-Specific Preprocessing

  • For Pima Indians Diabetes, preprocessing excluded identifier fields and removed observations with zero values in four specified clinical features, leaving 393 of 768 subjects.Source: S2 Dataset-Specific Preprocessing · p. 19
  • For the Bone Marrow Transplant: Children dataset, outcome-related and post-transplant time-to-event features were removed to reduce outcome leakage; survival status was retained only for post-clustering evaluation.Source: S2 Dataset-Specific Preprocessing · p. 19
  • For the Sachs protein-signaling dataset, PKA was dichotomized at its median, and the Shapiro–Wilk test used a random sample of 5,000 observations.Source: S2 Dataset-Specific Preprocessing · p. 19

S3 t-SNE Visualizations for Three Additional Datasets

  • Figure S1 presents t-SNE visualizations using K-means clustering for three datasets.Source: Figure S1 · p. 20

S4 t-SNE Visualizations Breast Cancer Wisconsin Dataset using K-means, HWD2 and GMM Clustering

The supplementary figures and tables provide visualizations and relative-performance summaries for the Breast Cancer Wisconsin dataset. The supplied captions identify the plotted clustering methods, label markers, and metrics, without stating comparative outcomes.

S4 t-SNE Visualizations Breast Cancer Wisconsin Dataset using K-means, HWD2 and GMM Clustering

  • Figure S2 presents t-SNE visualizations for K-means, HWD2, and GMM, marking predicted labels with open circles and true labels with crosses.Source: Figure S2 · p. 21

S5 Three-dimensional t-SNE Visualizations of the K-means Clustering Results for Breast Cancer Wisconsin Dataset

  • Figure S3 presents three-dimensional t-SNE visualizations of K-means clustering results for the Breast Cancer Wisconsin dataset.Source: Figure S3 · p. 22
  • Table S3 reports ARI changes relative to raw data and counts datasets with improved, tied, or worsened results.Source: Table S3 · p. 23
  • Table S4 reports NMI changes relative to raw data and counts datasets with improved, tied, or worsened results.Source: Table S4 · p. 23
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