Expectile Sparse Clustering for Mixed-Type Data with Data-Driven Ordinal Scoring
- 1Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, El Jadida 24000, Morocco
- 2LERSEM Laboratory, National School of Business and Management, Chouaib Doukkali University, Corner Avenue Ahmed Chaouki and Rue de Fès, BP.122, El Jadida 24000, Morocco
- 3AGEIS Laboratory, Grenoble Alpes University (UGA), UFR SHS, BP.47, CEDEX 09, 38040 Grenoble, France
Abstract
High-dimensional datasets combining continuous, ordinal, and nominal variables are increasingly prevalent in modern applications such as marketing analytics, biostatistics, and social sciences. Clustering in such settings is challenging because continuous variables may exhibit asymmetric distributional structure, ordinal variables have a natural ordering but no canonical numerical spacing, and irrelevant variables can obscure the underlying cluster structure. We propose ExSC-τθ, an adaptive sparse clustering framework for high-dimensional mixed-type data that addresses these three issues within a unified variable-selection framework. For continuous variables and Azzalini-scored ordinal variables, we introduce an expectile-based asymmetric dissimilarity parameterized by τ∈(0,1), which reduces to the squared Euclidean dissimilarity at τ=12 and allows for directional differences to receive unequal quadratic penalties. For ordinal variables, we extend Azzalini’s score transformation to the unsupervised setting by estimating the transformation parameters from a cluster-quality criterion, thereby obtaining data-driven non-equidistant numerical representations. For nominal variables, we employ a co-occurrence-based dissimilarity that incorporates dependence among categorical variables. The resulting type-specific dissimilarities are integrated through separate ℓ1-constrained variable-weight vectors and a variable-wise adjusted Davies–Bouldin objective. We establish screening consistency under explicit signal-strength, sparsity, dimensionality, and regularity conditions. Simulation studies covering different numbers of clusters and distributional heterogeneity demonstrate that the proposed framework is particularly effective for ordinal-variable recovery and for clustering structures in which symmetric dissimilarities are inadequate. Two real data applications, Steel Annealing and Credit Approval, illustrate the method, including the interpretability of the learned ordinal representation, the variable selection and the adaptive asymmetry parameter.
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BibTeX
@article{arjdal2026expectile,
title = {{Expectile Sparse Clustering for Mixed-Type Data with Data-Driven Ordinal Scoring}},
author = {Ismail Arjdal and Mohamed Alahiane and Echarif Elharfaoui and Mustapha Rachdi},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213966},
url = {https://www.mdpi.com/2227-7390/14/21/3966},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Expectile Sparse Clustering for Mixed-Type Data with Data-Driven Ordinal Scoring
AU - Ismail Arjdal
AU - Mohamed Alahiane
AU - Echarif Elharfaoui
AU - Mustapha Rachdi
PY - 2026
DA - 2026/10/30
JO - Mathematics
DO - 10.3390/math14213966
UR - https://www.mdpi.com/2227-7390/14/21/3966
AB - High-dimensional datasets combining continuous, ordinal, and nominal variables are increasingly prevalent in modern applications such as marketing analytics, biostatistics, and social sciences. Clustering in such settings is challenging because continuous variables may exhibit asymmetric distributional structure, ordinal variables have a natural ordering but no canonical numerical spacing, and irrelevant variables can obscure the underlying cluster structure. We propose ExSC-τθ, an adaptive sparse clustering framework for high-dimensional mixed-type data that addresses these three issues within a unified variable-selection framework. For continuous variables and Azzalini-scored ordinal variables, we introduce an expectile-based asymmetric dissimilarity parameterized by τ∈(0,1), which reduces to the squared Euclidean dissimilarity at τ=12 and allows for directional differences to receive unequal quadratic penalties. For ordinal variables, we extend Azzalini’s score transformation to the unsupervised setting by estimating the transformation parameters from a cluster-quality criterion, thereby obtaining data-driven non-equidistant numerical representations. For nominal variables, we employ a co-occurrence-based dissimilarity that incorporates dependence among categorical variables. The resulting type-specific dissimilarities are integrated through separate ℓ1-constrained variable-weight vectors and a variable-wise adjusted Davies–Bouldin objective. We establish screening consistency under explicit signal-strength, sparsity, dimensionality, and regularity conditions. Simulation studies covering different numbers of clusters and distributional heterogeneity demonstrate that the proposed framework is particularly effective for ordinal-variable recovery and for clustering structures in which symmetric dissimilarities are inadequate. Two real data applications, Steel Annealing and Credit Approval, illustrate the method, including the interpretability of the learned ordinal representation, the variable selection and the adaptive asymmetry parameter.
SN - 2227-7390
ER -