Unveil Linear Patterns of Dependence via K Regression Clustering
提出K回归聚类方法，将样本分为K个簇，每个簇内观测具有相同的线性依赖模式，不同簇结构不同。通过最小化簇内损失函数估计系数，证明估计量的一致性和渐近正态性，并设计针对回归聚类的BIC准则。模拟和临床试验亚组分析显示，ℓ1K回归在系数估计、簇数确定和分类上优于其他方法，适合大规模异质性数据分析。
Clustering is a fundamental problem in many scientific applications. This paper introduces the concept of K-regression, which divides a random sample of size n into K clusters such that the observations within each cluster exhibit an identical linear pattern of dependence, and the observations in different clusters exhibit distinctive structures of linear dependence. We estimate the coefficients of the clustering regressions through minimizing the within cluster ℓ1 and ℓ2 loss functions. From the asymptotic perspective, the resulting estimates obtained with either the ℓ1 or the ℓ2 loss are strongly consistent and asymptotically normal. From the non-asymptotic perspective, we further explore the conditions under which the models are identifiable and the algorithms are convergent. Furthermore, we propose a tailored Bayesian Information Criterion (BIC) designed specifically for regression-based clustering. Through extensive simulations and an application to clinical trial subgroup analysis, we demonstrate the effectiveness of K-regression. Numerical results highlight that, in the presence of heterogeneity, ℓ1K-regression outperforms alternative methods (including ℓ2K-regression) in coefficient estimation, cluster number determination, and subgroup classification while maintaining computational efficiency. These advantages make ℓ1K-regression particularly appealing for large-scale data analysis, especially when heterogeneous subpopulations are present.