In this paper, we study the high-dimensional correlated data with multi-level correlations. These data appear frequently in many fields, e.g., genes in gene pathways or stock in industry groups. It motivates us not only to exploit these clusters but also to distinguish the correlation levels. Besides, we analyze the data without pre-specified clustering information to covariates. A two-step method is proposed to address the above problems. The first step focuses on distinguishing the levels and clustering. We aim to divide covariates into sub-vectors, considering both grouping effect and varying correlation. In the second step, we propose a joint estimation and a modified coordinate descent algorithm. The proposed procedure estimates different correlated groups with different penalties. We provide the theoretical guarantees of this method. Numerical comparisons show that the method works effectively on the multi-level correlation structures. We also apply the proposed method to financial data and get interpretable results.

中文翻译：

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具有不同相关级别的集群的多重惩罚正则化

在本文中，我们研究了具有多层次相关性的高维相关数据。这些数据经常出现在许多领域，例如基因通路中的基因或行业组中的股票。它激励我们不仅要利用这些集群，还要区分相关级别。此外，我们对没有预先指定聚类信息的数据进行协变量分析。提出了一种两步法来解决上述问题。第一步侧重于区分级别和聚类。我们的目标是将协变量划分为子向量，同时考虑分组效应和变化的相关性。在第二步中，我们提出了一种联合估计和一种改进的坐标下降算法。建议的程序估计具有不同惩罚的不同相关组。我们提供了这种方法的理论保证。数值比较表明，该方法对多级相关结构有效。我们还将所提出的方法应用于财务数据并获得可解释的结果。