VARCLUST：使用降维对变量进行聚类
VARCLUST: clustering variables using dimensionality reduction
摘要：
VARCLUST算法是在假设给定聚类中的变量是被随机噪声破坏的少量隐变量的线性组合的情况下提出的。整个聚类任务被看作是选择的统计模型，这是由集群的数量，划分到这些集群的变量和“集群维数”，即向量的线性子空间跨越每个集群的尺寸定义的问题。最佳模型的选择使用近似贝叶斯准则的基础上的拉普拉斯近似和使用一个无信息的均匀先验的聚类数。为了解决在一个巨大的空间的可能的模型搜索的问题，我们提出了一个扩展的ClustOfVar算法，这是专门为子空间的维度只有1，这是类似的结构的$K$ -质心算法。我们提供了一个完整的方法与理论保证，广泛的数值实验，完整的数据分析和实施。该算法基于一致的贝叶斯信息准则（BIC）将变量分配到合适的聚类中，并通过PENALized SEmi-integrated Likestive准则（PESEL）估计每个聚类的维数，并证明了PESEL准则的一致性。此外，我们证明了我们的算法的每一次迭代导致增加的拉普拉斯近似的模型后验概率和提供的标准估计的集群的数量。与其他算法的数值比较表明，VARCLUST可能优于一些流行的机器学习工具的稀疏子空间聚类。我们还报告了实际数据分析的结果，包括TCGA乳腺癌数据和气象数据。所提出的方法在公开可用的R包变量中实现。
英文摘要：
VARCLUST algorithm is proposed for clustering variables under the assumption that variables in a given cluster are linear combinations of a small number of hidden latent variables, corrupted by the random noise. The entire clustering task is viewed as the problem of selection of the statistical model, which is defined by the number of clusters, the partition of variables into these clusters and the 'cluster dimensions', i.e. the vector of dimensions of linear subspaces spanning each of the clusters. The optimal model is selected using the approximate Bayesian criterion based on the Laplace approximations and using a non-informative uniform prior on the number of clusters. To solve the problem of the search over a huge space of possible models we propose an extension of the ClustOfVar algorithm which was dedicated to subspaces of dimension only 1, and which is similar in structure to the $K$-centroid algorithm. We provide a complete methodology with theoretical guarantees, extensive numerical experimentations, complete data analyses and implementation. Our algorithm assigns variables to appropriate clusterse based on the consistent Bayesian Information Criterion (BIC), and estimates the dimensionality of each cluster by the PEnalized SEmi-integrated Likelihood Criterion (PESEL), whose consistency we prove. Additionally, we prove that each iteration of our algorithm leads to an increase of the Laplace approximation to the model posterior probability and provide the criterion for the estimation of the number of clusters. Numerical comparisons with other algorithms show that VARCLUST may outperform some popular machine learning tools for sparse subspace clustering. We also report the results of real data analysis including TCGA breast cancer data and meteorological data. The proposed method is implemented in the publicly available R package varclust.
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