Statistically validated hierarchical clustering: Nested partitions in hierarchical trees

We develop an algorithm that is fast and scalable in the detection of a nested partition extracted from a dendrogram that is obtained from hierarchical clustering of a multivariate series. Our algorithm provides a -value for each clade observed in the hierarchical tree. The -value is obtained by computing many bootstrap replicas of the dissimilarity matrix and by performing a statistical test on each difference between the dissimilarity associated with a given clade and the dissimilarity of the clade of its parent node. We prove the efficacy of our algorithm with a set of benchmarks generated by a hierarchically nested factor model. We compare results obtained by our algorithm with those of Pvclust. Pvclust is a widely-used algorithm pursuing a global approach originally developed in the context of phylogenetic studies. In our numerical experiments, we focus on the role of multiple hypothesis test correction and the robustness of the algorithms to inaccuracies and errors of datasets. We verify that our algorithm is much faster than Pvclust algorithm and has a better scalability both in the number of elements and in the number of records of the investigated multivariate set. We also apply our algorithm to two empirical datasets, one related to a biological complex system and the other related to financial time-series. We prove that the clusters detected by our methodology are meaningful with respect to some consensus partitioning of the two datasets.