Clustering and hierarchy of financial markets data: advantages of the DBHT

We present a set of analyses aiming at quantifying the amount of information filtered by di↵erent hierarchical clustering methods on correlations between stock returns. In particular we apply, for the first time to financial data, a novel hierarchical clustering approach, the Directed Bubble Hierarchical Tree (DBHT), and we compare it with other methods including the Linkage and k-medoids. In particular by taking the industrial sector classification of stocks as a benchmark partition we evaluate how the di↵erent methods retrieve this classification. The results show that the Directed Bubble Hierarchical Tree outperforms the other methods, being able to retrieve more information with fewer clusters. Moreover, we show that the economic information is hidden at di↵erent levels of the hierarchical structures depending on the clustering method. The dynamical analysis also reveals that the di↵erent methods show di↵erent degrees of sensitivity to financial events, like crises. These results can be of interest for all the applications of clustering methods to portfolio optimization and risk hedging. Introduction Correlation-based networks have been extensively used in Econophysics as tools to filter, visualise and analyse financial market data [1], [2], [3], [4], [5], [6], [7], [8]. Since the seminal work of Mantegna on the Minimum Spanning Tree (MST) [1] they have provided insights into several aspects of financial markets including financial crises [9], [10], [11], [12], [13], [14], [15]. The MST is strictly related [16] to a hierarchical clustering algorithm, namely the Single Linkage (SL) [17]. Starting from a set of elements (e.g., stocks) and a related distance matrix (e.g., a convenient transformation of the stocks correlation matrix [1]), the SL performs an agglomerative algorithm that ends up with a tree (dendrogram) that arranges the elements in a hierarchical structure [16]. The filtering procedure linked to MST and SL has been succesfully applied to improve portfolio optimization [10]. Another hierarchical clustering method, the Average Linkage (AL), has been shown to be associated to a slightly di↵erent version of spanning tree [18], that the authors called the Average Linkage Minimum Spanning Tree. Another variant of Linkage methods, not associated to a spanning tree representation, is the Complete Linkage (CL) [17]. The MST is the first but not the only correlation-based filtered network studied in literature. In particular the Planar Maximally Filtered Graph (PMFG) is a further step from the MST, that is able to retain a higher amount of information [3], [4], [19], having less strict topology constraint allowing to keep a larger number of links. The PMFG has been proven to have interesting pratical applications, in particular in the field of investment strategies to hedge risk [20]. Since the MST has associated a clustering method, and the PMFG is a generalization of the MST, it could be raised the question whether the PMFG provides a clustering method that exploits this higher amount of information. In a recent work [21] it has been shown that this is the case: the Directed Bubble Hierarchical Tree (DBHT) is a novel hierarchical clustering method that takes advantage of the ar X iv :1 40 6. 04 96 v1 [ qfin .S T] 2 Ju n 20 14