Fig 1.
Demonstration that the average correlation evolves during time with large changes during periods of market instability.
The figure reports the average correlation for each time window Tk with k = 1, …, n (n = 100, each time window has length L = 1000 trading days), for both non-detrended (blue circles) and detrended log-returns (green squares). The average correlation is highly reduced by detrending the market mode.
Fig 2.
Visualization of the Planar Maximally Filtered Graph (PMFG) and DBHT clusters, for both non-detrended and detrended log-returns.
a) PMFG calculated on the entire period 1997–2012, using non-detrended log-returns. Stocks of the same color belong to the same DBHT cluster. b) PMFG calculated on the same data as in a), but using detrended log-returns. Stocks of the same color belong to the same DBHT cluster.
Fig 3.
Composition of DBHT clusters in terms of ICB supersectors.
The x-axis represents the cluster labels, the y-axis the number of stocks in each cluster. Each colour corresponds to an ICB supersector (legend on the left hand side). The clustering is obtained by using log-returns detrended by removing the market mode. See the Fig. S3 in S1 File for the case non-detrended.
Fig 4.
Composition of clustering in terms of ICB supersectors.
The x-axis represents the cluster labels, the y-axis the number of stocks in each cluster. Each colour corresponds to an ICB supersector (the legend is the same as in Fig. 3). The graphs show the results for a) SL clustering, b) for AL, c) for CL and d) for k-medoids. The clustering is obtained by using log-returns detrended by removing the market mode.
Fig 5.
Demonstration that different clustering methods show different degrees of disparity in the clustering structure.
The disparity measure y is shown for clusterings at different hierachical levels as function of Ncl in the dendrograms, for a) non-detrended log-returns and b) detrended log-returns.
Fig 6.
Demonstration that different clustering methods retrieve different amount of industrial sector information.
The Adjusted Rand Index ℛadj between clustering and ICB supersectors is shown for different number of clusters Ncl. In a) correlations are calculated on non-detrended log-returns, in b) are calculated on detrended log-returns. The vertical dashed line shows the value (Ncl = 19) correspondent to the actual number of ICB supersectors.
Fig 7.
Amount of ICB information retrieved by the clustering methods, in terms of ICB supersectors overexpressed by each cluster.
Each bar graph shows, varying the number of clusters Ncl, how many times () an ICB supersector is overexpressed by a cluster according to the Hypergeometric hypothesis test (i.e., number of null-hypothesis tests being rejected). Each colour shows the number of overexpressions for each ICB supersector. In graphs a)-e) the results for DBHT, AL, CL, SL and k-medoids clustering are shown respectively. The correlations are calculated on detrended log-returns. See Fig. S5 in S1 File for the non-detrended case.
Fig 8.
Dynamical evolution of the DBHT clustering.
Each plot refers to 100 moving time windows of length 1000 trading days. Specifically, in graph a) we plot the number of DBHT clusters, Ncl, for both log-returns non-detrended (red circles) and detrended by the market mode (blue squares), whereas the two dashed horizontal lines are the Ncl values obtained by taking the largest time window of 4026 trading days. Overall the non-detrended case shows a decreasing trend. In graph b) it is shown the disparity measures, y, again for the two sets of DBHT clustering (red dots non-detrended, blue dots detrended), the dashed horizontal lines being the y values from the 4026 length time window. In the non-detrended case the 2007 marks a transition to higher and more volatile values of y. Finally in graph c) it is shown the Adjusted Rand Index, ℛadj, measured at each time window between the detrended and non-detrended clusterings. A steady decreasing trend is evident.
Fig 9.
Dynamical evolution of the similarity between clustering and ICB.
It is shown the Adjusted Rand Index, ℛadj, calculated at each time window Tk (k = 1, …, n) between clustering and ICB partition, for a) DBHT, b) AL, c) CL, d) SL and e) k-medoids method. A drop in the similarity occurs for all the methods during the 2007–2008 crisis. The AL and SL show decreases also during other financial events. At each time window the number of clusters, Ncl, has been chosen in order to maximize the ℛadj itself: in f) we plot these Ncl values for each clustering method. It is evident as the maximum similarity clustering-ICB is reached at different hierarchical levels depending on the clustering method. The correlations are calculated on non-detrended log-returns.
Fig 10.
Dynamical evolution of the similarity between clustering and ICB, with detrended log-returns.
a)-f): Same graphs as in Fig. 9, but by using correlations on detrended log-returns.
Fig 11.
Test of robustness for the dynamical DBHT clustering.
a) Number of clusters Ncl as a function of the time t: the black squares correspond to the DBHT clusterings obtained by using the empirical (non-detrended) log-returns, the blue dots are the average over the 100 Ncl given by the 100 bootstrapping replica correlation matrices (see text for further details). The bar errors in the blue dot plot is the standard deviation calculated among the same set of 100 Ncl. As one can see the empirical Ncl is quite robust against the bootstrapping test. b) Same plot as in a), but by using detrended log-returns.