Michele Tumminello

A tool for filtering information in complex systems

We introduce a technique to filter out complex data sets by extracting a subgraph of representative links. Such a filtering can be tuned up to any desired level by controlling the genus of the resulting graph. We show that this technique is especially suitable for correlation-based graphs, giving filtered graphs that preserve the hierarchical organization of the minimum spanning tree but containing a larger amount of information in their internal structure. In particular in the case of planar filtered graphs (genus equal to 0), triangular loops and four-element cliques are formed. The application of this filtering procedure to 100 stocks in the U.S. equity markets shows that such loops and cliques have important and significant relationships with the market structure and properties.

Correlation, Hierarchies, and Networks in Financial Markets

We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative descriptions of asset price dynamics in financial markets. Specifically, we discuss how to define and obtain hierarchical trees, correlation based trees and networks from a correlation matrix. The hierarchical clustering and other procedures performed on the correlation matrix to detect statistically reliable aspects of it are seen as filtering procedures of the correlation matrix. We also discuss a method to associate a hierarchically nested factor model to a hierarchical tree obtained from a correlation matrix. The information retained in filtering procedures and its stability with respect to statistical fluctuations is quantified by using the Kullback-Leibler distance.

Dominating Clasp of the Financial Sector Revealed by Partial Correlation Analysis of the Stock Market

What are the dominant stocks which drive the correlations present among stocks traded in a stock market? Can a correlation analysis provide an answer to this question? In the past, correlation based networks have been proposed as a tool to uncover the underlying backbone of the market. Correlation based networks represent the stocks and their relationships, which are then investigated using different network theory methodologies. Here we introduce a new concept to tackle the above question—the partial correlation network. Partial correlation is a measure of how the correlation between two variables, e.g., stock returns, is affected by a third variable. By using it we define a proxy of stock influence, which is then used to construct partial correlation networks. The empirical part of this study is performed on a specific financial system, namely the set of 300 highly capitalized stocks traded at the New York Stock Exchange, in the time period 2001–2003. By constructing the partial correlation network, unlike the case of standard correlation based networks, we find that stocks belonging to the financial sector and, in particular, to the investment services sub-sector, are the most influential stocks affecting the correlation profile of the system. Using a moving window analysis, we find that the strong influence of the financial stocks is conserved across time for the investigated trading period. Our findings shed a new light on the underlying mechanisms and driving forces controlling the correlation profile observed in a financial market.

Evolution of Worldwide Stock Markets, Correlation Structure and Correlation Based Graphs

We investigate the daily correlation present among market indices of stock exchanges located all over the world in the time period Jan 1996 - Jul 2009. We discover that the correlation among market indices presents both a fast and a slow dynamics. The slow dynamics reflects the development and consolidation of globalization. The fast dynamics is associated with critical events that originate in a specific country or region of the world and rapidly affect the global system. We provide evidence that the short term timescale of correlation among market indices is less than 3 trading months (about 60 trading days). The average values of the non diagonal elements of the correlation matrix, correlation based graphs and the spectral properties of the largest eigenvalues and eigenvectors of the correlation matrix are carrying information about the fast and slow dynamics of correlation of market indices. We introduce a measure of mutual information based on link co-occurrence in networks, in order to detect the fast dynamics of successive changes of correlation based graphs in a quantitative way.

Spanning Trees and bootstrap reliability estimation in correlation based networks

We introduce a new technique to associate a spanning tree to the average linkage cluster analysis. We term this tree as the Average Linkage Minimum Spanning Tree. We also introduce a technique to associate a value of reliability to the links of correlation-based graphs by using bootstrap replicas of data. Both techniques are applied to the portfolio of the 300 most capitalized stocks traded on the New York Stock Exchange during the time period 2001–2003. We show that the Average Linkage Minimum Spanning Tree recognizes economic sectors and sub-sectors as communities in the network slightly better than the Minimum Spanning Tree. We also show that the average reliability of links in the Minimum Spanning Tree is slightly greater than the average reliability of links in the Average Linkage Minimum Spanning Tree.

Statistically Validated Networks in Bipartite Complex Systems

Many complex systems present an intrinsic bipartite structure where elements of one set link to elements of the second set. In these complex systems, such as the system of actors and movies, elements of one set are qualitatively different than elements of the other set. The properties of these complex systems are typically investigated by constructing and analyzing a projected network on one of the two sets (for example the actor network or the movie network). Complex systems are often very heterogeneous in the number of relationships that the elements of one set establish with the elements of the other set, and this heterogeneity makes it very difficult to discriminate links of the projected network that are just reflecting systems heterogeneity from links relevant to unveil the properties of the system. Here we introduce an unsupervised method to statistically validate each link of a projected network against a null hypothesis that takes into account system heterogeneity. We apply the method to a biological, an economic and a social complex system. The method we propose is able to detect network structures which are very informative about the organization and specialization of the investigated systems, and identifies those relationships between elements of the projected network that cannot be explained simply by system heterogeneity. We also show that our method applies to bipartite systems in which different relationships might have different qualitative nature, generating statistically validated networks in which such difference is preserved.

Identification of Clusters of Investors from Their Real Trading Activity in a Financial Market

We use statistically validated networks, a recently introduced method to validate links in a bipartite system, to identify clusters of investors trading in a financial market. Specifically, we investigate a special database allowing to track the trading activity of individual investors of the stock Nokia. We find that many statistically detected clusters of investors show a very high degree of synchronization in the time when they decide to trade and in the trading action taken. We investigate the composition of these clusters and we find that several of them show an over-expression of specific categories of investors.

Hierarchically nested factor model from multivariate data

We show how to achieve a statistical description of the hierarchical structure of a multivariate data set. Specifically, we show that the similarity matrix resulting from a hierarchical clustering procedure is the correlation matrix of a factor model, the hierarchically nested factor model. In this model, factors are mutually independent and hierarchically organized. Finally, we use a bootstrap-based procedure to reduce the number of factors in the model with the aim of retaining only those factors significantly robust with respect to the statistical uncertainty due to the finite length of data records.

Kullback-Leibler distance as a measure of the information filtered from multivariate data

We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically determine the expected values of the Kullback-Leibler distance of a sample correlation matrix from a reference model and we show that the expected values are known also when the specific model is unknown. We propose to make use of the Kullback-Leibler distance to estimate the information extracted from a correlation matrix by correlation filtering procedures. We also show how to use this distance to measure the stability of filtering procedures with respect to statistical uncertainty. We explain the effectiveness of our method by comparing four filtering procedures, two of them being based on spectral analysis and the other two on hierarchical clustering. We compare these techniques as applied both to simulations of factor models and empirical data. We investigate the ability of these filtering procedures in recovering the correlation matrix of models from simulations. We discuss such ability in terms of both the heterogeneity of model parameters and the length of data series. We also show that the two spectral techniques are typically more informative about the sample correlation matrix than techniques based on hierarchical clustering, whereas the latter are more stable with respect to statistical uncertainty.

Quantifying preferential trading in the e-MID interbank market

Interbank markets allow credit institutions to exchange capital for purposes of liquidity management. These markets are among the most liquid markets in the financial system. However, liquidity of interbank markets dropped during the 2007–2008 financial crisis, and such a lack of liquidity influenced the entire economic system. In this paper, we analyse transaction data from the e-MID market which is the only electronic interbank market in the Euro Area and US, over a period of 11 years (1999–2009). We adapt a method developed to detect statistically validated links in a network, in order to reveal preferential trading in a directed network. Preferential trading between banks is detected by comparing empirically observed trading relationships with a null hypothesis that assumes random trading among banks doing a heterogeneous number of transactions. Preferential trading patterns are revealed at time windows of 3-maintenance periods. We show that preferential trading is observed throughout the whole period of analysis and that the number of preferential trading links does not show any significant trend in time, in spite of a decreasing trend in the number of pairs of banks making transactions. We observe that preferential trading connections typically involve large trading volumes. During the crisis, we also observe that transactions occurring between banks with a preferential connection occur at larger interest rates than the complement set—an effect that is not observed before the crisis.