finacial filtered networks
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Financial Filtered NetworksUntangling financial data complexity
Tomaso Aste & Tiziana Di Matteo
Zurich 9 Jan 2014
UCL, London KCL, London
Aplollonian Networks
JS. Andrade, Jr., HJ Herrmann, R FS Andrade and LR. da Silva, PRL 94 (2005) 018702
M. Tumminello, TA, T. Di Matteo, and R. N. Mantegna, PNAS 102 (2005) 10421–10426
TA, R Gramatica, T Di Matteo PRE 86 (2012) 036109
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T2J. W. Alexander, “The combinatorial theory of complexes” Ann. Math. 31 (1930) 292.
Planar triangulationsPMFG
Filtered networks
Information and Big DataWe are witnessing interesting times rich of information, readily available for us all. Using, understanding and filtering such information has become one of the major tasks and a crucial bottleneck for scientific and industrial endeavors
Information content and flow are often associated with large degrees of interdependency that can be used to reduce data complexity
To filter information we must first understand the data inetr-dependency structure
This large amount of information must be filtered and meaning extracted
Linear measures• Correlations• Partial Correlations• Granger causality • Transfer EntropyNon-linear and kernel measures• Kernelized Granger/Geweke’s causality • Hilbert-Schmidt Normalised Conditional Independence Criterion (HSNCIC) • Transfer Entropy
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Interdependency
A Zaremba
Is the Consumer Price Index (US) causing interest rates (LIBOR)?
1 month lag 7 months lag
CPI->LIBOR
LIBOR->CPI
CPI->LIBOR
LIBOR->CPI
Kernelized Granger/Geweke’s causality Kernelized Granger/Geweke’s causality
Transfer Entropy
A Zaremba & TA, Measures of Causality in Complex Datasets with application to financial data http://arxiv.org/abs/1401.1457
Dependency and Causality:
Measuring and validating
The surface constraints the complexity of the network (the degree of interwoveness)
We can achieve this by embedding interdependency nets on surfaces
Information Filtering
Simplifying Complex Big Datasets
Planar surfaces are the simplest
To reduce complexity we must find ways to decrease the number of interrelations and gather data into clusters and hierarchical structures
Big data demand new algorithms
Sort similarities form the largest to the smallest
Connect the first two nodes on the top line of the list
Is the resulting graph planar?
Delete the top line from the list
Discard the edgeKeep the edge
Have we reached the maximum number of edges?
yes no
yes
no
http://www.mathworks.com/matlabcentral/fileexchange/27360
M. Tumminello, TA, T. Di Matteo, R.N. Mantegna, “A tool for filtering information in complex systems”, PNAS 102 (2005) 10421-10426.
Planar Maximally Filtered Graph
T2 (Apollonian) construction
G Previde Massara
The novelty of the method is that we do not longer rely on any particular ordering but at every stage we calculate the gain that would be obtained by adding any of the remaining vertices inside any triangle, complexity is O(n2) and results improve PMFG
Numerically efficient algorithm for big dataT1
T2
Non planar graph can be generated with this methodTA, R Gramatica and T. Di Matteo, Phys. Rev. E., 86 (2012) 036109.
Some 1000s times faster than PMFG and scalable to millions of vertices
Clique Tree construction for Markov Random Field inference modeling
Network Interdependence Modelling
Local Sparse Inverse Covariance (LoGo)
G Previde Massara
Wolfram Barfuß
The Join probability distribution factorizes over the cliques (for exponential classes) The T2 (Apollonian) construction generates a 4-clique tree!
In linear models (and kernel-linearized as well) interactions are associated with the inverse of the covariance, but noise makes the inverse meaningless. Local inversion on the clique-tree produce meaningful interactions.
Risk modelling
Efficient diversification / risk hedging
F Pozzi
Correlation networks can be used for efficient portfolio differentiation by selecting stocks from the periphery of the PMFG
Portfolio performance
Probability of negative returns
F. Pozzi, T. Di Matteo, and TA , “Spread of risk across financial markets: better to invest in the peripheries”, Scientific Reports 3 (2013) 1665.
We extract clusters and hierarchies form maximal planar graphs by using the fact that 3-cliques on Maximal Planar Graphs contain other cliques inside or/and they are contained inside the other cliques providing a natural hierarchy
Complexity reduction: DBHT
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W.M. Song, T. Di Matteo and T. Aste, “Hierarchical information clustering by means of topologically embedded graphs”, PLoS ONE, 7 (2012) e31929 Won-Min Song, T. Di Matteo, TA, Nested hierarchies in planar graphs, Discrete Applied Mathematics 159 (2011) 2135-2146.
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A deterministic method to capture both local clustering and global hierarchical organization without introducing any characteristic scale
WM Song
Directed Bubble Hierarchical Tree
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W.M. Song, T. Di Matteo and T. Aste, “Hierarchical information clustering by means of topologically embedded graphs”, PLoS ONE, 7 (2012) e31929
PMFGDBHT
Conclusions and PerspectivesBig data demand complexity reduction: information filtering
Networks are especially suited for this purpose because one can reduce size by clustering without loosing the information about the whole hierarchy
PMFG generated via T2 (Apollonian nets) are especially suited for this purpose
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