Financial Networks VI - Correlation Networks

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Sixth lecture of a PhD level course on "Financial Networks" at Center for Financial Research at Goethe University, Frankfurt.

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Financial Networks VI - Correlation Networks

  1. 1. Center for Financial Studies at the Goethe UniversityPhD Mini-courseFrankfurt, 25 January 2013Financial NetworksVI. Correlation Networks Dr. Kimmo Soramäki Founder and CEO FNA, www.fna.fi
  2. 2. AgendaV. Inferring Links• Prices and returns• Controlling for common factors• Correlation and dependence• Significant correlations• Multiple ComparisonsVI. Correlation Networks• Distance and Hierarchical Clustering• Minimum Spanning Tree & PMFG• Other filtering• Layout algorithms 2
  3. 3. Hierarchical structure in financial markets 3
  4. 4. Minimum Spanning TreeA spanning tree of a graph is a subgraph that:1. is a tree and2. connects all the nodes togetherLength of a tree is the sum of its links. Minimum spanning tree (MST) is a spanningtree with shortest length.MST reflects the hierarchical structure of the correlation matrix
  5. 5. MST and Hierarchical StructureSource: R.N. Mantegna (1999). Hierarchical structure in nancial markets,Eur. Phys. J. B 11, 193-197 5
  6. 6. 36Single Linkage Clustering• A method for hierarchical clustering• Clusters based on similarity or distance• SLINK algorithmR. Sibson (1973). SLINK: an optimally efficient algorithm for the single-link clustermethod. The Computer Journal (British Computer Society) 16 (1): 30–34. 6
  7. 7. Example# build network from correlationsbuildbycorrelationd -file daxreturns-2011-recon.csv -missing Alert -preservefalse# calculate distancecorrdistance -p correlation -method gower# calculate single linkage clisteringslink -p corrdistance# create heatmapsheatmap -sortv vertex_id -p correlation -symmetric true -cellsizedefault 13 -transition 0 -cellhover correlation -palette darkblue-lightgray-darkred -colordomain (-1)-1 -saveas daxheat-slink-Y 7
  8. 8. Unordered, Principal Ordered by Cluster, PrincipalComponent Removed Component Removed 8
  9. 9. Radial tree -layout• Calculates coordinates for radial layout as presented in Bachmaier, Brandes and Schlieper (2005)• The layout allows definition of each arc length• Specific parameters of command radialtreeviz: – Arc length property (-p) : Arc property defining arc length. Optional. – Root vertex (-rootvertex) : Id of root vertex. The root vertex is placed in the middle of the screen. Due to the repositioning of the tree, nodes may be placed outside the canvas in other than the first network. Optional. – Optimal rotation (-rotation) : Rotates layout to minimize sum of vertex distances between subsequent networks. Optional. By default false. – Scaling (-scale) : Scale of visualization: value/pixel.Christian Bachmaier, Ulrik Brandes, and Barbara Schlieper (2005). Drawing PhylogeneticTrees. Department of Computer & Information Science, University of 9Konstanz, Germany
  10. 10. Putting it all together# build network from correlationsbuildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -preserve false# calculate distancecorrdistance -p correlation -method gower# calculate single linkage clisteringminst -p corrdistance# drop arcs not in MSTdropa -e minst=false# calculate absolute correlationcalcap -e 1-abs(correlation) -saveas vizdistance# create heatmapsradialtreeviz -p vizdistance -vlabel vertex_id -vsize stdev -transition 3000 -ahover correlation -saveasdaxviz-MST 10
  11. 11. Asset Trees Size of node reflects volatility (variance) of returns Links between nodes reflect backbone correlations - short link = high correlation - long link = low correlation 11
  12. 12. Circle Tree -visualization• Calculates coordinates for circle tree layout as presented in Bachmaier, Brandes and Schlieper (2005)• As before but instead of radialtreeviz:circletreeviz -vlabel vertex_id -vsizestdev -transition 3000 -ahovercorrelation -saveas daxviz-MST-circle 12
  13. 13. Planar Maximally Filtered Graph Node size scales with degree• A complex graph with loops and cliques of up to 4 elements. It can be drawn on a planar surface without link crossings.• MST is contained in PMFG M. Tumminello, T. Ast, T. Di Matteo and R. N. Mantegna (2005). A Tool for Filtering Information in Complex Systems. PNAS vol. 102 no. 30 pp. 10421–10426 13
  14. 14. PMFG -command# build network from correlationsbuildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -preserve false# calculate distancecorrdistance -p correlation -method gower# calculate single linkage clisteringpmfg -p corrdistance# drop arcs not in MSTdropa -e pmfg=false# calculate 1-absolute correlationcalcap -e abs(correlation) -saveas vizdistance# calculate degreedegree# create heatmapsfrviz -vlabel vertex_id -vsize stdev -atransparency vizdistance -ahover correlation -transition 3000 -ahover correlation -arrows false -saveas daxviz-PMFG 14
  15. 15. Partial Correlation• Measures the degree of association between two random variables• What is the direct relationship between Adidas and Allianz, controlling for BASF, BAYER, ... ?• We build regression models for Adidas and Allianz and look at the correlation of their model residuals (i.e. wgat left unexplained by the other factors) -> Partial correltation 15
  16. 16. Example# build network from correlationsbuildbypartialcorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -preserve false# show as heatmapheatmap -sortv vertex_id -p partial_correlation -symmetric true -cellsizedefault 13 -transition 0 -cellhover partial_correlation -palettedarkblue-lightgray-darkred -colordomain (-1)-1 -saveas daxheat-partial-Y 16
  17. 17. Correlations Partial Correlations 17
  18. 18. NETS• Network Estimation for Time- Series• Forthcoming paper by Barigozzi and Brownlees• Estimates an unknown network structure from multivariate data• Captures both comtemporenous and serial dependence (partial correlations and lead/lag effects) 18
  19. 19. Correlation filtering PMFGBalance between too much and too littleinformationOne of many methods to create networksfrom correlation/distance matrices – PMFGs, Partial Correlation Networks, Influence Networks, Granger Influence Network Causality, NETS, etc.New graph, information-theory, economics& statistics -based models are beingactively developed 19
  20. 20. Sammon’s ProjectionProposed by John W. Sammon in IEEE Transactions on Computers 18: 401–409(1969)A nonlinear projection method to map ahigh dimensional space onto a space oflower dimensionality. Example: Iris Setosa Iris Versicolor Iris Virginica 20
  21. 21. Example# build network from correlationsbuildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -preserve false# calculate distancecorrdistance -p correlation -method gower# Calculate sammonlayoutsammonlayouta -p corrdistance -saveerror true# Sum up errorsumaforv -p error -saveas error# create heatmapssammonaviz -p corrdistance -vlabel vertex_id -vsize error -transition 3000 -ahover error-saveas daxviz-Sammon-Y 21
  22. 22. Node size reflectserror in layout
  23. 23. Tutorials• Tutorial 1 – Loading Networks into FNA• Tutorial 2 – Managing Data in FNA• Tutorial 3 – Network Summary Measures• Tutorial 4 – Centrality Measures• Tutorial 5 – Connectedness and Components• Tutorial 6 – Network Visualization• Tutorial 7 – Correlation Networks• Tutorial 8 – Payment System Simulations• Tutorial 9 – Analyzing Cross-Border Banking Exposures 23
  24. 24. Blog, Library and Demos at www.fna.fiDr. Kimmo Soramäkikimmo@soramaki.netTwitter: soramaki

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