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

1. Center for Financial Studies at the Goethe University
PhD Mini-course
Frankfurt, 25 January 2013
Financial Networks
VI. Correlation Networks
Dr. Kimmo Soramäki
Founder and CEO
FNA, www.fna.fi

2. Agenda
V. Inferring Links
• Prices and returns
• Controlling for common factors
• Correlation and dependence
• Significant correlations
• Multiple Comparisons
VI. Correlation Networks
• Distance and Hierarchical Clustering
• Minimum Spanning Tree & PMFG
• Other filtering
• Layout algorithms
2

3. Hierarchical structure in financial markets
3

4. Minimum Spanning Tree
A spanning tree of a graph is a subgraph that:
1. is a tree and
2. connects all the nodes together
Length of a tree is the sum of its links. Minimum spanning tree (MST) is a spanning
tree with shortest length.
MST reflects the hierarchical structure of the correlation matrix

5. MST and Hierarchical Structure
Source: R.N. Mantegna (1999). Hierarchical structure in nancial markets,
Eur. Phys. J. B 11, 193-197 5

6. 36
Single Linkage Clustering
• A method for hierarchical clustering
• Clusters based on similarity or distance
• SLINK algorithm
R. Sibson (1973). SLINK: an optimally efficient algorithm for the single-link cluster
method. The Computer Journal (British Computer Society) 16 (1): 30–34. 6

7. Example
# build network from correlations
buildbycorrelationd -file daxreturns-2011-recon.csv -missing Alert -preserve
false
# calculate distance
corrdistance -p correlation -method gower
# calculate single linkage clistering
slink -p corrdistance
# create heatmaps
heatmap -sortv vertex_id -p correlation -symmetric true -cellsizedefault 13 -
transition 0 -cellhover correlation -palette darkblue-lightgray-darkred -
colordomain (-1)-1 -saveas daxheat-slink-Y
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8. Unordered, Principal Ordered by Cluster, Principal
Component Removed Component Removed
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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 Phylogenetic
Trees. Department of Computer & Information Science, University of
9
Konstanz, Germany

10. Putting it all together
# build network from correlations
buildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -preserve false
# calculate distance
corrdistance -p correlation -method gower
# calculate single linkage clistering
minst -p corrdistance
# drop arcs not in MST
dropa -e minst=false
# calculate absolute correlation
calcap -e 1-abs(correlation) -saveas vizdistance
# create heatmaps
radialtreeviz -p vizdistance -vlabel vertex_id -vsize stdev -transition 3000 -ahover correlation -saveas
daxviz-MST
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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
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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 -vsize
stdev -transition 3000 -ahover
correlation -saveas daxviz-MST-circle
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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
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14. PMFG -command
# build network from correlations
buildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -preserve false
# calculate distance
corrdistance -p correlation -method gower
# calculate single linkage clistering
pmfg -p corrdistance
# drop arcs not in MST
dropa -e pmfg=false
# calculate 1-absolute correlation
calcap -e abs(correlation) -saveas vizdistance
# calculate degree
degree
# create heatmaps
frviz -vlabel vertex_id -vsize stdev -atransparency vizdistance -ahover correlation -transition 3000 -ahover correlation -
arrows false -saveas daxviz-PMFG
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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
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16. Example
# build network from correlations
buildbypartialcorrelationd -file daxreturns-2011.csv -missing Alert -
savestdev -preserve false
# show as heatmap
heatmap -sortv vertex_id -p partial_correlation -symmetric true -
cellsizedefault 13 -transition 0 -cellhover partial_correlation -palette
darkblue-lightgray-darkred -colordomain (-1)-1 -saveas daxheat-
partial-Y
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17. Correlations Partial Correlations
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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)
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19. Correlation filtering PMFG
Balance between too much and too little
information
One of many methods to create networks
from correlation/distance matrices
– PMFGs, Partial Correlation
Networks, Influence Networks, Granger Influence Network
Causality, NETS, etc.
New graph, information-theory, economics
& statistics -based models are being
actively developed
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20. Sammon’s Projection
Proposed by John W. Sammon in IEEE Transactions on Computers 18: 401–409
(1969)
A nonlinear projection method to map a
high dimensional space onto a space of
lower dimensionality. Example:
Iris Setosa
Iris Versicolor
Iris Virginica
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21. Example
# build network from correlations
buildbycorrelationd -file daxreturns-2011.csv -missing Alert -savestdev -savereturns -
preserve false
# calculate distance
corrdistance -p correlation -method gower
# Calculate sammonlayout
sammonlayouta -p corrdistance -saveerror true
# Sum up error
sumaforv -p error -saveas error
# create heatmaps
sammonaviz -p corrdistance -vlabel vertex_id -vsize error -transition 3000 -ahover error
-saveas daxviz-Sammon-Y
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22. Node size reflects
error in layout

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
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24. Blog, Library and Demos at www.fna.fi
Dr. Kimmo Soramäki
kimmo@soramaki.net
Twitter: soramaki