# Source code for netrd.reconstruction.correlation_spanning_tree

```
"""
correlation_spanning_tree.py
----------------------------
Graph reconstruction algorithm based on Mantegna, R. N. (1999). Hierarchical structure in
financial markets. The European Physical Journal B-Condensed Matter and Complex Systems,
11(1), 193-197. DOI https://doi.org/10.1007/s100510050929
https://link.springer.com/article/10.1007/s100510050929
author: Matteo Chinazzi
Submitted as part of the 2019 NetSI Collabathon.
"""
from .base import BaseReconstructor
import numpy as np
import networkx as nx
from scipy.sparse.csgraph import minimum_spanning_tree
[docs]class CorrelationSpanningTree(BaseReconstructor):
"""Minimum spanning tree connecting the sensors."""
[docs] def fit(self, TS, distance='root_inv', **kwargs):
r"""Create a minimum spanning tree connecting the sensors.
The empirical correlation matrix is used to first compute a
distance matrix and then to create a minimum spanning tree
connecting all the sensors in the data. This method implements the
methodology described in [1]_ and applied in the context of creating
a graph connecting the stocks of a portfolio of generated by
looking at the correlations between the daily time series of stock
prices.
The results dictionary also stores the distance matrix (computed
from the correlations) as `'distance_matrix'`.
Parameters
----------
TS (np.ndarray)
:math:`N \times L` array consisting of :math:`L` observations
from :math:`N` sensors.
distance (str)
'inv_square' calculates distance as :math:`1-corr_{ij}^2`
as in [1]_. 'root_inv' calculates distance as
:math:`\sqrt{2 (1-corr_{ij})}` [2]_.
Returns
-------
G (nx.Graph)
A reconstructed graph with :math:`N` nodes.
Examples
--------
.. code:: python
import numpy as np
import networkx as nx
from matplotlib import pyplot as plt
from netrd.reconstruction import CorrelationSpanningTree
N = 25
T = 300
M = np.random.normal(size=(N,T))
print('Create correlated time series')
market_mode = 0.4*np.random.normal(size=(1,T))
M += market_mode
sector_modes = {d: 0.5*np.random.normal(size=(1,T)) for d in range(5)}
for sector_mode, vals in sector_modes.items():
M[sector_mode*5:(sector_mode+1)*5,:] += vals
print('Link node colors to sectors')
colors = ['b','r','g','y','m']
node_colors = [color for color in colors for __ in range(5)]
print('Network reconstruction step')
cst_net = CorrelationSpanningTree()
G = cst_net.fit(M)
print('Plot reconstructed spanning tree')
fig, ax = plt.subplots()
nx.draw(G, ax=ax, node_color=node_colors)
References
----------
.. [1] Mantegna, R. N. (1999). Hierarchical structure in financial
markets. The European Physical Journal B-Condensed Matter
and Complex Systems, 11(1), 193-197. DOI
https://doi.org/10.1007/s100510050929
https://link.springer.com/article/10.1007/s100510050929
.. [2] Bonanno, G., Caldarelli, G., Lillo, F. & Mantegna,
R. N. (2003) Topology of correlation-based minimal spanning
trees in real and model markets. Physical Review E 68.
.. [3] Vandewalle, N., Brisbois, F. & Tordoir, X. (2001) Non-random
topology of stock markets. Quantitative Finance 1, 372–374.
"""
N, L = TS.shape
C = np.corrcoef(TS) # Empirical correlation matrix
D = (
np.sqrt(2 * (1 - C)) if distance == 'root_inv' else 1 - np.square(C)
) # Distance matrix
self.results['distance_matrix'] = D
MST = minimum_spanning_tree(D) # Minimum Spanning Tree
G = nx.from_scipy_sparse_matrix(MST)
self.results['graph'] = G
return G
```