Theory

Partial Correlation Planar Algorithm

The Partial Correlation Planar Graph (PCPG) 1 is based on partial correlation which measures the effect that a random variable \(Z\) has on the correlation between two other random variables \(X\) and \(Y\). The partial correlation is defined in terms of the Pearson correlations \(\rho(\cdot, \cdot)\) between the three variables as

\[\rho(X, Y: Z)=\frac{\rho(X, Y)-\rho(X, Z) \rho(Y, Z)}{\sqrt{\left[1-\rho^{2}(X, Z)\right]\left[1-\rho^{2}(Y, Z)\right]}}.\]

A small value of \(\rho(X, Y: Z)\) may be ambiguous, as this could be due to the correlations among the three variables being small; or due to variable \(Z\) having a strong effect on the correlation between \(X\) and \(Y\), which is generally the interesting case. In order to discriminate between these two cases the correlation influence or influence of variable \(Z\) on the pair of elements \(X\) and \(Y\) is used. This is defined as

\[d(X, Y: Z) \equiv \rho(X, Y)-\rho(X, Y: Z).\]

Finally, the metric on which the PCPG is built is the average influence of variable \(Z\) on the correlations between \(X\) and all other variables in the system. This is given by

\[d(X: Z)=\langle d(X, Y: Z)\rangle_{Y \neq X}.\]

An important detail is that, in general, \(d(X: Z) \neq d(Z: X)\). The largest among these two quantities indicates the main direction of influence between \(X\) and \(Z\), as influence is generally bidirectional. The difference between these two values are often small, which makes a bootstrap procedure necessary in order to asses the confidence in the direction of the average influence, as well as the average influence values.

The construction algorithm of a PCPG network starts with a list of the \(N(N-1)\) average influence values in decreasing order and an empty graph of \(N\) nodes and no edges, where \(N\) is the number of variables in the system. We then cycle through the sorted list, starting with the largest average influence value found, e.g. \(d(J: I)\). The edge \(I \to J\) is included in the network if and only if the resulting network is still planar and the edge \(J \to I\) has not been included already.

We stop adding edges if adding the next edge in the list would break the planarity of the graph. This procedure ensures two things: (i) only the largest among \(d(X: Z)\) and \(d(Z: X)\) will be included in the network, and (ii) the final network has \(3(N-2)\) edges. The end result of this procedure is what we refer to as the PCPG network, \(G\).

Naturally, we also obtain the average influence \(d\) associated to each edge in \(G\), as well as the network’s adjacency matrix \(\mathbf{A}\) defined as

\[\begin{split}A_{I, J} = \begin{cases} 1 & \text{if} \ \text{edge} \ I \to J \in G, \\ 0 & \text{otherwise}. \end{cases}\end{split}\]

References

1

Kenett DY, Tumminello M, Madi A, Gur-Gershgoren G, Mantegna RN, Ben-Jacob E (2010) Dominating Clasp of the Financial Sector Revealed by Partial Correlation Analysis of the Stock Market. PLoS ONE 5(12): e15032. <https://doi.org/10.1371/journal.pone.0015032>