Computes the O-information (\(\Omega\)) for a subset of variables from their correlation matrix under the Gaussian assumption.
Value
A list containing:
- Omega
O-information value.
- TC
Total Correlation.
- DTC
Dual Total Correlation.
- type
Interaction type:
"redundant","synergistic", or"neutral".
Details
O-information is an information-theoretic measure proposed by Rosas et al. (2019) and introduced to psychometric hypergraph analysis by Marinazzo et al. (2024). It quantifies the balance between redundant and synergistic higher-order statistical dependencies among three or more variables.
The O-information is defined as:
$$ \Omega = TC - DTC $$
where \(TC\) is the Total Correlation and \(DTC\) is the Dual Total Correlation.
Total Correlation (TC) measures the amount of collective constraint among variables and is defined as:
$$ TC(X_1,\ldots,X_n) = \sum_{i=1}^{n} H(X_i) - H(X_1,\ldots,X_n) $$
where \(H(\cdot)\) denotes Shannon entropy.
Dual Total Correlation (DTC) measures the amount of shared randomness among variables and is defined as:
$$ DTC(X_1,\ldots,X_n) = H(X_1,\ldots,X_n) - \sum_{i=1}^{n} H(X_i \mid X_{-i}) $$
where \(X_{-i}\) denotes all variables except \(X_i\).
For multivariate Gaussian variables, TC and DTC can be computed directly from the covariance (or correlation) matrix. Given a correlation matrix \(\Sigma\) and precision matrix \(C=\Sigma^{-1}\),
$$ TC = -\frac{1}{2}\log |\Sigma| $$
and
$$ DTC = \frac{1}{2} \left( \log |\Sigma| + \sum_i \log C_{ii} \right). $$
Interpretation of \(\Omega\):
\(\Omega > 0\): redundancy-dominated interaction ("redundant"), indicating that the system is dominated by low-order constraints.
\(\Omega < 0\): synergy-dominated interaction ("synergistic"), indicating a predominance of higher-order constraints within the system of interest.
\(\Omega \approx 0\): balanced or neutral interaction.
In psychometric hypergraphs, subsets with positive O-information are represented as redundant hyperedges, whereas subsets with negative O-information are represented as synergistic hyperedges.
This implementation follows the Gaussian formulation described in Marinazzo et al. (2024), where O-information is computed directly from the determinant of the correlation matrix and its inverse. This approach avoids explicit entropy estimation and is computationally efficient for psychometric applications involving large numbers of candidate hyperedges.
References
Rosas, F. E., Mediano, P. A. M., Gastpar, M., & Jensen, H. J. (2019). Quantifying high-order interdependencies via multivariate extensions of the mutual information. Physical Review E, 100(3), 032305. https://doi.org/10.1103/PhysRevE.100.032305
Marinazzo, D., Van Roozendaal, J., Rosas, F. E., Stella, M., Comolatti, R., Colenbier, N., Stramaglia, S., & Rosseel, Y. (2024). An information-theoretic approach to build hypergraphs in psychometrics. Behavior Research Methods, 56(7), 8057-8079. https://doi.org/10.3758/s13428-024-02471-8