A Heuristic for Completing Covariance And Correlation Matrices
This paper presents a heuristic for completing partially speciﬁed covariance (and cor-relation) matrices. In many statistical and stochastic models, covariance (and corre-lation) matrices play a prominent role and are, for example, part of the model spec-iﬁcation.
In practical applications, and especially in high dimensional models, it is common that not all covariances are known due to, for example, data limitations. In such situations, the covariance matrix must be completed in order to use the statisti-cal or stochastic model at hand. The already speciﬁed covariances generically imply a dependence between variables between which the covariance is unspeciﬁed. A criterion that can then be used to complete the covariance matrix is to introduce as little extra dependence between variables as possible which is equivalent with entropy maximization. Such completion problems can be solved with global optimization al-gorithms which unfortunately are slow for large matrices. And thus, this makes global optimization less suitable for applications in which the computation time is important. Another difﬁculty is that the initially speciﬁed covariances can be inconsistent in the sense that no valid completion exists. In this paper, we present a heuristic for complet-ing partially speciﬁed covariance (and correlation) matrices that is: fast (also for large matrices), tries to introduce as little extra dependence between variables as possible and also corrects for inconsistencies in the initially speciﬁed covariances. Finally, we present error measures for the performance of the heuristic.