The generalized variance is a scalar value which generalizes variance for multivariate random variables. It was introduced by Samuel S. Wilks.
The generalized variance is defined as the determinant of the covariance matrix, det ( Σ ) {\displaystyle \det(\Sigma )} . It can be shown to be related to the multidimensional scatter of points around their mean.
Minimizing the generalized variance gives the Kalman filter gain.
References
Kocherlakota, S.; Kocherlakota, K. (2004). "Generalized Variance". Encyclopedia of Statistical Sciences. Wiley Online Library. doi:10.1002/0471667196.ess0869. ISBN 0471667196. Retrieved 30 October 2019. 0471667196 ↩
Proof that the Kalman gain minimizes the generalized variance, Eviatar Bach https://arxiv.org/abs/2103.07275 https://arxiv.org/abs/2103.07275 ↩