  The accurate estimation of the noise covariance matrix (NCM) in a dynamic
system is critical for state estimation and control, as it has a major
influence in their optimality. Although a large number of NCM estimation
methods have been developed, most of them assume the noises to be white.
However, in many real-world applications, the noises are colored (e.g., they
exhibit temporal autocorrelations), resulting in suboptimal solutions. Here, we
introduce a novel brain-inspired algorithm that accurately and adaptively
estimates the NCM for dynamic systems subjected to colored noise. Particularly,
we extend the Dynamic Expectation Maximization algorithm to perform both online
noise covariance and state estimation by optimizing the free energy objective.
We mathematically prove that our NCM estimator converges to the global optimum
of this free energy objective. Using randomized numerical simulations, we show
that our estimator outperforms nine baseline methods with minimal noise
covariance estimation error under colored noise conditions. Notably, we show
that our method outperforms the best baseline (Variational Bayes) in joint
noise and state estimation for high colored noise. We foresee that the accuracy
and the adaptive nature of our estimator make it suitable for online estimation
in real-world applications.
