# Adaptive Noise Covariance Estimation under Colored Noise using Dynamic Expectation Maximization

arXiv: https://arxiv.org/abs/2308.07797 · HTML: https://mlsanity.com/paper/2308.07797/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Builds on (most cited first, 5 of 5)

- [Active Inference in Robotics and Artificial Agents: Survey and Challenges](https://mlsanity.com/paper/2112.01871/) (2021)
- [Adaptive robot body learning and estimation through predictive coding](https://mlsanity.com/paper/1805.03104/) (2018)
- [Free Energy Principle for State and Input Estimation of a Quadcopter Flying in Wind](https://mlsanity.com/paper/2109.12052/) (2021)
- [Towards Stochastic Fault-tolerant Control using Precision Learning and Active Inference](https://mlsanity.com/paper/2109.05870/) (2021)
- [Free Energy Principle for the Noise Smoothness Estimation of Linear Systems with Colored Noise](https://mlsanity.com/paper/2204.01796/) (2022)

## Built on by (most cited first, 2 of 2)

- [Seeing Through Uncertainty: Free-Energy-Inspired Real-Time Adaptation for Robust Visual Navigation](https://mlsanity.com/paper/2403.01977/) (2024)
- [Online Generalised Predictive Coding](https://mlsanity.com/paper/2605.02675/) (2026)

## Abstract (arXiv)

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.