Editing Kalman filter (section)

=== Estimation of the noise covariances Q<sub>''k''</sub> and R<sub>''k''</sub> ===
Practical implementation of a Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub>.  Extensive research has been done to estimate these covariances from data.  One practical method of doing this is the ''autocovariance least-squares (ALS)'' technique that uses the time-lagged [[autocovariance]]s of routine operating data to estimate the covariances.<ref>{{cite thesis |url=http://jbrwww.che.wisc.edu/theses/rajamani.pdf |last=Rajamani |first=Murali |type=PhD Thesis |title=Data-based Techniques to Improve State Estimation in Model Predictive Control |location=University of Wisconsin–Madison |date=October 2007 |access-date=2011-04-04 |archive-url=https://web.archive.org/web/20160304194938/http://jbrwww.che.wisc.edu/theses/rajamani.pdf |archive-date=2016-03-04  }}</ref><ref>{{cite journal |last1=Rajamani |first1=Murali R. |last2=Rawlings |first2=James B. |title=Estimation of the disturbance structure from data using semidefinite programming and optimal weighting |journal=Automatica |volume=45 |issue=1 |pages=142–148 |year=2009 |doi=10.1016/j.automatica.2008.05.032 |s2cid=5699674 }}</ref>  The [[GNU Octave]] and [[Matlab]] code used to calculate the noise covariance matrices using the ALS technique is available online using the [[GNU General Public License]].<ref>{{cite web |url=https://sites.engineering.ucsb.edu/~jbraw/software/als/index.html |title=Autocovariance Least-Squares Toolbox |publisher=Jbrwww.che.wisc.edu |access-date=2021-08-18 }}</ref> Field Kalman Filter (FKF), a Bayesian algorithm, which allows simultaneous estimation of the state, parameters and noise covariance has been proposed.<ref>{{cite conference |url= https://www.researchgate.net/publication/312029167|title= Field Kalman Filter and its approximation|last1= Bania|first1= P.|last2= Baranowski|first2=J.|publisher=IEEE |date=12 December 2016|pages= 2875–2880|location= Las Vegas, NV, USA|conference= IEEE 55th Conference on Decision and Control (CDC)}}</ref> The FKF algorithm has a recursive formulation, good observed convergence, and relatively low complexity, thus suggesting that the FKF algorithm may possibly be a worthwhile alternative to the Autocovariance Least-Squares methods. Another approach is the ''Optimized Kalman Filter'' (''OKF''), which considers the covariance matrices not as representatives of the noise, but rather, as parameters aimed to achieve the most accurate state estimation.<ref name=":0">{{Cite journal |last1=Greenberg |first1=Ido |last2=Yannay |first2=Netanel |last3=Mannor |first3=Shie |date=2023-12-15 |title=Optimization or Architecture: How to Hack Kalman Filtering |url=https://proceedings.neurips.cc/paper_files/paper/2023/hash/9dfcc83c01e94d02c751c47517855c9f-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=36 |pages=50482–50505|arxiv=2310.00675 }}</ref> These two views coincide under the KF assumptions, but often contradict each other in real systems. Thus, OKF's state estimation is more robust to modeling inaccuracies.