Editing Kalman filter (section)

=== Optimality and performance ===
The Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated), and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters.<ref>{{Cite book|last1=Bar-Shalom|first1=Yaakov|title=Estimation with Applications to Tracking and Navigation|last2=Li|first2=X.-Rong|last3=Kirubarajan|first3=Thiagalingam|date=2001|publisher=John Wiley & Sons, Inc.|isbn=0-471-41655-X|location=New York, USA|pages=319 ff|doi=10.1002/0471221279}}</ref> 
Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in the section above. More generally, if the model assumptions do not match the real system perfectly, then optimal state estimation is not necessarily obtained by setting '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub> to the covariances of the noise. Instead, in that case, the parameters '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub> may be set to explicitly optimize the state estimation,<ref name=":0" /> e.g., using standard [[supervised learning]].

After the covariances are set, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the [[Innovation (signal processing)|innovations]] measures filter performance. Several different methods can be used for this purpose.<ref>Three optimality tests with numerical examples are described in  {{cite book|doi=10.3182/20120711-3-BE-2027.00011|chapter=Optimality Tests and Adaptive Kalman Filter|title=16th IFAC Symposium on System Identification|series=IFAC Proceedings Volumes|volume=45|issue=16|pages=1523–1528|year=2012|last1=Peter|first1=Matisko|isbn=978-3-902823-06-9}}</ref> If the noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature.<ref>{{cite journal|doi=10.1016/0005-1098(95)00069-9|title=The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions|journal=Automatica|volume=31|issue=10|pages=1513–1517|year=1995|last1=Spall|first1=James C.}}</ref><ref>{{cite journal|doi=10.1109/TAC.2003.821415|title=Use of the Kalman Filter for Inference in State-Space Models with Unknown Noise Distributions|journal=IEEE Transactions on Automatic Control|volume=49|pages=87–90|year=2004|last1=Maryak|first1=J.L.|last2=Spall|first2=J.C.|last3=Heydon|first3=B.D.|s2cid=21143516}}</ref>