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The Unscented Kalman Filter for Nonlinear Estimation

Second-order ,therelationshipbetweentheKalmanFilter(K F)andRecursiveLeastSquares(RLS)isgivenin [3].TheuseoftheEKFfortrainingneuralnetwo rkshasbeendevelopedbySinghalandWu[9]andP uskoriousandFeldkamp[8].DualEstimationAs pecialcaseofmachinelearningariseswhenthe inputisunobserved, ,weagainconsideradiscrete-timenonlineard ynamicsystem,(6)(7) ,weintroducetheUnscentedKalmanFilter(UKF ) ,inSection4, ,are-cursiveestimationforcanbeexpressedi ntheform(see[6]),predictionofpredictiono f(8)Thisrecursionprovidestheoptimalminim ummean-squarederror(MMSE)estimateforassu mingthepriorestimateandcurrentobservatio nareGaussianRandomVari-ables(GRV). (9)(10)(11)wheretheoptimalpredictionofis writtenas,andcorrespondstotheexpectation ofanonlinearfunctionoftherandomvariables and(similarinterpretationfortheoptimalpr ediction).Theoptimalgaintermisexpresseda safunctionofposteriorcovariancematrices( with).

ple pointscompletely capture the true mean and covariance of the GRV, and when propagated through the true non-linear system, captures the posterior mean and covariance accuratelytothe3rdorder(Taylorseries expansion)forany nonlinearity. TheEKF, in contrast,onlyachievesfirst-order accuracy. Remarkably,the computationalcomplexityofthe

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