Accounting for Location Error in Kalman Filters: Integrating ... - PLOS

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Accounting for Location Error in Kalman Filters: Integrating Animal. Borne Sensor Data into Assimilation Schemes. Aritra Sengupta, Scott D. Foster, Toby A.
Accounting for Location Error in Kalman Filters: Integrating Animal Borne Sensor Data into Assimilation Schemes Aritra Sengupta, Scott D. Foster, Toby A. Patterson, Mark Bravington

Appendix S3: Second-Order Correction to the Posterior Expectation Here we will derive the second-order correction to the expectation of the posterior distribution [Yt |ξξt , Y≤t−1 ]. We have:   E(Yt |ξξt , Y≤t−1 ) = EXt |ξξt E (Yt |Xt , Y≤t−1 ) (



 ∂ ≈ EXt |ξξt E (Yt |Xt = ξt , Y≤t−1 ) + (Xt − ξt ) E (Yt |Xt , Y≤t−1 ) ∂Xt Xt =ξξt )   1 ∂2 (Xt − ξt ) E (Yt |Xt , Y≤t−1 ) + (Xt − ξt )> 2 ∂Xt> ∂Xt Xt =ξξt ) (   1 ∂2 = E (Yt |Xt = ξt , Y≤t−1 ) + trace V (Xt |ξξt ) . E (Yt |Xt , Y≤t−1 ) 2 ∂Xt> ∂Xt Xt =ξξt >

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