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  • Nonlinear Estimation

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    2 Literature review on quadrature based nonlinear estimation (cont'd...)

    2.5 Generation of deterministic sampling points and weights

    Sparse grid Gauss Hermite filter (SGHF)

    $[X,W]=SGQ[n,L,\hat{p}]$

    $X$: SGQ point set with the element of $X_i$, $W$: weight sequence with element of $W_i$, $\hat{p}$ is the univariate SGQ point set.

    $\bullet$ FOR the points $\hat{p_i} \in \hat{p}$,

    Determine the corresponding weights $\hat{w_s} \in \hat{w}$, for univariate quadrature rules $I_l$ as described in section $2.4$.

    $\bullet$ FOR $q=L-n:L-1$, determine $N^{n}_{q}$

    $\bullet$ FOR each element $\Xi=(i_1,...,i_n)$ in $N^{n}_{q}$, form $X_{i_1}\otimes X_{i_2} \otimes...\otimes X_{i_n}$.

    $\bullet$ IF the point is new, add it to $X$, assign a new index $i$ to this point and calculate the weight of $X_i$ as \begin{equation*} W_i=(-1)^{L-1-q}C_{n-1}^{L-1-q} \Pi_{j=1}^{n}\hat{w_{sj}} \end{equation*} where $(\hat{w_{sj}} \in \hat{w})$

    $\bullet$ ELSE (if the point already exists) update the old weight by \begin{equation*} W_i=W_i+(-1)^{L-1-q}C_{n-1}^{L-1-q} \Pi_{j=1}^{n}\hat{w_{sj}} \end{equation*} END IF
    END FOR
    END FOR
    END FOR

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