Engg. tutorials

Gauss-Hermite Filters make use of the Gauss-Hermite Quadrature rule
for integrating the intractable integrals encountered in non linear Bayesian Filtering.
This rule for integration was already available in literature [Hilderbrand 2008]
[Krylov 2005] ,
but it was used in estimation problems by the work of Ito and Xiong [Ito 2000] .
In this filtering technique, the unknown probability density function is
approximated as Gaussian. This is done by defining a set of weights and their
corresponding Gauss-Hermite quadrature points.

Consider an integral of any function $f(x)$
\begin{equation} I=\int_{-\infty}^{\infty}f(x)e^{-x^{2}}dx \end{equation}
It can be evaluated numerically with N quadrature points
\begin{equation}
I\approx\sum_{j=1}^{N}f(q_{j})w_{j}
\end{equation}
where $q_{j}$ and $w_{j}$ are the $j_{th}$ quadrature point and it's corresponding weight.
The quadrature points and its weights can be evaluated using a symmetric tridiagonal matrix
$J$ with $J_{j,j+1}=\sqrt{j/2};1\leq j \leq(N-1)$.The quadrature points are obtained as $q_{j}=\sqrt{2}\Psi_{j}$,
where $\Psi_{j}$ is the $j_{th}$ eigenvalue of matrix $J$.
The $j_{th}$ weight $w_{j}$ can be defined as $w_{j}=k_{j1}^{2}$, where $k_{j1}$ is the first
element of the $j_{th}$ normalized eigenvector of $J$ [Arasaratnam 2009]
[Chalasani 2012] .
This method was actually proposed by Golub *et. al.* [Golun 1969] and was later used by
Arasaratnam *et.al.* [Arasaratnam 2009] for filtering applications.

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