Engg. tutorials

Gaussian filters for non linear filtering problems has been studied exclusively in [Ito 2000].
A systematic construction of Gaussian filters and efficient numerical integration for filter development was presented.
Introduction of GHF using quadrature rule for efficient numerical integration has been proposed in this work.
In [Singer 2006], a generalization of the GHF has been proposed. Thus, the usual GHF was considered as a special case.
Here filter density was represented using a Hermite expansion with leading Gaussian term.
Higher order moments were computed and then introduced into Hermite expansion.
Gauss-Hermite integration was used for update steps, so that method yields closed moment equations.

By using statistical linear regression (SLR), a new version of Quadratic Kalman Filter has been introduced in [Arasaratnam 2009] by linearizing the process and measurement functions. This was done by using a set of Gauss-Hermite quadrature points which parameterize the Gaussian density. It was then extended to nonlinear discrete systems with non Gaussian noise. For this, a bank of parallel filters named Gaussian sum-quadrature filter was used to approximate the predicted and posterior densities using a finite set of weighted sums of Gaussian densities. The weights were obtained form the residuals of the Quadrature Kalman Filters.

There are mainly two variants of Gauss-Hermite filter, which can efficiently reduce
the computational cost in filtering algorithms. They are 1) Sparse-grid Gauss-Hermite filter (SGHF)and
2) Multiple Quadrature Kalman filter(MSQKF). In these filters, the intractable integrals are
approximated numerically by using Gauss-Hermite quadrature rule which is defined for the single dimension integral.
The GHF utilizes product rule to extend the single dimensional rule to the multidimensional rule,
but its computational cost increases exponentially with increasing dimension and hence it
suffers from the *curse of dimensionality problem*. The SGHF is an extension of GHF, which utilizes
the Smolyak's rule [Gerstner 1998] to extend the single dimensional quadrature rule for the multidimensional systems.
It reduces the computational cost considerably. MSQKF uses state space partitioning technique and runs filters in parallel.
Thus, it features a bank of filters which compute posterior densities simultaneously.

In SGHF, the Gauss-Hermite quadrature rule used for solving single dimensional intractable integrals is extended to multidimensional rule by using Smolyak's rule. This significantly reduces the number of quadrature points and hence, the curse of dimensionality problem inherent in GHF.

The word sparse-grid filtering was first mentioned in [Balaji 2008] . But the method used was based on sparse tensor product which, in no way reduced the curse of dimensionality problem. To alleviate this problem, sparse-grid theory was first used in [Hess 2008] [Smolyak 1963] for numerical integration. In [Hess 2008], a detailed study of univariate quadrature rule and it's expansion to multi-dimensional problems was presented using sparse-grid theory. It gives a thorough insight into the evolution of multivariate quadrature rules and the curse of dimensionality problem associated with it (product rule). The study shows that the integration rule using sparse-grid theory is exact for complete polynomials of given order in multiple dimensions. Like the product rule, it combines the univariate quadrature rule, which makes it very general and easy to implement. But, the computational costs do not increase exponentially but at a considerably slower rate. This idea, proposed in [Smolyak 1963], is a powerful tool for multivariate extensions of univariate operators. This approach has been widely used in numerical mathematics [Bungartz 2004] [Wasilkowski 1995] .

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