Engg. tutorials

To compute the integral, $I(f)$, first we need to compute
\begin{equation}
\int_{U_n}f(CrZ+\mu)ds(Z)
\end{equation}
This integral can be approximately calculated by third degree fully symmetrical spherical radial cubature rule.
By considering zero mean and unity variance, it can be approximated as
\begin{equation}
\int_{U_n}f(rZ)ds(Z)=\dfrac{2\sqrt{\pi^n}}{2n\Gamma(n/2)} \sum_{i=1}^{2n}f[ru]_i
\end{equation}
where $[u]_i(i=1,2,...,2n)$ are the cubature points located at the intersections of unit hyper-sphere and it's axes.
For example, in single dimension, the two cubature points will be on $+1$ and $-1$. For two dimensions,
the four cubature points will be on $(+1,0)$, $(-1,0)$, $(0,+1)$ and $(0,-1)$. For Gaussian distribution with non
zero mean and non unity covariance, the cubature points will be located at ($C[u]_i + \mu$).
For a third-degree spherical radial rule, it takes only $2n$ number of points for approximation of the
intractable integrals. *i.e* computational cost increases linearly with increase in system dimension.
Hence, this algorithm avoids the *curse of dimensionality problem*.

Continuous-Discrete Cubature Kalman Filter has been introduced in [Arasaratnam 2010] .
This work dealt with extending CKF to deal with state space models of the continuous-discrete type.
Ito-Taylor expansion of order 1.5 [Ito 2000] was used to transform the process equation in stochastic
ordinary differential form to stochastic difference equation.This transformation proved to be more accurate
in extracting the information about system states in discrete time. Assuming all the conditional probabilities
to be Gaussian, third-degree cubature rule was used to numerically compute the Gaussian-weighted integrals.
To ensure numerical stability square root version of the algorithm has been used. Cubature Kalman Filters were
extended to nonlinear smoothing problems [Arasaratnam 2011] ; named as fixed-interval cubature kalman smoother (FI-CKS).
State estimation of a smoother algorithm is always accurate than that of corresponding filter counterparts.
Here, cubature integration theory is applied to already existing integration based smoothing theory.
All conditional densities were assumed to be Gaussian. Readily available densities and Gaussian lemmas
were used [Sarkka 2008, 2010] for algorithm development. Square-root version of the algorithm
has been implemented for higher numerical stability, *i.e*, square-roots of error covariances are
propagated in the algorithm

In [Jia 2013] , a new class of CKF called High-degree cubature kalman filter has been introduced. The third degree spherical-radial rule used by CKF makes it more stable than the unscented kalman filter (UKF), but less accurate than the Gauss-Hermite filter (GHF). To make CKF more accurate, it's high degree variant is proposed with arbitrary degree in evaluating the spherical and radial integrals. This makes high- degree CKF's to achieve accuracy and stability performances in-par with Gauss-Hermite filters, with lesser computational cost. Here arbitrary degree spherical-radial cubature rule is used to compute Gaussian integrals. A new version of cubature kalman filter has been introduced in [Bhaumik 2013]. In this technique, spherical radial cubature and Gauss-Laguerre quadrature rule was used for nonlinear state estimation problems. The filter was named as Cubature quadrature kalman filter (CQKF) and was proved to have more accuracy than CKF.

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