Error Propagation through Linearized Statistics

The Sigma-Point Approach (SPA) allows for the estimation of the mean and variance of a random variable at the output of a system modeled by a nonlinear function $f: \mathbb{R}^{n} \mapsto \mathbb{R}^m$.

To estimate the mean and variance, the input random variable $\mathbf{x} \in \mathbb{R}^{n}$ is approximated by $2n+1$ points $\bm{\mathcal{X}}_i$, referred to as sigma points, each weighted by a weight $w_i$, in order to achieve a distribution with mean and variance $\bar{\mathbf{x}}$ and $\bm{\Sigma}_{\mathbf{x}}$ respectively, that is, parameters exactly equal to those of $\mathbf {x}$.

One way to obtain a set of points whose distribution has the same mean and variance as the original distribution is to take $2n+1$ sigma-points and their respective weights as follows:

$$\begin{array}{rl} {\bm{\mathcal{X}}}_0&= \bar{\mathbf{x}}\\... ...ft( \sqrt{ \bm{\Sigma}_{\mathbf{x}} } \right)_i \\ \end{array}$$ (2.37)

where $\zeta$ is a scaling factor that accounts for how spread out the sigma points are relative to the mean $\bar{\mathbf{x}}$. Associated with each sigma point is a pair of weights $w_i^{m}$ and $w_i^{c}$ used in the calculation of the mean and covariance, respectively.

Unlike the Monte Carlo methods, the sigma points are chosen deterministically to best represent the statistics of the variable.

Once the sigma-points are obtained, they are transformed (unscented transformation) through the function $f$ into transformed sigma points

$${\bm{\mathcal{Y}}}_i = f({\bm{\mathcal{X}}}_i) \quad \scriptstyle i=0,\ldots,2n$$ (2.38)

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From these points, it is possible to calculate the mean and variance of the output variable using

$$\begin{array}{l} \bar{\mathbf{y}} \approx \sum_{i=0}^{2n} w... ...{y}})(\bm{\mathcal{Y}}_i - \bar{\mathbf{y}})^{\top} \end{array}$$ (2.39)

for each point $i=0, \ldots, 2n$. The mean and variance obtained in this way provide a good approximation of the mean and variance of the input distribution transformed by the function $f$.

The problem addressed by the Sigma Point approach is inherently ill-defined because there exist infinite probability distributions that share the same mean and covariance. The Unscented Transform (UT) (JU97), one of the possible Sigma-Point Approaches, sets the values as $\zeta = \sqrt{n + \lambda}$, where $n$ is the dimension of the space and $\lambda$ is a number defined as $\lambda = \alpha^2 (n + \kappa) - n$ with $\alpha \in ]0.001, 1]$ being a small positive number and $\kappa$ typically set to $0$ or $3-n$. In some articles, $\alpha=1$ and $\kappa=3-n$ are used for Gaussian distributions.

In the unscented transformation, the sigma points are weighted points, and the weights differ in the calculation of the mean and the covariance matrix. The unscented transformation therefore sets these weights to

$$\begin{array}{l} w^{m}_0 = \frac{\lambda}{n + \lambda} \\ ... ...eta) \\ w_i =w_{i+n} = \frac{1}{2 (n + \lambda)} \end{array}$$ (2.40)

The difference between the weights $w^{m}_i$ and $w^{c}_i$ lies solely in the central term. The value $\beta=2$ is fixed for Gaussian distributions.

It is important to emphasize that the variants of the sigma-point approaches have their weights calculated differently.