Error Propagation through Linearized Statistics

The Sigma-Point Approach (Sigma-Point Approach, or SPA) estimates 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 $\bm{\mathcal{X}}_i$, called sigma points, each weighted by a weight $w_i$, so as to obtain 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 corresponding weights as follows:

$$\begin{array}{rl} {\bm{\mathcal{X}}}_0&= \bar{\mathbf{x}}\\... ...ft( \sqrt{ \bm{\Sigma}_{\mathbf{x}} } \right)_i \\ \end{array}$$ (2.47)

where $\zeta$ is a scalar factor that accounts for how widely the sigma points are spread around the mean $\bar{\mathbf{x}}$. Each sigma point is associated with a pair of weights, $w_i^{m}$ and $w_i^{c}$, used to compute the mean and covariance, respectively.

Unlike Monte Carlo methods, sigma points are selected deterministically so as to represent the statistics of the variable as accurately as possible.

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

$${\bm{\mathcal{Y}}}_i = f({\bm{\mathcal{X}}}_i) \quad \scriptstyle i=0,\ldots,2n$$ (2.48)

The mean and variance of the output variable can then be computed from these points as

$$\begin{array}{l} \bar{\mathbf{y}} \approx \sum_{i=0}^{2n} w... ...{y}})(\bm{\mathcal{Y}}_i - \bar{\mathbf{y}})^{\top} \end{array}$$ (2.49)

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

The problem addressed by the Sigma-Point Approach is nevertheless ill-posed, because infinitely many probability distributions share the same mean and covariance. The Unscented Transform (UT) (JU97), one of the possible Sigma-Point Approaches, fixes the values $\zeta = \sqrt{n + \lambda}$, where $n$ is the dimension of the space and $\lambda$ is a number defined as $\lambda = \alpha^2 (n + \kappa) - n$, with $\alpha \in ]0.001, 1]$ a small positive number and $\kappa$ usually set to $0$ or $3-n$. In some papers, $\alpha=1$ and $\kappa=3-n$ are used for Gaussian distributions.

In the unscented transform as well, the sigma points are weighted, with different weights used to compute the mean and covariance matrix. The unscented transform therefore sets these weights to

$$\begin{array}{l} w^{m}_0 = \frac{\lambda}{n + \lambda} \\ ... ...eta) \\ w_i =w_{i+n} = \frac{1}{2 (n + \lambda)} \end{array}$$ (2.50)

The difference between the weights $w^{m}_i$ and $w^{c}_i$ lies only in the central term. For Gaussian distributions, $\beta=2$ is set.

It should be noted that variants of the sigma-point approaches compute these weights differently.