Error Propagation through Linearized Statistics

The Sigma-Point Approach (Sigma-Point Approach or SPA) makes it possible to estimate 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 select $2n+1$ 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 in computing 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 obtained, the sigma points 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 mean and variance obtained in this way provide a good approximation of the mean and variance of the input distribution after transformation by the function $f$.

The problem addressed by the Sigma-Point Approach is nevertheless ill-posed, because infinitely many probability distributions can have the same mean and covariance. The Unscented Transform (UT) (JU97), one of the possible Sigma-Point Approaches, sets 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, and the weights used to compute the mean differ from those used to compute the 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. $\beta=2$ is set for Gaussian distributions.

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