Definition of the Sigma Points

• The sigma points: $X^{(i)}\in {\mathbb{R}}^n,i=1,\dots ,p$
• The associated weights: $W^{(i)}\in {\mathbb{R}}^n,i=1,\dots ,p$, such that: ${\sum }_{i=1}^p{W}^{(i)}=1$
• $p\in N$ is choose arbitrarily, more sigma points means more accuracy, but more computational cost.
• The minimum number of sigma points that can be taken is $p=2n$.
• $n$ is the dimension of $x(t)$, we can write $x(t)\in {\mathbb{R}}^n$
• Sigma points are selected such that their mean $\mu$ and covariance $P$ are the same as the pdf.
• Also the sigma points are distributed symmetrically with respect to the mean of the pdf.

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~Example

Let’s see a generic example on how to calculate the sigma points: Where:

• $p$ and $n$ were defined before.
• $m$ is the mean of $x(t|t)$.
• $P$ is the covariance matrix of $x(t|t)$.
• $M$ is the just a place holder matrix to understand the notation, below the explanations of the notations used.
• When we say:

$${\left(\sqrt{n\cdot P}\right)}_i$$

we mean the $i$-th column of $\sqrt{n\cdot P}$

• To calculate $\sqrt{n\cdot P}$ you can use the sqrtm function on MATLAB.