Editing Kalman filter (section)

==== Sigma points ====
For a [[Random variable|random]] vector <math>\mathbf{x}=(x_1, \dots, x_L)</math>, sigma points are any set of vectors
:<math> \{\mathbf{s}_0,\dots, \mathbf{s}_N \}=\bigl\{\begin{pmatrix} s_{0,1}& s_{0,2}&\ldots& s_{0,L} \end{pmatrix}, \dots, \begin{pmatrix} s_{N,1}& s_{N,2}&\ldots& s_{N,L} \end{pmatrix}\bigr\}</math>
attributed with

* first-order weights <math>W_0^a, \dots, W_N^a</math> that fulfill
# <math> \sum_{j=0}^N W_j^a=1 </math> 
# for all <math>i=1, \dots, L</math>: <math> E[x_i]=\sum_{j=0}^N W_j^a s_{j,i} </math>
* second-order weights <math>W_0^c, \dots, W_N^c</math> that fulfill 
# <math> \sum_{j=0}^N W_j^c=1 </math>  
# for all pairs <math> (i,l) \in \{1,\dots, L\}^2: E[x_ix_l]=\sum_{j=0}^N W_j^c s_{j,i}s_{j,l} </math>.

A simple choice of sigma points and weights for <math>\mathbf{x}_{k-1\mid k-1}</math> in the UKF algorithm is
:<math>\begin{align}
\mathbf{s}_0&=\hat \mathbf{x}_{k-1\mid k-1}\\
-1&<W_0^a=W_0^c<1\\
\mathbf{s}_j&=\hat \mathbf{x}_{k-1\mid k-1} + \sqrt{\frac{L}{1-W_0}} \mathbf{A}_j, \quad j=1, \dots, L\\
\mathbf{s}_{L+j}&=\hat \mathbf{x}_{k-1\mid k-1} - \sqrt{\frac{L}{1-W_0}} \mathbf{A}_j, \quad j=1, \dots, L\\
W_j^a&=W_j^c=\frac{1-W_0}{2L}, \quad j=1, \dots, 2L
\end{align}
</math>
where <math>\hat \mathbf{x}_{k-1\mid k-1}</math> is the mean estimate of <math>\mathbf{x}_{k-1\mid k-1}</math>. The vector <math>\mathbf{A}_j</math> is the ''j''th column of <math>\mathbf{A}</math> where <math>\mathbf{P}_{k-1\mid k-1}=\mathbf{AA}^\textsf{T}</math>. Typically, <math>\mathbf{A}</math> is obtained via [[Cholesky decomposition]] of <math>\mathbf{P}_{k-1\mid k-1}</math>. With some care the filter equations can be expressed in such a way that <math>\mathbf{A}</math> is evaluated directly without intermediate calculations of <math>\mathbf{P}_{k-1\mid k-1}</math>.  This is referred to as the ''square-root unscented Kalman filter''.<ref>{{cite book |last1=Van der Merwe |first1=R. |last2=Wan |first2=E.A. |title=2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221) |chapter=The square-root unscented Kalman filter for state and parameter-estimation |date=2001 |volume=6 |pages=3461–3464 |doi=10.1109/ICASSP.2001.940586|isbn=0-7803-7041-4 |s2cid=7290857 }}</ref>

The weight of the mean value, <math>W_0</math>, can be chosen arbitrarily.

Another popular parameterization (which generalizes the above) is 
:<math>\begin{align}
\mathbf{s}_0&=\hat \mathbf{x}_{k-1\mid k-1}\\
W_0^a&= \frac{\alpha^2\kappa-L}{\alpha^2\kappa}\\
W_0^c&= W_0^a + 1-\alpha^2+\beta \\
\mathbf{s}_j&=\hat \mathbf{x}_{k-1\mid k-1} + \alpha\sqrt{\kappa} \mathbf{A}_j, \quad j=1, \dots, L\\
\mathbf{s}_{L+j}&=\hat \mathbf{x}_{k-1\mid k-1} - \alpha\sqrt{\kappa} \mathbf{A}_j, \quad j=1, \dots, L\\
W_j^a&=W_j^c=\frac{1}{2\alpha^2\kappa}, \quad j=1, \dots, 2L.
\end{align}
</math>

<math>\alpha</math> and <math>\kappa</math> control the spread of the sigma points.  <math>\beta</math> is related to the distribution of <math>x</math>. Note that this is an overparameterization in the sense that any one of <math>\alpha</math>, <math>\beta</math> and <math>\kappa</math> can be chosen arbitrarily.

Appropriate values depend on the problem at hand, but a typical recommendation is <math>\alpha = 1</math>, <math>\beta = 0</math>, and <math>\kappa \approx 3L/2</math>.{{cn|date=January 2025}} If the true distribution of <math>x</math> is Gaussian, <math>\beta = 2</math> is optimal.<ref>{{Cite book |doi=10.1109/ASSPCC.2000.882463 |chapter-url=http://www.lara.unb.br/~gaborges/disciplinas/efe/papers/wan2000.pdf |chapter=The unscented Kalman filter for nonlinear estimation |title=Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373) |page=153 |year=2000 |last1=Wan |first1=E.A. |last2=Van Der Merwe |first2=R. |isbn=978-0-7803-5800-3 |citeseerx=10.1.1.361.9373 |s2cid=13992571 |access-date=2010-01-31 |archive-date=2012-03-03 |archive-url=https://web.archive.org/web/20120303020429/http://www.lara.unb.br/~gaborges/disciplinas/efe/papers/wan2000.pdf |url-status=dead }}</ref>