Transforms of RVs

2020-06-04 – 2020-10-23
Maybe related 🤷:

I have a nonlinear transformation of a random process. What is its distribution?

Stochastic Itō-Taylor expansion

See stochastic taylor expansion. tl;dr: More trouble than it is worth.

Linearization

As seen in the Ensemble Kalman Filter.

Unscented transform

The great invention of Uhlmann and Julier is unscented transform, which uses a ‘\(\sigma\)-point approximation.’

In the context of Kalman filtering,

What the Unscented Transform does is to replace the mean vector and its associated error covariance matrix with a special set of points with the same mean and covariance. In the case of the mean and covariance representing the current position estimate for a target, the UT is applied to obtain a set of points, referred to as sigma points, to which the full nonlinear equations of motion can be applied directly. In other words, instead of having to derive a linearized approximation, the equations could simply be applied to each of the points as if it were the true state of the target. The result is a transformed set of points, and the mean and covariance of that set represents the estimate of the predicted state of the target.

See, e.g., Roth, Hendeby, and Gustafsson (2016).

References

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Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. “Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences 13 (3). https://doi.org/10.11113/mjfas.v13n3.633.
Collard, Fabrice, and Michel Juillard. 2001. “Accuracy of Stochastic Perturbation Methods: The Case of Asset Pricing Models.” Journal of Economic Dynamics and Control 25 (6-7): 979–99. https://ideas.repec.org/a/eee/dyncon/v25y2001i6-7p979-999.html.
Gustafsson, Fredrik, and Gustaf Hendeby. 2008. “On Nonlinear Transformations of Stochastic Variables and Its Application to Nonlinear Filtering.” In 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 3617–20. https://doi.org/10.1109/ICASSP.2008.4518435.
———. 2012. “Some Relations Between Extended and Unscented Kalman Filters.” IEEE Transactions on Signal Processing 60 (2): 545–55. https://doi.org/10.1109/TSP.2011.2172431.
Hendeby, Gustaf, and Fredrik Gustafsson. 2007. “On Nonlinear Transformations of Gaussian Distributions,” 3. https://users.isy.liu.se/en/rt/fredrik/reports/07SSPut.pdf.
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