Transforms of RVs


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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