Unscented transform

The unscented transform (UT) is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics. The most common use of the unscented transform is in the nonlinear projection of mean and covariance estimates in the context of nonlinear extensions of the Kalman filter. In an interview, its creator Jeffrey Uhlmann explained that he came up with the name after noticing unscented deodorant on a coworker's desk.

Many filtering and control methods represent estimates of the state of a system in the form of a mean vector and an associated error covariance matrix. As an example, the estimated 2-dimensional position of an object of interest might be represented by a mean position vector, ${\displaystyle [x,y]}$, with an uncertainty given in the form of a 2x2 covariance matrix giving the variance in ${\displaystyle x}$, the variance in ${\displaystyle y}$, and the cross covariance between the two. A covariance that is zero implies that there is no uncertainty or error and that the position of the object is exactly what is specified by the mean vector.

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