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. Its creator Jeffrey Uhlmann explained that "unscented" was an arbitrary name that he adopted to avoid it being referred to as the “Uhlmann filter.”
Background 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, [ x , y ] {\displaystyle [x,y]} , with an uncertainty given in the form of a 2x2 covariance matrix giving the variance in x {\displaystyle x} , the variance in y {\displaystyle y} , and the cross covariance between the two. A variance 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. The mean and covariance representation only gives the first two moments of an underlying, but otherwise unknown, probability distribution. In the case of a moving object, the unknown probability distribution might represent the uncertainty of the object's position at a given time. The mean and covariance representation of uncertainty is mathematically convenient because any linear transformation T {\displaystyle T} can be applied to a mean vector m {\displaystyle m} and covariance matrix M {\displaystyle M} as T m {\displaystyle Tm} and T M T T {\displaystyle TMT^{\mathrm {T} }} . This linearity property does not hold for moments beyond the first raw moment (the mean) and the second central moment (the covariance), so it is not generally possible to determine the mean and covariance resulting from a nonlinear transformation because the result depends on all the moments, and only the first two are given. Although the covariance matrix is often treated as being the expected squared error associated with the mean, in practice the matrix is maintained as an upper bound on the actual squared error. Specifically, a mean and covariance estimate ( m , M ) {\displaystyle (m,M)} is conservatively maintained so that the covariance matrix M {\displaystyle M} is greater than or equal to the actual squared error associated with m {\displaystyle m} . Mathematically this means that the result of subtracting the expected squared error (which is not usually known) from M {\displaystyle M} is a semi-definite or positive-definite matrix. The reason for maintaining a conservative covariance estimate is that most filtering and control algorithms will tend to diverge (fail) if the covariance is underestimated. This is because a spuriously small covariance implies less uncertainty and leads the filter to place more weight (confidence) than is justified in the accuracy of the mean. Returning to the example above, when the covariance is zero it is trivial to determine the location of the object after it moves according to an arbitrary nonlinear function f ( x , y ) {\displaystyle f(x,y)} : just apply the function to the mean vector. When the covariance is not zero the transformed mean will not generally be equal to f ( x , y ) {\displaystyle f(x,y)} and it is not even possible to determine the mean of the transformed probability distribution from only its prior mean and covariance. Given this indeterminacy, the nonlinearly transformed mean and covariance can only be approximated. The earliest approximation was to linearize the nonlinear function and apply the resulting Jacobian matrix to the given mean and covariance. This is the basis of the extended Kalman Filter (EKF), and although it was known to yield poor results in many circumstances, there was no practical alternative for many decades.
Motivation In 1994 Jeffrey Uhlmann noted that the EKF takes a nonlinear function and partial distribution information (in the form of a mean and covariance estimate) of the state of a system but applies an approximation to the known function rather than to the imprecisely-known probability distribution. He suggested that a better approach would be to use the exact nonlinear function applied to an approximating probability distribution. The motivation for this approach is given in his doctoral dissertation, where the term unscented transform was first defined:
… excerpt ends here. Continue reading the full article.
