A more robust unscented transform

Abstract

The unscented transformation is extended to use extra test points beyond the minimum necessary to determine the second moments of a multivariate normal distribution. The additional test points can improve the estimated mean and variance of the transformed distribution when the transforming function or its derivatives have discontinuities.

Significance, interpretation, and quantification of uncertainty in prognostics and remaining useful life prediction

“…5. Unscented transform sampling: Unscented transform sampling [6,70] is a sampling approach that focuses on estimating the mean and variance of R accurately, instead of the entire probability distribution of R. Certain pre-determined sigma points are selected in the X-space and these sigma points are used to generate corresponding realizations of R. Using weighted averaging principles, the mean and variance of R are calculated.…”

Survey of maneuvering target tracking: III. Measurement models

“…First, as demonstrated in [15,69,66,71,72,73], the accuracy of this transformation is surprisingly good 9 in view of the fact that a quite small number of sample points are used. Second, the moments are approximated directly in this approach, while in the linearization (or Taylor series expansion) based techniques, the transformation is approximated first and then moments are computed accordingly.…”

“…More accurate measurement conversions can be obtained in a similar manner by using higher-order transformations (e.g., those match the skew and kurtosis of , as developed in [15,66]) or more carefully designed sigma points [15,66,71]. Following the original development of the above symmetric second-order sigma points, several enhanced sets of sigma points 8 It was recommended in [15] with theoretical justification that Ò ¿ be used for a Gaussian distribution.…”

“…Briefly, [73] proposes an enhancement by scaling the pattern of sigma points; to match the mean and covariance of an Ò-dimensional vector , [72] proves Ò · ½ to be the minimum number of required sigma points and derives a minimal set of asymmetric sigma points that minimizes the skew; a more robust implementation of the second-order transformation, utilizing more sigma points by augmenting with its noise, was used in [69] and later more fully explored in [71], along with an associated square root.…”

Reliability estimation using unscented transformation

“…While some remarks on this topic are made in [18], [19] for low-dimensional spaces, the optimal selection of sampling points in high-dimensional spaces for non-analytic functions along with rigorous error analysis is still an open problem and topic for future research to our best knowledge.…”