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” though others have indicated that "unscented" is a contrast to "scented" intended as a euphemism for "stinky" 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, , with an uncertainty given in the form of a 2x2 covariance matrix giving the variance in , the variance in , 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. 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 can be applied to a mean vector and covariance matrix as and . 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 is conservatively maintained so that the covariance matrix is greater than or equal to the actual squared error associated with . Mathematically this means that the result of subtracting the