from each point so that the resulting set has a mean of zero, then computing the covariance of the zero-mean set of points and applying its inverse to each point so that the covariance of the set will be equal to the identity. Uhlmann showed that it is possible to conveniently generate a symmetric set of sigma points from the columns of and the zero vector, where is the given covariance matrix, without having to compute a matrix inverse. It is computationally efficient and, because the points form a symmetric distribution, captures the third central moment (the skew) whenever the underlying distribution of the state estimate is known or can be assumed to be symmetric. He also showed that weights, including negative weights, can be used to affect the statistics of the set. Julier also developed and examined techniques for generating sigma points to capture the third moment (the skew) of an arbitrary distribution and the fourth moment (the kurtosis) of a symmetric distribution. Example The unscented transform is defined for the application of a given function to any partial characterization of an otherwise unknown distribution, but its most common use is for the case in which only the mean and covariance is given. A common example is the conversion from one coordinate system to another, such as from a Cartesian coordinate frame to polar coordinates. Suppose a 2-dimensional mean and covariance estimate, , is given in Cartesian coordinates with: and the transformation function to polar coordinates, , is: Multiplying each of the canonical simplex sigma points (given above) by and adding the mean, , gives: Applying the transformation function to each of the above points gives: The mean of these three transformed points, , is the UT estimate of the mean in polar coordinates: The UT estimate of the covariance is: where each squared term in the sum is a vector outer product. This gives: This can be compared to the linearized mean and covariance: The absolute difference between the UT and linearized estimates in this case is relatively small, but in filtering applications the cumulative effect of small errors can lead to unrecoverable divergence of the estimate. The effect of the errors are exacerbated when the covariance is underestimated because this causes the filter to be overconfident in the accuracy of the mean. In the above example it can be seen that the linearized covariance estimate is smaller