expected squared error (which is not usually known) from 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 : just apply the function to the mean vector. When the covariance is not zero the transformed mean will not generally be equal to 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 for the unscented transform 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: Consider the following intuition: With a fixed number of parameters it should be easier to approximate a given distribution than it is to approximate an arbitrary nonlinear function/transformation. Following this intuition, the goal is to find a parameterization that captures the mean and covariance information while at the same time permitting the direct propagation of the information through an arbitrary set of nonlinear equations. This can be accomplished by generating a discrete distribution having the same first and second (and possibly higher) moments, where each point in the discrete approximation can be directly transformed. The mean and covariance of the transformed ensemble can