A transformation-based derivation of the Kalman filter and an extensive unscented transform

In the unscented Kalman filter (UKF), the state vector is typically augmented with process and measurement noise in order to approximate the joint predictive distribution of state and observation. For that, the unscented transform is used. As its point selection mechanism changes the higher order moments between the random variables, statistical independence is not preserved. In this work, we show how statistical independence can be preserved by representing independent variables by separate point-sets. In addition to that, we show how the Kalman filter (KF) can be derived based on a particular type of linear transform that allows for a more uniform treatment of KF and UKF.