Unscented Transformations

Another method which I’ve been implementing for parameter uncertainty estimation is the unscented transformation (which as one might expect is the basis for the unscented Kalman filter). Since I’ve been recently discussing the inherent ineffectiveness of first order Taylor uncertainty propagation, you all should be well aware of the potential flaws in its predictions. For “hard” nonlinearities with jump discontinuities, we encounter strange linearizations which propagate though the calculations and effect the final answer. One method that I’ve been working on and discussing is another form of linearization (describing functions analysis), but the unscented transformation is an entirely different approach. Rather than linearizing the nonlinearities, we utilize these functions to capture the statistics of sigma points.

The above plot displays the general idea of the unscented transformation. A general function (f(x) = sin(x1) + cos(x2)) is given with the input variables having normal distributions. Since the variables are random variables, the output of this transformation is also going to be a random variable. If we assume that the output of the function will also be normal (a decent assumption in this case for small variance), we can propagate the uncertainty though the function and compute a normal output. The unscented transformation selects statistically significant “sigma points” (shown above as pink dots), and these points are all fed through the function, f(x). The output for each of the sigma points are then analyzed and the statistics for the outputs are computed (mean and variance for a normal distribution). The plots above show how the unscented transformation fairs against other prediction methods (the ellipsoids are centered on the mean of the prediction and the radii indicate the standard deviation prediction). As can be seen, the unscented transformation more readily approximates the actual statistics (as computed by Monte Carlo) and radically outperforms the Taylor first order prediction.