I could not resist trying PEF estimation anyway. After all, maybe a bad equation shows up much like a bad data point. It is gratifying that bad regression equations are thrown out of the matrix used to determine .
Unfortunately, I found the program became extremely unstable, diverging rapidly. To achieve convergence I needed to make the noise small. The filters I got were never close to correct.
It seems the best method to deal with the PEF problem is as currently described in TDF. The way TDF handles such noise is via weighting functions. Generally when computing a prediction-error filter, there are ample equations compared to unknowns. When any regression equation has a large residual, I abandon that equation by applying a zero weight to it. One method is to change the weighting function as the regression proceeds. This is risky. The more reliable method is to solve the problem with fixed weights, then use the residuals to determine new weights to solve the problem a second time.
Figure 1 (a) Simple data with bursty noise. (b) Cleaned by L2 method of TDF (c) Huber method.