Estimating the noise variance

 

Choosing $\sigma^2$ is a subjective matter; or at least how we choose $\sigma^2$could be the subject of a lengthy philosophical analysis. Perhaps that is why so much of the literature ignores this question. Without any firm theoretical basis, I chose $\vert\sigma \vert$ to be approximately the noise level. I estimated this as follows.

The simplest method of choosing $\sigma^2$is to find the average v² in the plane and then choose some arbitrary fraction of it, say 10% of the average. Although this method worked in Figure 2, I prefer another. I chose $\sigma^2$ to be the median value of v². (In other words, we conceptually prepare a list of the numbers v²; then we sort the list from smallest to largest; and finally we choose the value in the middle. In reality, median calculation is quicker than sorting.)

Notice that Figure 2 uses more initial crosstalk than Figure 1. Without the extra crosstalk I found that the first iteration worked so well, the second one was not needed. Thus I could not illustrate the utility of nonlinear estimation without more crosstalk.