Minimizing the Cauchy function

Cauchy function A good trick (I discovered accidentally) is to use the weight  

$$\bold W \eq \ {\bf diag} \left( {1\over{\sqrt{1+r_i^2/\bar r^2}}} \right)$$ (16)

Sergey Fomel points out that this weight arises from minimizing the Cauchy function:  

$$E(\bold r) \eq \sum_i \ \log (1+r_i^2/\bar r^2)$$ (17)

A plot of this function is found in Figure .

 

cauchy
Figure 2 The coordinate is m. Top is Cauchy measures of m-1. Bottom is the same measures of the data set (1,1,2,3,5). Left, center, and right are for $\bar r = (2, 1, .2)$.

Because the second derivative is not positive everywhere, the Cauchy function introduces the possibility of multiple solutions, but because of the good results we see in Figure , you might like to try it anyway. Perhaps the reason it seems to work so well is that it uses mostly residuals of ``average size,'' not the big ones or the small ones. This happens because $\Delta \bold m$ is made from $\bold F'$ and the components of $\bold W^2\bold r$ which are a function $r_i/(1+r_i^2/\bar r^2)$that is maximum for those residuals near $\bar r$.

Module irls supplies two useful weighting functions that can be interchanged as arguments to the reweighted scheme . irlsweighting functions for iterative reweighting