The estimation of negentropy is difficult, as mentioned above, and therefore this contrast function remains mainly a theoretical one. In practice, some approximation have to be used. Here we introduce approximations that have very promising properties, and which will be used in the following to derive an efficient method for ICA.
The classical method of approximating negentropy is
using higher-order moments, for example as follows :
To avoid the problems encountered with the preceding approximations of
negentropy, new approximations were developed in
. These approximation were based on the
In general we obtain the following approximation:
In the case where we use only one nonquadratic function G, the
But the point here is that by choosing G wisely, one obtains
approximations of negentropy that are much better than the one given by
(23). In particular, choosing G that does not grow too
fast, one obtains more robust estimators.
The following choices of G have proved very useful:
Thus we obtain approximations of negentropy that give a very good compromise between the properties of the two classical nongaussianity measures given by kurtosis and negentropy. They are conceptually simple, fast to compute, yet have appealing statistical properties, especially robustness. Therefore, we shall use these contrast functions in our ICA methods. Since kurtosis can be expressed in this same framework, it can still be used by our ICA methods. A practical algorithm based on these contrast function will be presented in Section 6.