Statistical Methods for Astronomical Data Analysis by Asis Kumar Chattopadhyay & Tanuka Chattopadhyay

Statistical Methods for Astronomical Data Analysis by Asis Kumar Chattopadhyay & Tanuka Chattopadhyay

Author:Asis Kumar Chattopadhyay & Tanuka Chattopadhyay
Language: eng
Format: epub
Publisher: Springer New York, New York, NY


7.3.2 Approximation of Negentropy

One drawback of negentropy is that it is very difficult to compute. That’s why it needs to be approximated (Hyvarinen et al. 2001). The approximation is given by:

(7.14)

where G is a non-quadratic function. In particular, G should be so chosen that it does not grow too fast. Two popular choices of G are:

(7.15)

where 1 ≤ a ≤ 2 is some suitable constant, which is often taken equal to 1.

7.3.3 The FastICA Algorithm

There are many algorithms which do ICA like FastICA, ProDen (Hastie and Tibshirani 2003), KernelICA, etc. The fastICA algorithm is a commonly used one, including industrial applications. This algorithm was developed by Hyvarinen and Oja (2000). In this method the independent components are estimated one by one. This algorithm converges very fast and is very reliable. This algorithm is also very easy to use. Our objective is to maximize J(S). Now this is equivalent to maximizing E[G(WZ)] as given in Eq. (7.13) under the constraint | | W | |  = 1. For the sake of notational and computational complicity, we consider one particular component. We are interested in finding out the optima of E[G(W K T Z)] under the constraint | | W | |  = 1, where W K T Z is the kth component of WZ. This optimization problem can be solved by the Lagrange multiplier method. The objective function is



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