In the ICA model in Eq. (4), it is easy to see that the following ambiguities will hold:
1. We cannot determine the variances (energies) of the independent components.
The reason is that, both and being unknown, any scalar multiplier in one of the sources si could always be cancelled by dividing the corresponding column of by the same scalar; see eq. (5). As a consequence, we may quite as well fix the magnitudes of the independent components; as they are random variables, the most natural way to do this is to assume that each has unit variance: . Then the matrix will be adapted in the ICA solution methods to take into account this restriction. Note that this still leaves the ambiguity of the sign: we could multiply the an independent component by -1 without affecting the model. This ambiguity is, fortunately, insignificant in most applications.
2. We cannot determine the order of the independent components.
The reason is that, again both and being unknown, we can freely change the order of the terms in the sum in (5), and call any of the independent components the first one. Formally, a permutation matrix and its inverse can be substituted in the model to give . The elements of are the original independent variables sj, but in another order. The matrix is just a new unknown mixing matrix, to be solved by the ICA algorithms.