I have read that it is not possible to verify the variance and order of independent component extracted from ICA.

So, I reconstructed the spectra from matrix A and ST(independent component) then apply the PCA to this data. I found that the loadings of PCA have very similar characteristic with the independent component of ICA and I could get the explained variance and their order from this result. I'm just wondering whether what I did is correct or not?


Remember that PCA makes two more or less arbitrary assumptions that ICA does not make:

  • the loadings are defined to be orthonormal, and
  • are ordered by decreasing variance

So the order is arbitrary but sometimes convenient. Orthonormal loadings mean that the scores are just rotated and/or flipped, but not stretched. This is sometimes important, but often also more a convenient convention.

ICA does not make these assumptions, and thus leaves the decomposition ambiguous to scaling factors: You can multiply your mixture matrix $A$ with a factor $f$ and at the same time the source matrix $S$ by $\frac{1}{f}$ and the decomposition is still valid. However, this not only means that the scale of $A$ changes, but its variance changes accordingly by $f^2$. This is the reason for "no variance in ICA".

In chemometric context that means e.g. that ICA doesn't care whether your concentrations are in g/l or mol/l - as long as the pure component spectra use the same scale.

Another way to illustrate the ambiguity: consider fluorescence emission spectra. From the spectra alone you cannot tell whether you have strong solutions but a weak excitation source or dilute solutions but a strong excitation source. Neither can you tell the quantum efficiency of your fluorophores.

PCA basically puts both variations into the scores, and the PC variance is thus a "combined" variance that includes the absolute physical scale of concentration and excitation source intensity and quantum efficiency. For some questions it does make sense to talk about this combined variance, but if you need the scales of either source or mixture, you need external information to resolve the ambiguity (you may say that your model is underdetermined).
PCA does not get you around this basic fact.

You mention explained variance. You can certainly look at the total variance of your data $X$, and compare to explained and/or residual variance. Whether you do this via PCA or directly does not matter.


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