I am trying to understand how the ICA by A. Hyvärinen, J. Karhunen, E. Oja (2001) works in practice. In particular, I have problems trying to understand its application to factor-analysis-kind of models. In their book (2001) and in another paper, the authors refer to the fact that they apply ICA to the cash flows of 40 stores/shops of which they have 140 time-observations each ("Kiviluoto, Oja, 1998, Independent component analysis for parallel financial time series"). They say they reduce first the dimension of teh data (the 40 time-series) to only 4 using PCA through which they also whiten the data. Then, they apply their FastICA algorithm to extract the sources from those new 4 time-series. The idea is to extract the underlying sources common to all the 40 shops and investigate them.
I am trying something similar using the FastICA provided in scikit-learn in Python. Using the fit_transform method, I should get back the "fitted" values for the original time series, but actually the ouput is 4 time series only. Now my question is: can I actually apply the ICA method by choosing a number of sources (much) smaller than the signals we observe in the data (in this case, the cash flows of shops) and then, once I have my mixing matrix, multiply it with the matrix of sources to get back fitted values (or predictions or transformed values) for my initial input matrix?