Sources identification using shifted non-negative matrix factorization combined with semi-supervised clustering.

Non-negative matrix factorization (NMF) is a well-known unsupervised learning method that has been successfully used for blind source separation of non-negative additive signals.NMF method requires the number of the original sources to be known a priori. Recently, we reported a method, we called NMFk, which by coupling the original NMF multiplicative algorithm with a custom semi-supervised clustering allows us to estimate the number of the sources based on the robustness of the reconstructed solutions. Here, an extension of NMFk is developed, called ShiftNMFk, which by combining NMFk with previously formulated ShiftNMF algorithm, Akaike Information Criterion (AIC), and a custom procedure for estimating the source locations is capable of identifying: (a) the number of the unknown sources, (b) the eventual delays in the signal propagation, (c) the locations of the sources, and (d) the speed of propagation of each of the signals in the medium. Our new method is a natural extension of NMFk that can be used for sources identification based only on observational data. We demonstrate how our novel method identifies the components of synthetic data sets, discuss its limitations, and present a Julia language implementation of ShiftNMFk algorithm.