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Factor analysis is broadly used as a powerful unsupervised machine learningtool for reconstruction of hidden features in recorded mixtures of signals. Inthe case of a linear approximation, the mixtures can be decomposed by a varietyof model-free Blind Source Separation (BSS) algorithms. Most of the availableBSS algorithms consider an instantaneous mixing of signals, while the case whenthe mixtures are linear combinations of signals with delays is less explored.Especially difficult is the case when the number of sources of the signals withdelays is unknown and has to be determined from the data as well. To addressthis problem, in this paper, we present a new method based on NonnegativeMatrix Factorization (NMF) that is capable of identifying: (a) the unknownnumber of the sources, (b) the delays and speed of propagation of the signals,and (c) the locations of the sources. Our method can be used to decomposerecords of mixtures of signals with delays emitted by an unknown number ofsources in a nondispersive medium, based only on recorded data. This is thecase, for example, when electromagnetic signals from multiple antennas arereceived asynchronously; or mixtures of acoustic or seismic signals recorded bysensors located at different positions; or when a shift in frequency is inducedby the Doppler effect. By applying our method to synthetic datasets, wedemonstrate its ability to identify the unknown number of sources as well asthe waveforms, the delays, and the strengths of the signals. Using Bayesiananalysis, we also evaluate estimation uncertainties and identify the region oflikelihood where the positions of the sources can be found.

Nonnegative Matrix Factorization for identification of unknown number of sources emitting delayed signals