Decomposing time series data by a non-negative matrix factorization algorithm with temporally constrained coefficients

The non-negative matrix factorization algorithm (NMF) decomposes a data matrix into a set of non-negative basis vectors, each scaled by a coefficient. In its original formulation, the NMF assumes the data samples and dimensions to be independently distributed, making it a less-than-ideal algorithm for the analysis of time series data with temporal correlations. Here, we seek to derive an NMF that accounts for temporal dependencies in the data by explicitly incorporating a very simple temporal constraint for the coefficients into the NMF update rules. We applied the modified algorithm to 2 multi-dimensional electromyographic data sets collected from the human upper-limb to identify muscle synergies. We found that because it reduced the number of free parameters in the model, our modified NMF made it possible to use the Akaike Information Criterion to objectively identify a model order (i.e., the number of muscle synergies composing the data) that is more functionally interpretable, and closer to the numbers previously determined using ad hoc measures.