Non-negative Matrix Factorization under equality constraints—a study of industrial source identification

This work is devoted to the factorization of an observation matrix into additive factors, respectively a contribution matrix G and a profile matrix F which enable to identify many pollution sources. The search for G and F is achieved through Non-negative Matrix Factorization techniques which alternatively look for the best updates on G and F.These methods are sensitive to noise and initialization, and—as for any blind source separation method—give results up to a scaling factor and a permutation. A Weighted Non-negative Matrix Factorization extension has also been proposed in the literature, so that different standard deviations of the data matrix components are taken into account. However, some estimated profile components may be inconsistent with practical experience. To prevent this issue, we propose an informed Non-negative Matrix Factorization, where some components of the profile matrix are set to zero or to a constant positive value. A special parametrization of the profile matrix is developed in order to freeze some profile components and to let free the other ones.The problem amounts to solve a family of quadratic sub-problems. A Maximization Minimization strategy leads to some global analytical expressions of both factors.These techniques are used to estimate source contributions of airborne particles from both industrial and natural influences. The relevance of the proposed approach is shown on a real dataset.