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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.

In this paper, we propose two weighted Non-negative Matrix Factorization (NMF) methods using a β-divergence cost function. This divergence is used as a dissimilarity measure which can be tuned by the parameter β. The weights allow to deal with the uncertainty associated to each data sample. Our first approach consists of generalizing weighted NMF methods proposed with specific divergences or norms to the βdivergence. In our second approach, we assume that some components of the factorization are known and we use them to inform our NMF algorithm. We thus consider a specific parameterization which involves these constraints. In particular, we propose specific multiplicative update rules for the minimization of this parameterization with a weighted divergence. Lastly, some experiments on simulated mixtures of particulate matter sources show the relevance of these approaches.

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