Linear unmixing of multivariate observations: A structural model

In many fields of science there are multivariate observations that may be assumed to be generated by a (physical) linear mixing process of contributions from different sources. If the compositions of the sources are constant for different observations, then these observations are, up to a random error term, nonnegative linear combinations of a fixed set of so-called source profiles that characterize the sources. The goal of linear unmixing is to recover both the source profiles and the source activities (also called scores) from a multivariate dataset. We present a new parametric mixing model that assumes a multivariate lognormal distribution for the scores. This model is proved to be identifiable. Moreover, consistency and asymptotic normality of the maximum likelihood estimator (MLE) are established in special cases. To calculate the MLE, we propose the combination of two variants of the Monte Carlo EM algorithm. The proposed model is applied to simulated datasets and to a set of air pollution measurements. In addition to the basic model, several extensions are discussed.