Linear Unmixing of Multivariate Observations: A Structural Model

Marcel Wolbers and Werner Stahel

August 2002

Abstract: In many fields of science there aremultivariate observations which are generated by a (physical) linear mixing process of contributions from different sources.If it is assumed that the composition of the sourcesis constant for different observations, these observations are, up to measurement error, non-negative linear combinations of a fixed set of so-called source profiles whichcharacterize the sources. The goal of linear unmixing is to recoverboth the source profiles and the source activities (also called scores)from a multivariate dataset.We present a new parametric mixing model which assumes a multivariatelognormal distribution for the scores. This model is proved to be identifiable. To calculate the MLE we propose the combination of two variants of the MCEM algorithm. The proposed model is appliedto simulated datasets and to air pollution measurements from Zurich.In addition to the basic model we discuss several extensions.

Keywords: linear mixing model, source apportionment, latent variables, identifiability, MCEM algorithm

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