Current factor analysis receptor models are ill-posed

Abstract

Current factor models lack sufficient physical constraints to guarantee a unique, physically valid solution; in this sense they are ill-posed. Any realistic factor model must obey certain natural physical constraints, for example, the predicted source contributions and elemental compositions must be non-negative. Five such constraints are given in the paper. As shown by a simple example with only two sources and three elements, these natural constraints are insufficient to define a unique factor model. The same is shown to be true for a more complex example with seven sources and 10 elements. Since the examples use simulated data without observational or other errors, they prove that current factor models are, in general, biased in the statistical sense. The examples also show that the bias, or systematic error, can be very large. Thus, while factor analysis continues to be a valuable screening tool for unexpected sources, in the hands of the inexperienced it could lead to serious errors in source apportionment and derived source compositions.