Dealing with near collinearity in chemical mass balance receptor models

The amount of airborne particulate pollution attributable to various sources can be estimated from a linear least squares (LLS) fit of concentrations of chemical elements observed at a receptor to the known elemental composition of the particles emitted by the sources. The resulting least squares problem often displays a high degree of ill-conditioning and associated inflation of the uncertainties in the estimates. Because of the physical constraints of the problem, variable selection and ridge regression cannot be used to remedy the ill-conditioning. In particular, it is shown that a stable ridge regression solution is equivalent to assuming sources of airborne particulates with negative elemental composition. A method is developed which defines, for a specific level of acceptable uncertainty, three classes of sources; those which can be estimated accurately by LLS, those which cannot be so estimated and those which cannot be accurately estimated individually but participate in linear combinations which can be accurately estimated. A technique is presented which determines those linear combinations of source contributions of minimum variance.