A numerical method for obtaining multiple linear regression parameters with physically realistic signs and magnitudes: Applications to the determination of equilibrium constants from solubility data

Abstract

A common problem in experimental geochemistry is the derivation of equilibrium constants from solubility experiments. A simple method of deriving these equilibrium constants, multiple linear regression, often results in the appearance of negative values. This has been a significant obstacle to continuing research in this field. The problem occurs for the most part because of significant correlations among the “independent” concentration variables. These correlations are an inescapable result of the nature of the experiments and the physical model being fitted. Ridge regression is an appropriate modification to simple linear regression which overcomes this difficulty. Ridge regression results in a simple procedure to obtain physically plausible, yet statistically rigorous stability constants. Of course, other problems may further degrade the quality of derived equilibrium constants, e.g. uncertainty in activity coefficients and no purely statistical method can overcome these types of problems. However, ridge regression is an effective procedure to overcome the multicolinearity which is the main cause of negative equilibrium constants. We demonstrate the use of ridge regression with a general mathematical model and then illustrate its use in the determination of iron-chloro complex equilibrium constants from solubility studies of pyrite-pyrrhotite-magnetite in NaCl solutions at 250°C. Ridge regression may also be of use in other geochemical problems where one must estimate parameters with a physical interpretation and where the independent variables are significantly intercorrelated.