The Use of Least-Squares for XPS Peak Parameters Estimation: Part 3. Multicollinearity, ill-conditioning and constraint induced bias

This paper addresses the problem of ill-conditioning in XPS or UPS regressions that was first discussed by Wertheim. Ill-conditioning is addressed in terms of multicollinearities in the information matrix, and two diagnostics (the variance inflation factor and the condition number) are suggested to identify it. It is shown that constraining the parameters can remove ill-conditioning, whether or not the constraints are physically meaningful. The questions of bias, standard deviation, reliability of regression estimates, variance inflation, nuisance parameters and identifiability of parameters are also discussed.