Principal component regression that minimizes the sum of the squares of the relative errors: Application in multivariate calibration models

Abstract

Relative errors are typically used in chemometrics to evaluate the performance of a multivariate predictive model. However, these models are not obtained through the criterion of minimizing relative errors, as would be expected in a model whose response is the concentration of an analyte. There are no studies in chemometrics on the use of a principal component regression that minimizes the sum of the squares of the relative errors. This work proposes a model, which serves this purpose. The suggested model, wPCR, has been applied to 7 datasets with 12 predicted responses, 10 of which are multivariate calibrations of analytes in complex mixtures based on instrumental signals coming from various analytical techniques. As PCR and wPCR are methods seeking to optimize different criteria, each one achieves a better performance with respect to its own criterion. Therefore, the new model wPCR leads to better results insofar as the relative errors are considered, especially for the smallest responses. In this sense, the wPCR model also outperforms PCR with logarithmic transformation of the response (logPCR). In addition, as for the performance of the method using Joint Confidence Regions for the intercept and the slope of the accuracy line, it is shown that the application of wPCR does not introduce bias, neither constant nor proportional for the models built, nor a systematic alteration of the achievable accuracy.