A simple, all-purpose nonlinear algorithm for univariate calibration

Abstract

The goal of univariate calibration is the estimation of some unknown concentration or amount and its uncertainty, from the sample’s response to a probe measurement, by reference to a set of known samples whose responses are similarly measured. For this purpose, a simple nonlinear algorithm has been devised, in which the unknown sample is included as the (n + 1)-th data point in the data set and is fitted directly to the unknown concentration while the n calibration points are fitted to the calibration function. For example, in straight-line calibration, y = a + bx, the response y₀ of the unknown is fitted to y₀ = a + bx₀. The standard error in the estimated concentration x₀ is obtained directly from the variance–covariance matrix for the fit. The method handles homo- and heteroscedastic data equally well, and is easily extended to more complex linear calibration functions (e.g., polynomials in x) and to nonlinear functions (exponentials, logs). Its implementation is illustrated for a typical microcomputer data analysis program.