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The objective of this chapter is to present some fundamental issues that every atomic spectrometry practitioner should keep in mind when developing a calibration. However familiar this may sound it is definitely a critical step, as important as the development of proper experimental measurement conditions. The fundamentals of the least squares fit are presented and discussed, with special emphasis on the practical limitations we currently experience in laboratories. Many discussions will be devoted to validation of the model and to clarification of some misconceptions that appear sometimes in the literature. The widely applied standard additions method is reviewed and discussed to show that extrapolation is a risky practice that can be avoided very easily. Polynomials of order two (quadratic) provide the analyst with a readily available means of matching points from a curved calibration to a function suitable for interpolation. However, extra caution is necessary to avoid lack of fit, and even small extrapolations beyond the calibrated range are unwise. Finally, two appendixes are included to discuss in some detail the correct use of the Mandel’s test for linearity and how to compare two regression lines (typically, the aqueous calibration fit and the standard additions one).

This chapter presents the most widely applied and, probably, satisfactory multivariate regression method used nowadays: partial least squares (PLS). Graphical explanations of many concepts are given to complement the more formal mathematical background. Several approaches to solving current problems are suggested. The development of a satisfactory regression model can alleviate the typical laboratory workload (preparation of many standards, solutions with concomitants, etc.) but only when a strict and serious job is performed with the PLS methodology. Iteration is the key word here as the analyst has to iterate the data within the software capabilities. Validation is essential, as can never be stressed sufficiently enough, and it will be explained here in detail. Two approaches to deal with the new concepts of ‘limit of detection’ and ‘limit of quantification’ (these terms will be used although they have been superseded) given by International Organization for Standardization (ISO) and the European Union (EU) are presented. Finally, a comprehensive review of practical applications that have used PLS within the atomic spectrometry field is presented.

The huge efforts made currently by atomic spectroscopists to resolve interferences and optimise instrumental measuring devices to increase accuracy and precision have led to a point where many of the difficulties that need to be solved nowadays cannot be described by simple classical linear regression methods and not even by other advanced linear regression methods. Typical situations where these can fail involve spectral non‐linearities. This chapter introduces two relatively recent regression methodologies which, in contrast to classical programming, work with rules rather than with well‐defined and fixed algorithms: artificial neural networks (ANNs), a fairly established technique nowadays, and the support vector machine (SVM), which is emerging as a powerful method to perform both classification and regression tasks.

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