Editing Chemometrics (section)

===Multivariate calibration===
Many chemical problems and applications of chemometrics involve [[calibration]].  The objective is to develop models which can be used to predict properties of interest based on measured properties of the chemical system, such as pressure, flow, temperature, [[infrared spectroscopy|infrared]], [[Raman spectroscopy|Raman]],<ref>{{Cite journal |last1=Barton |first1=Bastian |last2=Thomson |first2=James |last3=Lozano Diz |first3=Enrique |last4=Portela |first4=Raquel |date=September 2022 |title=Chemometrics for Raman Spectroscopy Harmonization |url=http://journals.sagepub.com/doi/10.1177/00037028221094070 |journal=Applied Spectroscopy |language=en |volume=76 |issue=9 |pages=1021–1041 |doi=10.1177/00037028221094070 |pmid=35622984 |bibcode=2022ApSpe..76.1021B |s2cid=249129065 |issn=0003-7028|url-access=subscription }}</ref> [[NMR|NMR spectra]] and [[mass spectrometry|mass spectra]].  Examples include the development of multivariate models relating 1) multi-wavelength spectral response to analyte concentration, 2) molecular descriptors to biological activity, 3) multivariate process conditions/states to final product attributes.  The process requires a calibration or training data set, which includes reference values for the properties of interest for prediction, and the measured attributes believed to correspond to these properties.  For case 1), for example, one can assemble data from a number of samples, including concentrations for an analyte of interest for each sample (the reference) and the corresponding infrared spectrum of that sample.  Multivariate calibration techniques such as partial-least squares regression, or principal component regression (and near countless other methods) are then used to construct a mathematical model that relates the multivariate response (spectrum) to the concentration of the analyte of interest, and such a model can be used to efficiently predict the concentrations of new samples.

Techniques in multivariate calibration are often broadly categorized as classical or inverse methods.<ref name="Martens1989" /><ref name="Franke2002">{{cite book |first=J. |last=Franke |editor1-first=John M |editor1-last=Chalmers |chapter=Inverse Least Squares and Classical Least Squares Methods for Quantitative Vibrational Spectroscopy |title=Handbook of Vibrational Spectroscopy |publisher=Wiley |year=2002 |location=New York |isbn=978-0471988472 |doi=10.1002/0470027320.s4603 }}</ref>  The principal difference between these approaches is that in classical calibration the models are solved such that they are optimal in describing the measured analytical responses (e.g., spectra) and can therefore be considered optimal descriptors, whereas in inverse methods the models are solved to be optimal in predicting the properties of interest (e.g., concentrations, optimal predictors).<ref name="Brown2004">{{cite journal |first=C. D. |last=Brown |title=Discordance between Net Analyte Signal Theory and Practical Multivariate Calibration |journal=Analytical Chemistry |volume=76 |year=2004 |issue=15 |pages=4364–4373 |doi=10.1021/ac049953w |pmid=15283574}}</ref>  Inverse methods usually require less physical knowledge of the chemical system, and at least in theory provide superior predictions in the mean-squared error sense,<ref name="krutchkoff1969">{{cite journal |first=R. G. |last=Krutchkoff |title=Classical and inverse regression methods of calibration in extrapolation |journal=Technometrics |volume=11 |issue=3 |year=1969 |pages=11–15 |doi=10.1080/00401706.1969.10490714 }}</ref><ref name="Kowalski1984">{{cite book |first=W. G. |last=Hunter |chapter=Statistics and chemistry, and the linear calibration problem |title=Chemometrics: mathematics and statistics in chemistry |editor-first=B. R. |editor-last=Kowalski |publisher=Riedel |location=Boston |year=1984 |isbn=978-9027718464 }}</ref><ref name="Tellinghuisen2000">{{cite journal |first=J. |last=Tellinghuisen |title=Inverse vs. classical calibration for small data sets |journal=Fresenius' J. Anal. Chem. |volume=368 |year=2000 |issue=6 |pages=585–588 |doi=10.1007/s002160000556 |pmid=11228707 |s2cid=21166415 }}</ref> and hence inverse approaches tend to be more frequently applied in contemporary multivariate calibration.

The main advantages of the use of multivariate calibration techniques is that fast, cheap, or non-destructive analytical measurements (such as optical spectroscopy) can be used to estimate sample properties which would otherwise require time-consuming, expensive or destructive testing (such as [[Liquid chromatography–mass spectrometry|LC-MS]]).  Equally important is that multivariate calibration allows for accurate quantitative analysis in the presence of heavy interference by other analytes.  The selectivity of the analytical method is provided as much by the mathematical calibration, as the analytical measurement modalities.  For example, near-infrared spectra, which are extremely broad and non-selective compared to other analytical techniques (such as infrared or Raman spectra), can often be used successfully in conjunction with carefully developed multivariate calibration methods to predict concentrations of analytes in very complex matrices.