One of my research interests is in chemometrics, specifically in multivariate calibration.
The response “y” comes from the concentration (mole fraction) of a chemical mixture, consisting of three different chemical components (water, 3-amino-1-propanol, 1,2-ethanediol).
The ternary plot for a total of m = 34 sample mixtures (represented by dots) is provided in Fig. 1. The interior dots are mixtures containing all three components, the edge dots are mixtures containing only two components, and the corner dots are pure (containing a single component).
Corresponding to each of the 34 sample mixtures, there exists an extremely rich near-infrared (NIR) spectroscopy regressor (signal) information, taken under different temperature conditions; the figure below displays NIR signal regressors taken at two different sample temperatures 30 °C and 70 °C (out of 12 different temperature conditions) for each of the m = 34 mixtures.
(Although not an awe-inspiring magnitude of change, this small “nuisance” effect of temperature on spectra can in fact significantly influence accurate prediction of the mole fraction of the chemical mixtures.)
Multivariate calibration refers to building a mathematical model that relates the multivariate response (spectrum) to the concentration of the chemical mixture: such a model can be used to predict the concentrations of new mixtures based on (fast and cheap) NIR spectroscopy.
My specific interest here is to develop a simple, tractable yet flexible semiparametric functional regression model in the class of projection pursuit regression, specifically to model the association between temperature and the shape of spectra.
The primary goal of this regression is then, first, to identify a set of “prototypes” of spectra that is most associated with the temperature variations, and, second, by accounting for these temperature effects, to make reliable future (external) prediction on the mole fraction of the chemical mixtures based on their spectra, i.e., to perform a reliable multivariate calibration.