The multivariate calibration problem is a problem of predicting the concentration in an unknown sample, <i>c</i><sub>un</sub>, from the response vector of an unknown sample, <b>r</b><sub>un</sub> (<i>J</i> responses). The predicting equation can be arranged in the form <i>ĉ</i><sub>un</sub> = <b>r</b><sub>un</sub><sup>T</sup><b>R</b><sup>+</sup><b>c.</b> (1) <b>R</b><sup>+</sup> is the pseudo-inverse of the calibration set matrix of responses, <b>R</b>, whose column indices correspond to the <i>J</i> sensors or wavelengths and row indices correspond to the <i>I</i> samples (individuals), and <b>c</b> is the vector of concentrations for the <i>I</i> samples of the analyte in each of the calibration samples. Derivation of Eq. 1 is described in Ref. 1. The PLS regression involves solution of the predicting equation.
Avraham Lorber and Bruce R. Kowalski, "A Note on the Use of the Partial Least-Squares Method for Multivariate Calibration," Appl. Spectrosc. 42, 1572-1574 (1988)