The multivariate calibration problem is a problem of predicting the concentration in an unknown sample, cun, from the response vector of an unknown sample, run (J responses). The predicting equation can be arranged in the form ĉun = runTR+c. (1) R+ is the pseudo-inverse of the calibration set matrix of responses, R, whose column indices correspond to the J sensors or wavelengths and row indices correspond to the I samples (individuals), and c is the vector of concentrations for the 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)