Abstract

The multivariate calibration problem is a problem of predicting the concentration in an unknown sample, <i>c</i>_un, from the response vector of an unknown sample, <b>r</b>_un (<i>J</i> responses). The predicting equation can be arranged in the form <i>ĉ</i>_un = <b>r</b>_un^T<b>R</b>⁺<b>c.</b> (1) <b>R</b>⁺ 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.