Second-order bilinear calibration: the effects of vectorising the data matrices of the calibration set

In a groundbreaking paper, Linder and Sundberg [Chemometr. Intell. Lab. Syst. 42 (1998) 159] developed a statistical framework for the calibration of second-order bilinear data. Within this framework, they formulated three different predictor construction methods [J. Chemom. 16 (2002) 12], namely the so-called naı̈ve method, the bilinear least squares (BLLS) method, and a refined version of the latter that takes account of the calibration uncertainty. Elsewhere [J. Chemom. 15 (2001) 743], a close relationship is established between the naı̈ve method and the generalized rank annihilation method (GRAM) by comparing expressions for prediction variance. Here it is proved that the BLLS method can be interpreted to work with vectorised data matrices, which establishes an algebraic relationship with so-called unfold partial least squares (PLS) and unfold principal component regression (PCR). It is detailed how these results enable quantifying the effects of vectorising bilinear second-order data matrices on analytical figures of merit and variance inflation factors.