Using the L 1 norm to select basis set vectors for multivariate calibration and calibration updating

• Published in: Journal of chemometrics 2016-02, Vol.30 (3), p.109-120
• Main Authors: Shahbazikhah, Parviz, Kalivas, John H., Andries, Erik, O'Loughlin, Trevor
• Format: Article
• Language: English
• ISSN: 0886-9383; 1099-128X
• DOI: 10.1002/cem.2778

With projection based calibration approaches, such as partial least squares (PLS) and principal component regression (PCR), the calibration space is spanned by respective basis vectors (latent vectors). Up to rank k basis vectors are formed where k  ≤ min( m , n ) with m and n denoting the number of calibration samples and measured variables. The user needs to decide how many and which respective basis vectors (tuning parameters). To avoid the second issue, basis vectors are selected top‐down starting with the first and sequentially adding until model criteria are satisfied. Ridge regression (RR) avoids the issues by using the full set of basis vectors. Another approach is to select a subset from the total available. The presented work develops a process based on the L 1 vector norm to select basis vectors. Specifically, the L 1 norm is used to select singular value decomposition (SVD) basis set vectors for PCR (LPCR). Because PCR, PLS, RR, and others can be expressed as linear combination of the SVD basis vectors, the focus is on selection and comparison using the SVD basis set. Results based on respective tuning parameter selections and weights applied to the SVD basis vectors for LPCR, top‐down PCR, correlation PCR (CPCR), PLS, and RR are compared for calibration and calibration updating using spectroscopic data sets. The methods are found to predict equivalently. In particular, the L 1 norm produces similar results to those obtained by the well‐studied CPCR process. Thus, the new method provides a different theoretical framework than CPCR for selecting basis vectors. Copyright © 2016 John Wiley & Sons, Ltd. The L 1 vector norm minimization penalty (originally developed for variable section) is adapted to select singular value (SVD) basis set vectors for principal component regression (LPCR). The methods LPCR, top‐down PCR, correlation PCR, partial least squares, and ridge regression are compared for calibration and calibration updating using spectroscopic data sets. Comparison is based on model quality measures and how respective methods weight the common SVD basis set. No unique advantages are obtained with LPCR.