Figure 2

Illustration of the HC-PLSR approach. The HC-PLSR "pipeline" starts with calibration of an initial global polynomial PLSR using all observations in the calibration set. This global PLSR model provides PLS scores and loadings, which constitute the basis for separation of the calibration set observations into groups by fuzzy C-means (FCM) clustering [44, 45]. Local PLSR models are then calibrated in each cluster. Predictions of response variables for new observations (or test set observations) are done by a) selecting the local model for the most probable cluster based on classification or by b) computing the regression coefficients as a weighted sum of the local models, where the weights are estimated cluster membership probabilities from the classification.

*See Additional file 1: Appendix 1, Eq. A4 and A6.