3.2 Mathematics of PLS Partial least squares (PLS) finds the maximum variance in the predictor variables (X) and finds the correlation factors between and the predicted variables (Y) that have X maximum variance. In PLS, two linear combinations are generated from the and X Y respectively and the maximum covariance between and is calculated. Consider an X Y X matrix of size and an matrix N×K N×M Y The following descriptions are mainly based on [1,4,5,7]. The scores of X, ta (a=1, 2, A=the number of PLS components) are calculated as linear combinations of …, the original variables with the weights The mathematical expression is w*ka. or (3.1) where number of variables). The predictor variables, are expressed k=(1, K=the X X, …, as: (3.2) where is the residuals. eik X Similarly, for predicted variables if the scores of are and the weights Y, Y ua cam: or (3.3) Since scores are good predictors of in PLS, then: X Y or (3.4) where F represents the error between observed values and the predicted response. Using equation (3.1), the equation (3.2) is also expressed as
3.2 Mathematics of PLS Partial least squares (PLS) finds the maximum variance in the predictor variables (X) and finds the correlation factors between X and the predicted variables (Y) that have maximum variance. In PLS, two linear combinations are generated from the X and Y respectively and the maximum covariance between X and Y is calculated. Consider an X matrix of size N×K and an N×M matrix Y The following descriptions are mainly based on [1,4,5,7]. The scores of X, ta (a=1, 2, …, A=the number of PLS components) are calculated as linear combinations of the original variables with the weights w*ka. The mathematical expression is or (3.1) where k=(1, …, K=the number of X variables). The predictor variables, X, are expressed as: (3.2) where eik is the X residuals. Similarly, for predicted variables Y, if the scores of Y are ua and the weights cam: or (3.3) Since scores X are good predictors of Y in PLS, then: or (3.4) where F represents the error between observed values and the predicted response. Using equation (3.1), the equation (3.2) is also expressed as
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