Minitab uses the nonlinear iterative partial least squares (NIPALS) algorithm developed by Herman Wold1 to solve problems associated with ill-conditioned data. PLS reduces the number of predictors by extracting uncorrelated components based on the covariance between the predictor and response variables. PLS is similar to principal components regression and ridge regression, but varies in its computational method.
The PLS algorithm produces a sequence of models, where each consecutive model contains one additional component. Components are calculated one at a time, starting with the standardized x- and y-matrix. Subsequent components are calculated from the x- and y-residual matrix; iterations stop upon reaching the maximum number of components or when x-residuals become the zero matrix. If the number of components equals the number of predictors, the PLS model equals the least squares regression model. Cross-validation is used to identify the number of components that minimizes prediction error.
PLS performs decomposition on both predictors and responses simultaneously. After Minitab determines the number of components and calculates the loadings, it calculates the regression coefficients for each predictor. For more detailed information on PLS and NIPALS see234.