Prediction intervals of an alternative formulation of partial least squares algorithm

The prediction interval is an important property when applies the partial least squares (PLS) to virtual sensor applications. In this work, we propose a new formulation of PLS, such that after projecting out score vectors, the estimated coefficient matrix is obtained as the product of pseudo inverse of the predictor matrix and corresponding weighting matrices. The new formulation, which facilitates the calculation of Jacobian matrix and can be extended to multivariate PLS, is proved to be equivalent to the nonlinear iterative partial least squares (NIPALS). The prediction interval of the algorithm is developed based on the Jacobian of singular vectors. Industrial case studies demonstrate the utility of the algorithm for univariate PLS.