Partial constrained least squares (PCLS) and application in soft sensor

With the combination of extracting latent variables and setting constrained samples, partial constrained least squares (PCLS) is proposed and applied for soft sensor. Similar to constrained least squares (CLS), for the purpose of generating matrix equations with Lagrange multiplier, PCLS assigns some samples in calibration set as constrained ones and others as non-constrained ones. Then, this regression coefficients vector for calibration can be obtained by solving matrix equations with partial least squares (PLS). In moving window method of soft sensor, the sample to be predicted is highly related to the samples in previous adjacent sequential time points, thus those samples can be set as constrained ones and other samples not close to those sequential time points in the window as non-constrained ones. Based on the constrained and non-constrained samples, PCLS can be applied to calibrating a model and estimating the predicted sample. Two batches of datasets containing Sulfur Recovery Unite (SRU) and simulated datasets generated by random walk were tested by the proposed method. The results showed that PCLS is the generation of CLS, while CLS is the special case of PCLS when the number of latent variables equals the total number of variables and constrained samples. Meanwhile, in contrast with least squares (LS), PLS and CLS, PCLS can result in smaller prediction errors. Furthermore, four simulated datasets (SIM1, SIM2, SIM3 and SIM4) with trend and/or random walk, or, without trend and/or random walk, showed PCLS can be applied to the datasets when the samples in sequential time points are correlated to those in the previous adjacent sequential time points.