Nonlinear projection to latent structures method and its applications

Projection to latent structures or partial least squares (PLS) is one of the most powerful linear regression techniques to deal with noisy and highly correlated data. To address the inherent nonlinearity present in the data from many industrial applications, a number of approaches have been proposed to extend it to the nonlinear case, and most of them rely on the nonlinear approximation capability of neural networks. However, these methods either merely introduce nonlinearities to the inner relationship model within the linear PLS framework or suffer from training a complicated network. In this paper, starting from an equivalent presentation of PLS, both nonlinear latent structures and nonlinear reconstruction are obtained straightforwardly through two consecutive steps. First, a radial basis function (RBF) network is utilized to extract the latent structures through linear algebra methods without the need of nonlinear optimization. This is followed by developing two feed-forward networks to reconstruct the original predictor variables and response variables. Extraction of multiple latent structures can be achieved in either sequential or parallel fashion. The proposed methodology thus possesses the capability of explicitly characterizing the nonlinear relationship between the latent structures and the original predictor variables while exhibiting fast convergence speed. This approach is appealing in diverse applications such as developing soft sensors and statistical process monitoring. It is assessed through both mathematical example and monitoring of a simulated batch polymerization reactor.