A data-driven soft-sensing approach using probabilistic latent variable model with functional data framework

Functional principal component analysis (FPCA) and functional partial least squares (FPLS) are two mainstream functional data analysis (FDA) methods, which have been commonly used to extract deep information hidden in the original data space. However, the process data always contain random noise, which affects the performance of FDA models. To overcome this issue, two functional probabilistic latent variable models (FPLVMs), including functional probabilistic principal component analysis (FPPCA) and functional probabilistic partial least squares (FPPLS) are proposed in this work. First, the process data are converted into functional data using the FDA. Subsequently, a log-likelihood function considering the noise factor and functional latent variables is designed. Finally, the regression model parameters are estimated using an expectation–maximisation algorithm. In contrast to FPPCA, FPPLS decomposes the process data and the key variable with constrained latent variables, which is similar to the partial least squares (PLS) and the principal component analysis (PCA). Moreover, the degeneration mechanism from FPLVMs into probabilistic latent variable models and latent variable models is discussed. An adaptive strategy with functional covariance is used to satisfy the online predictive capabilities of the model. Finally, the proposed approach is validated using a numerical case, the Tennessee Eastman process and an industrial o-xylene distillation column for evaluation.