Bayesian Principal Component Analysis

Principal component analysis (PCA) is a dimensionality reduction modeling technique that transforms a set of process variables by rotating their axes of representation. Maximum Likelihood PCA (MLPCA) is an extension that accounts for different noise contributions in each variable. Neither PCA nor its extensions utilize external information about the model or data such as the range or distribution of the underlying measurements. Such prior information can be extracted from measured data and can be used to greatly enhance the model accuracy. This paper develops a Bayesian PCA (BPCA) modeling algorithm that improves the accuracy of estimating the parameters and measurements by incorporating prior knowledge about the data and model. The proposed approach integrates modeling and feature extraction by simultaneously solving parameter estimation and data reconciliation optimization problems. Methods for estimating the prior parameters from available data are discussed. Furthermore, BPCA reduces to PCA or MLPCA when a uniform prior is used. Several examples illustrate the benefits of BPCA versus existing methods even when the measurements violate the assumptions about their distribution.