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Unspecific missing of values in real chemical and biological industries have been found. Regardless of the incompleteness of the measured sample, a monitoring system should be designed to tackle the missing data problem and be applied to on-line systems immediately. A calibration method of a factor analysis (FA) model for incomplete data sets is proposed. And a prediction method based on the calibrated model is suggested in order to estimate missing values in incomplete calibration sets and incomplete test sets. An expectation and maximization (EM) algorithm is used to calibrate the model and expectation of conditional density is used to predict the model result. The proposed method is compared with the well-known iterative singular values decomposition (iSVD) method, i.e. a principal component analysis (PCA) based method; and a simple data set is tested as an illustrative example. The proposed method gives better estimation results for the missing values than the well-known PCA based method. There are several advantages of the proposed method over the PCA based projection methods: (1) data pretreatment is not an essential step since the FA model is scale invariant whereas the PCA model is not, (2) since the proposed method utilizes probability information of all variables directly, to apply it as a statistical process monitoring technique is preferable to others, and (3) the single model can be extended to a mixture of such models by the virtue of the EM algorithm.

Calibration, Prediction and Process Monitoring Model Based on Factor Analysis for Incomplete Process Data