Figure 6.10. SPE and T2 charts for continuous process monitoring based on PCA.
puts that describe product quality are collected less frequently since these measurements are expensive and time consuming. Although it is possible to measure some quality variables on-line by means of sophisticated devices, measurements are generally made off-line in the quality control laboratory. Process data contain important information about both the quality of the product and the performance of the process operation. PLS models can be used in two ways:
Quality monitoring. The correlation between the process variables and the quality variables can be determined through the PLS model. This statistical model provides information for estimating product quality from process data.
Statistical process control. PLS model can also be used to quickly detect process upsets and unexpected behavior. When an assignable cause is detected, necessary actions can be taken to prevent any damage to process performance and/or product quality.
The traditional statistical modeling methods such as Multiple Linear Regression (MLR) fail to handle process data that are correlated and collinear. PLS, as a projection method, offers a suitable solution for modeling such data.
The first step in the development of a PLS model is to determine the variables that will be considered as process variables X and as indicator of product quality Y. This selection is dependent on the measurements available and the objectives of monitoring. The reference set used to develop the multivariate monitoring chart will determine the variations considered to be part of normal operation and ideally includes all variations leading to desired process performance. If the reference set variation is too small, the procedure will cause frequent alarms, and if it is too large the sensitivity of the monitoring scheme to the abnormal operation will be poor. The normal operating data are collected from past successful process history. The reference data set selected should include the range of process variables that yield desired product quality. If the PLS model is developed for monitoring certain process conditions, the reference data set should include data collected under these conditions. Data for various batch runs are then stacked together to form the reference set that represents normal behavior of the process.
Since PLS technique is sensitive to outliers and scaling, outliers should be removed and data should be scaled prior to modeling. After data pre-treatment, another decision to be made is the determination of the number of latent variables (PLS dimensions) to be retained in the model. Cumulative prediction sum of squares (CUMPRESS) vs number of latent variables or prediction sum of squares (PRESS) vs number of latent variables plots are used for this purpose. It is usually enough to consider the first few PLS
Variables 12... k
Variables 12... k
Figure 6.11. Data arrangement in PLS for continuous SPM.
dimensions for monitoring while for prediction more PLS dimensions are needed in order to improve the precision of the predictions.
Once the PLS model is built, squared prediction error (SPE) can be calculated for either the X or the Y block model (Eqs. 6.38 and 6.39)
where x and y are predicted observations in X and Y using the PLS model, respectively, i and j denote observations and variables in X or Y, respectively.
x and y in Eqs. 6.38 and 6.39 are calculated for new observations as follows:
Was this article helpful?