Batchmaturity

Purpose

Batch process model and monitoring, identifying outliers.

Synopsis

model = batchmaturity(x,ncomp_pca,options);

model = batchmaturity(x,y,ncomp_pca,options);

model = batchmaturity(x,y,ncomp_pca,ncomp_reg,options);

pred = batchmaturity(x,model,options);

pred = batchmaturity(x,model);

Description

Analyzes multivariate batch process data to quantify the acceptable variability of the process variables during normal processing conditions as a function of the percent of batch completion. The resulting model can be used on new batch process data to identify measurements which indicate abnormal processing behavior (See the pred.inlimits field for this indicator.)

The progression through a batch is described in terms of the "Batch Maturity" which is often defined in terms of percentage of completion. The resulting model contains confidence limits on PCA scores which are a function of batch maturity and reflect the normal range of variability for the given cross-section of progression through the batch.

Methodology:

Given multivariate X data and a Y variable which represents the corresponding state of batch maturity (BM) build a model by:

1. Build a PLS model on X and Y using specified preprocessing. Use its self-prediction of Y, ypred, as the indicator of BM.
2. Simplify the X data by performing PCA analysis (with specified preprocessing). We now have PC scores and a measure of BM (ypred) for each sample.
3. Sort the samples to be in order of increasing BM. Calculate a running-mean of each PC's ordered scores ("smoothed score means"). Calculate deviations of scores from the smoothed means for each PC.
4. Form a set of equally-spaced BM values over the range (BMstart, BMend). For each BM point, find the n samples which have BM closest to that value.
5. For each BM point, calculate low and high score limit values corresponding to the cl/2 and 1-cl/2 percentiles of the n sample score deviations just selected (repeat for each PC). Add the smoothed scores to these limits to get the actual limits for each PC at each BM point. These BM points and corresponding low/high score limits constitute a lookup table for score limits for each PC in terms of BM value.
6. The score limits lookup table contains upper and lower score limits for each PC, for every equally-spaced BM point over the BM range.
7. The batch maturity model contains the PLS and PCA sub-models and the score limits lookup table. It is applied to a new batch processing dataset, X1, by applying the PLS sub-model to get BM (ypred), then applying the PCA sub-model to get scores. The upper and lower score limits (for each PC) for each sample are obtained by using the sample's BM value and querying the score limits lookup table. A sample is considered to be an inlier if its score values are within the score limits for each PC.

Fig. 1 shows an example of the Batch Maturity Scores Plot (obtained from the BATCHMATURITY Analysis window's Scores Plot). This shows the second PC scores upper and lower limit as a function of BM as calculated form the "Dupont_BSPC" demo dataset using cl = 0.9 and step 2 only from batches 1 to 36. These batches had normal processing conditions so the shaded zone enclosed by the limit lines indicates the range where a measured sample's PC=2 scores should occur if processing is evolving "normally". Similar plots result for the other modeled PCs. The data points shown are the PCA model scores, which are accessible from the batchmaturity model or pred's t field.

• Fig. 1. Batchmaturity Scores Plot.
• 

Inputs

• x = X-block (2-way array class "double" or "dataset").
• y = Y-block (vector class "double" or "dataset").
• ncomp_pca = Number of components to to be calculated in PCA model (positive integer scalar).
• ncomp_reg = Number of latent variables for regression method.

Outputs

• model = standard model structure containing the PCA and Regression model (See MODELSTRUCT).
• pred = prediction structure contains the scores from PCA model for the input test data as pred.t.

Model and pred contain the following fields which relate to score limits and whether samples are within normal ranges or not:

limits : struct with fields:

cl: value used for cl option

bm: (1 x bmlookuppts) bm values for score limits

low: (nPC x bmlookuppts) lower score limit of inliers

high: (nPC x bmlookuppts) upper score limit of inliers

median: (nPC x bmlookuppts) median trace of scores

inlimits : (nsample x nPC) logical indicating if samples are inliers.

t : (nsample x nPC) scores from the PCA submodel.

t_reduced : (nsample x nPC) scores scaled by limits, with limits -> +/- 1 at upper/lower limit, -> 0 at the median score.

submodelreg : regression model built to predict bm. Only PLS currently.

submodelpca : PCA model used to calculate X-block scores.

Options

options = a structure array with the following fields:

• regression_method : [ {'pls'} ] A string indicating type of regression method to use. Currently, only 'pls' is supported.
• preprocessing : { [] } preprocessing structure goes to both PCA and PLS. PLS Y-block preprocessing will always be autoscale.
• zerooffsety : [ 0 | {1}] transform y resetting to zero per batch
• stretchy : [ 0 | {1}] transform y to have range=100 per batch
• cl : [ 0.90 ] Confidence limit (2-sided) for moving limits (defined as 1 - Expected fraction of outliers.)
• nearestpts : [{25}] number nearby scores used in getting limits
• smoothing : [{0.05}] smoothing of limit lines. Width of window used in Savgol smoothing as a fraction of BM range.
• bmlookuppts : [{1001}] number of equally-spaced points in BM lookup table. Default gives lookup values spaced every 0.1% over the BM range.
• plots : [ 'none' | 'detailed' | {'final'} ] governs production of plots when model is built. 'final' shows standard scores and loadings plots. 'detailed' gives individual scores plots with limits for all PCs.
• waitbar : [ 'off' | {'auto'} ] governs display of waitbar when calculating confidence limits ('auto' shows waitbar only when the calculation will take longer than 15 seconds)

See Also

batchfold, batchdigester