Nonlinear logistic regression mixture experiment modeling for binary data using dimensionally reduced components

Abstract

This article motivates, presents, and illustrates an approach using nonlinear logistic regression (NLR) for modeling binary response data from a mixture experiment when the components can be partitioned into groups used to form dimensionally reduced components (DRCs). A DRC is formed from a linear combination of the components in a group having similar roles and/or effects of the same sign, where the linear combinations over all groups are normalized so that the DRC proportions sum to one. Linear combinations of a particular form provide for quantifying the effects of the remaining components in a group relative to a chosen component. This reason, plus dimensional reduction, are the primary motivations for the proposed DRC mixture experiment modeling approach. NLR is required because models expressed in terms of the DRCs are nonlinear in the parameters that specify the linear combinations. A method for obtaining nonparametric tolerance limits on the probability of a "success" for the binary response variable using a bootstrap approach is also presented.Finally, the article shows how DRCs provide for visualizing data and modeling results that otherwise would be impossible. A real database on the presence or absence of nepheline crystals in simulated nuclear waste glass is used to illustrate the NLR modeling and nonparametric tolerance limit approaches. The methodology is general and can be applied to other applications.

A data‐driven approach for predicting nepheline crystallization in high‐level waste glasses

“…20 For the NLR-DRC model, depending on the threshold value used, either false positive or false negative rates are too high. 39 The maximum Al 2 O 3 concentration allowed prior to predicting nepheline formation by our model is 32 wt.%, whereas the maximum Al 2 O 3 concentration allowed when the decision is based on ANN model is 28.24 wt.%, which was the highest allowed until the present study. 20 Also, the model presented here provides an analysis of model reliability and false prediction character, i.e., if there is a wrong prediction, which kind, false negative or false positive, is the greater probability, based on the sample's position in the feature space.…”

“…The previous models, ANN, 1 SM, 20 NLR-DRC, 39 and this model all have very similar accuracies, around 92%. The model presented here is more balanced in terms of rate of false negatives, 28 out of 212 or 13.2%, and false positives, 35 out of 535 or 6.5%, and both of these rates are within acceptable limits.…”

“…The model presented here is more balanced in terms of rate of false negatives, 28 out of 212 or 13.2%, and false positives, 35 out of 535 or 6.5%, and both of these rates are within acceptable limits. 20,39 On the other hand, the SM model has a false negative rate that is higher than the desirable. 20 For the NLR-DRC model, depending on the threshold value used, either false positive or false negative rates are too high.…”

“…20 In the second, instead of a polynomial curve, 20 a logistic regression model was used to predict nepheline formation and instead of submixtures, dimensionally reduced components are used. 39 Both models were tested on the same dataset of 747 glass dataset. The polynomial had a misclassification rate of 8.3%, and the logistic regression model a misclassification rate of 8.2%.…”

“…In the current work, we use an extended version of the glass dataset from SM studies 20,39 and develop a nonparametric, predictive model for nepheline formation. We discover the existence of an "overlapping" region, i.e., a region of composition space where samples with similar compositions exhibited different crystallization behavior, that limits the accuracy of the model.…”