Computational intelligence models to predict porosity of tablets using minimum features

Figures & data

Table 1 Characteristics of MCC tablets and the various compaction conditions

Statistical parameters	Die compaction force (KN)	MCC proportion (%)	Average granule range (μm)	Roll compaction force (kN/cm)	Porosity (%)
(A) Experimental parameters for training data
Minimum	2	25	80	Fixed at 11.02;	0.17
Maximum	14	100	900	not used in the	0.46
Median	6	50	408	training models	0.26
Mean	6.84	54	481.9		0.27
(B) Experimental parameters for external validation data
Minimum	2	Fixed at 100%	80	0	0.14
Maximum	15		350	8	0.47
Median	8		350	4	0.21
Mean	8.14		296	4.8	0.24

Abbreviation: MCC, microcrystalline cellulose.

Table 2 NRMSE for 10cv and external validation tests for all CI methods

Data set	Multivariate linear regression (lm)	Cubist (Cubist)	RF (randomForest)	Monmlp (monmlp)	Symbolic regression (rgp)
Average of 10cv	15	3	3	1	4
External (unknown to models)	–	3	7	6	3

Abbreviations: 10cv, tenfold cross-validation; CI, computational intelligence; NRMSE, normalized root-mean-square error; RF, random forest.

Table 3 MLR model statistical parameters

	Intercept	Die compaction force (kN)	MCC proportion	Granule fraction range
Estimate	3.486e–01	−2.041e–02	1.242e–01	−1.094e–05
Standard error	8.743e–03	8.284e–04	1.289e–02	8.426e–06
t-value	39.874	−24.634	9.633	−1.299
P-value	<2e–16	<2e–16	1.22e–14	0.198

Note: Adjusted R-squared: 0.8899.

Abbreviations: MCC, microcrystalline cellulose; MLR, multiple linear regression.

Figure 1 Predicted vs observed graph for MLR.

Abbreviation: MLR, multiple linear regression.

Figure 2 Predicted vs observed graph for ANN.

Abbreviation: ANN, artificial neural network.

Figure 3 Predicted vs observed graph for rgp (NRMSE: 4%).

Abbreviations: NRMSE, normalized root-mean-square error; rgp, r-genetic programming.

Table 4 RMSE and NRMSE for external validation data set using rgp model

System	RMSE	NRMSE
Cheekplates (4 kN)	0.0314	10
Cheekplates (8 kN)	0.0401	14
Rim rolls (4 kN)	0.0309	9
Rim rolls (8 kN)	0.0387	13
Feed powder	0.0254	7

Abbreviations: NRMSE, normalized root-mean-square error; RMSE, root-mean-square error.

Figure 4 Model performance on external data set.

Figure 5 Surface plot showing the influence of average granule fraction size on porosity based on the monmlp model.

Abbreviation: monmlp, Monotone Multi-Layer Perception Neural Network.

Figure 6 Surface plot showing the influence of MCC and die compaction force on porosity based on rgp model.

Abbreviations: MCC, microcrystalline cellulose; rgp, r-genetic programming.

Table 5 Performance metrics for different input combinations of training and test data sets

Data set and input numbers	MLRa	Decision treesa
Training data with three inputs	8% (0.89)	3.4% (0.97)
Training data with two inputs	8% (0.89)	3.4% (0.96)
Test data with five inputs	11% (0.90)	1% (0.99)
Test data with three inputs	10% (0.90)	1% (0.99)
Test data with two inputs	11.7% (0.78)	5.2% (0.95)

Notes:

a NRMSE and R² are presented. All experiments were done in 10cv mode of data representation. MLR and decision trees were implemented in R using lm command and Cubist package, respectively.

Abbreviations: 10cv, tenfold cross-validation; MLR, multiple linear regression; NRMSE, normalized root-mean-square error.

Figure 7 Correlation plots for (A) training and (B) test data.

Abbreviations: Avg, average; MCC, microcrystalline cellulose.