Kuhn & Johnson – Applied Predictive Modeling: Problems 6.2 & 6.3
Ariba Mandavia
2026-04-12
Problem 6.2 – Permeability Data
(a) Load the Data
library(AppliedPredictiveModeling)
library(caret)
library(pls)
library(elasticnet) # for ridge / enet
library(MASS) # for ridge regression
library(tidyverse)
data(permeability)
# Dimensions
cat("fingerprints dimensions:", dim(fingerprints), "\n")## fingerprints dimensions: 165 1107## permeability length: 165The fingerprints matrix has 165
compounds described by 1107 binary predictors.
The response permeability is a continuous numeric vector of
length 165.
(b) Remove Near-Zero Variance Predictors
Binary fingerprint predictors are often sparse — many substructures
appear in only a handful of molecules. We use nearZeroVar()
from caret to identify and remove these low-information
predictors.
## Predictors before filtering: 1107## Near-zero variance predictors removed: 719fingerprints_filtered <- fingerprints[, -nzv_idx]
cat("Predictors remaining after filtering:", ncol(fingerprints_filtered), "\n")## Predictors remaining after filtering: 388After removing near-zero variance predictors, 388 predictors remain for modeling.
(c) Train/Test Split, Pre-processing, and PLS Tuning
We split 80 % of compounds into training and 20 % into testing, then tune a Partial Least Squares (PLS) model via 10-fold cross-validation.
set.seed(614)
train_idx <- createDataPartition(permeability, p = 0.80, list = FALSE)
X_train <- fingerprints_filtered[train_idx, ]
X_test <- fingerprints_filtered[-train_idx, ]
y_train <- permeability[train_idx]
y_test <- permeability[-train_idx]
# Pre-process: center and scale inside resampling via caret
ctrl <- trainControl(
method = "cv",
number = 10
)
set.seed(614)
pls_model <- train(
x = X_train,
y = y_train,
method = "pls",
tuneLength = 20,
trControl = ctrl,
preProcess = c("center", "scale")
)
pls_model## Partial Least Squares
##
## 133 samples
## 388 predictors
##
## Pre-processing: centered (388), scaled (388)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 120, 121, 120, 120, 119, 119, ...
## Resampling results across tuning parameters:
##
## ncomp RMSE Rsquared MAE
## 1 12.53358 0.3553953 9.567597
## 2 11.61019 0.4890810 8.091447
## 3 11.80130 0.4757242 8.778872
## 4 12.19362 0.4352315 9.219110
## 5 12.28300 0.4387307 9.390196
## 6 12.08771 0.4535351 9.055823
## 7 12.08226 0.4509900 9.120314
## 8 11.95363 0.4632350 9.268706
## 9 11.93529 0.4698718 9.422742
## 10 12.09437 0.4693474 9.568933
## 11 12.29766 0.4546334 9.622594
## 12 12.44555 0.4481673 9.786040
## 13 12.62766 0.4410592 9.850552
## 14 12.90519 0.4196692 10.058005
## 15 13.01441 0.4128496 10.109928
## 16 13.15024 0.4127095 10.329062
## 17 13.29641 0.4080838 10.480771
## 18 13.51509 0.3942294 10.772092
## 19 13.33906 0.4070753 10.558082
## 20 13.45224 0.4002228 10.674754
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was ncomp = 2.
best_lv <- pls_model$bestTune$ncomp
best_r2 <- max(pls_model$results$Rsquared)
cat("Optimal number of latent variables:", best_lv, "\n")## Optimal number of latent variables: 2## Resampled R² at optimal LV: 0.4891The optimal PLS model uses 2 latent variable(s) with a resampled cross-validated R² of approximately 0.489.
(d) Test Set Prediction and R²
pls_pred <- predict(pls_model, newdata = X_test)
# R² on test set
ss_res <- sum((y_test - pls_pred)^2)
ss_tot <- sum((y_test - mean(y_test))^2)
test_r2 <- 1 - ss_res / ss_tot
test_rmse <- RMSE(pls_pred, y_test)
cat("Test set R²: ", round(test_r2, 4), "\n")## Test set R²: 0.3539## Test set RMSE: 13.5836tibble(Observed = y_test, Predicted = pls_pred) %>%
ggplot(aes(Observed, Predicted)) +
geom_point(color = "#2C5C8F", alpha = 0.7, size = 2.5) +
geom_abline(slope = 1, intercept = 0, linetype = "dashed", color = "firebrick") +
labs(
title = "PLS – Observed vs. Predicted (Test Set)",
subtitle = paste0("Test R² = ", round(test_r2, 3)),
x = "Observed Permeability",
y = "Predicted Permeability"
) +
theme_minimal(base_size = 13)
The test set R² is 0.354, which is lower than the resampled training estimate — typical for a model evaluated on truly held-out data.
(e) Alternative Models
We try Ridge Regression and Elastic Net as alternative linear models discussed in Chapter 6.
Ridge Regression
set.seed(614)
ridge_model <- train(
x = X_train,
y = y_train,
method = "ridge",
tuneLength = 15,
trControl = ctrl,
preProcess = c("center", "scale")
)
ridge_pred <- predict(ridge_model, newdata = X_test)
ridge_r2 <- 1 - sum((y_test - ridge_pred)^2) / sum((y_test - mean(y_test))^2)
ridge_rmse <- RMSE(ridge_pred, y_test)
cat("Ridge – Test R²: ", round(ridge_r2, 4), "\n")## Ridge – Test R²: 0.4963## Ridge – Test RMSE: 11.9942Elastic Net
set.seed(614)
enet_model <- train(
x = X_train,
y = y_train,
method = "enet",
tuneLength = 15,
trControl = ctrl,
preProcess = c("center", "scale")
)
enet_pred <- predict(enet_model, newdata = X_test)
enet_r2 <- 1 - sum((y_test - enet_pred)^2) / sum((y_test - mean(y_test))^2)
enet_rmse <- RMSE(enet_pred, y_test)
cat("Elastic Net – Test R²: ", round(enet_r2, 4), "\n")## Elastic Net – Test R²: 0.4588## Elastic Net – Test RMSE: 12.4324(f) Recommendation
The Ridge Regression achieved the highest test set R² of 0.496.
Would I recommend replacing the permeability lab experiment?
Not yet. While the model explains a meaningful portion of variance in permeability, an R² below ~0.75–0.80 still leaves substantial unexplained variance — a concern for a pharmaceutical application where prediction errors could mean pursuing non-viable drug candidates or discarding promising ones. A few considerations:
- Cost-benefit: If the lab assay is expensive and slow, even an imperfect model could be used to screen out clearly poor candidates before lab testing, reducing overall cost without replacing the experiment entirely.
- Data size: With only 165 compounds, the model may not generalize well to chemically diverse new molecules.
- Recommendation: Use the model as a pre-screening filter, not as a direct replacement for the lab measurement. As more data accumulates, model accuracy should improve.
Problem 6.3 – Chemical Manufacturing Process Data
(a) Load the Data
data(ChemicalManufacturingProcess)
# The dataset is a single data frame: first column is yield, rest are predictors
chem_df <- ChemicalManufacturingProcess
cat("Dimensions:", dim(chem_df), "\n")## Dimensions: 176 58## Response: yield (column 1)## Predictors: columns 2 – 58The dataset contains 176 manufacturing runs with 1 response (Yield) and 57 predictors (12 biological + 45 process).
(b) Impute Missing Values
# Count missing values per predictor
missing_summary <- chem_df %>%
summarise(across(everything(), ~ sum(is.na(.)))) %>%
pivot_longer(everything(), names_to = "Variable", values_to = "Missing") %>%
filter(Missing > 0) %>%
arrange(desc(Missing))
cat("Predictors with missing values:", nrow(missing_summary), "\n")## Predictors with missing values: 28We use k-Nearest Neighbor (kNN) imputation via
preProcess() from caret, which is well-suited for this type
of continuous numeric data.
# Separate response and predictors
yield <- chem_df$Yield
predictors <- chem_df %>% dplyr::select(-Yield)
# kNN imputation (k=5 neighbors)
impute_obj <- preProcess(predictors, method = "knnImpute", k = 5)
pred_imputed <- predict(impute_obj, predictors)
cat("Missing values after imputation:", sum(is.na(pred_imputed)), "\n")## Missing values after imputation: 0After kNN imputation, there are 0 missing values in the predictor matrix.
(c) Train/Test Split, Pre-processing, and Model Tuning
We use an Elastic Net model, which simultaneously performs shrinkage (like ridge) and variable selection (like lasso) — well-suited to this problem with correlated predictors of mixed importance.
set.seed(63)
train_idx2 <- createDataPartition(yield, p = 0.80, list = FALSE)
X_train2 <- pred_imputed[ train_idx2, ]
X_test2 <- pred_imputed[-train_idx2, ]
y_train2 <- yield[ train_idx2]
y_test2 <- yield[-train_idx2]
ctrl2 <- trainControl(method = "cv", number = 10)
set.seed(63)
enet_chem <- train(
x = X_train2,
y = y_train2,
method = "enet",
tuneLength = 20,
trControl = ctrl2,
preProcess = c("center", "scale")
)
enet_chem## Elasticnet
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 131, 130, 130, 130, 131, 129, ...
## Resampling results across tuning parameters:
##
## lambda fraction RMSE Rsquared MAE
## 0.0000000000 0.05 1.186108 0.6147880 0.9464485
## 0.0000000000 0.10 1.131212 0.6291962 0.9149650
## 0.0000000000 0.15 1.396410 0.5482895 1.0307046
## 0.0000000000 0.20 1.785212 0.5017366 1.1474710
## 0.0000000000 0.25 2.169950 0.4570006 1.2640063
## 0.0000000000 0.30 2.630179 0.4283351 1.3969039
## 0.0000000000 0.35 2.962627 0.4131297 1.5022143
## 0.0000000000 0.40 3.501100 0.4027609 1.6605875
## 0.0000000000 0.45 4.283376 0.3975417 1.8692519
## 0.0000000000 0.50 4.629154 0.3954879 1.9658172
## 0.0000000000 0.55 4.813848 0.3944340 2.0173018
## 0.0000000000 0.60 5.029671 0.3939696 2.0780288
## 0.0000000000 0.65 5.265348 0.3965452 2.1435145
## 0.0000000000 0.70 5.535822 0.4044852 2.2100021
## 0.0000000000 0.75 5.878113 0.3916772 2.3121040
## 0.0000000000 0.80 6.200252 0.3700210 2.4217128
## 0.0000000000 0.85 6.683295 0.3622836 2.5628459
## 0.0000000000 0.90 7.192252 0.3578218 2.7081781
## 0.0000000000 0.95 7.847778 0.3552880 2.8870927
## 0.0000000000 1.00 8.336394 0.3532957 3.0174684
## 0.0001000000 0.05 1.217514 0.6067799 0.9766506
## 0.0001000000 0.10 1.119281 0.6372945 0.9075049
## 0.0001000000 0.15 1.184786 0.6084910 0.9386593
## 0.0001000000 0.20 1.446029 0.5466735 1.0426481
## 0.0001000000 0.25 1.787256 0.4971469 1.1421239
## 0.0001000000 0.30 2.256608 0.4539964 1.2800889
## 0.0001000000 0.35 2.641824 0.4389425 1.3914526
## 0.0001000000 0.40 2.924071 0.4276684 1.4793189
## 0.0001000000 0.45 3.933359 0.4138696 1.7499053
## 0.0001000000 0.50 4.481777 0.4050432 1.9031085
## 0.0001000000 0.55 4.836093 0.3997405 2.0076281
## 0.0001000000 0.60 5.117117 0.3976395 2.0891472
## 0.0001000000 0.65 5.338259 0.3963177 2.1514644
## 0.0001000000 0.70 5.566175 0.3952836 2.2149197
## 0.0001000000 0.75 5.794625 0.3946125 2.2784660
## 0.0001000000 0.80 5.987644 0.3949633 2.3319189
## 0.0001000000 0.85 6.158954 0.3974844 2.3769119
## 0.0001000000 0.90 6.419971 0.4032102 2.4376950
## 0.0001000000 0.95 6.758463 0.3976828 2.5250837
## 0.0001000000 1.00 7.275880 0.3766970 2.6801222
## 0.0001467799 0.05 1.225644 0.6064609 0.9834487
## 0.0001467799 0.10 1.119681 0.6380129 0.9049313
## 0.0001467799 0.15 1.179142 0.6108015 0.9364938
## 0.0001467799 0.20 1.372086 0.5618095 1.0179678
## 0.0001467799 0.25 1.685743 0.5113035 1.1116720
## 0.0001467799 0.30 2.163798 0.4625382 1.2527618
## 0.0001467799 0.35 2.556453 0.4423037 1.3654092
## 0.0001467799 0.40 2.834946 0.4331458 1.4503989
## 0.0001467799 0.45 3.798447 0.4197879 1.7093548
## 0.0001467799 0.50 4.336548 0.4108204 1.8590943
## 0.0001467799 0.55 4.782835 0.4036562 1.9868102
## 0.0001467799 0.60 5.052960 0.3997018 2.0667124
## 0.0001467799 0.65 5.287089 0.3981186 2.1336258
## 0.0001467799 0.70 5.514842 0.3968170 2.1974570
## 0.0001467799 0.75 5.750422 0.3956389 2.2631539
## 0.0001467799 0.80 5.979914 0.3950460 2.3267366
## 0.0001467799 0.85 6.155770 0.3952349 2.3748989
## 0.0001467799 0.90 6.370050 0.3977342 2.4299118
## 0.0001467799 0.95 6.642514 0.4032959 2.4932970
## 0.0001467799 1.00 7.031994 0.3986727 2.5924600
## 0.0002154435 0.05 1.233984 0.6075134 0.9937932
## 0.0002154435 0.10 1.120361 0.6392658 0.9019755
## 0.0002154435 0.15 1.174629 0.6121090 0.9343811
## 0.0002154435 0.20 1.247354 0.5903820 0.9753086
## 0.0002154435 0.25 1.596458 0.5227810 1.0850731
## 0.0002154435 0.30 2.073855 0.4713886 1.2248748
## 0.0002154435 0.35 2.454173 0.4479132 1.3352410
## 0.0002154435 0.40 2.725787 0.4377531 1.4174638
## 0.0002154435 0.45 3.550275 0.4257805 1.6394921
## 0.0002154435 0.50 4.140398 0.4169760 1.8022658
## 0.0002154435 0.55 4.629866 0.4091566 1.9390493
## 0.0002154435 0.60 4.965547 0.4034881 2.0370184
## 0.0002154435 0.65 5.215321 0.3999876 2.1103753
## 0.0002154435 0.70 5.428620 0.3986862 2.1702093
## 0.0002154435 0.75 5.661630 0.3973712 2.2349441
## 0.0002154435 0.80 5.899915 0.3961691 2.3014857
## 0.0002154435 0.85 6.116117 0.3952579 2.3608315
## 0.0002154435 0.90 6.343428 0.3953208 2.4215598
## 0.0002154435 0.95 6.573370 0.3973439 2.4805817
## 0.0002154435 1.00 6.821532 0.4020555 2.5399918
## 0.0003162278 0.05 1.245848 0.6079478 1.0062526
## 0.0003162278 0.10 1.121299 0.6401457 0.8987805
## 0.0003162278 0.15 1.171374 0.6122070 0.9327262
## 0.0003162278 0.20 1.163956 0.6157750 0.9327256
## 0.0003162278 0.25 1.520425 0.5341429 1.0611962
## 0.0003162278 0.30 1.958086 0.4818745 1.1884075
## 0.0003162278 0.35 2.340780 0.4570534 1.3021228
## 0.0003162278 0.40 2.606751 0.4424332 1.3821450
## 0.0003162278 0.45 3.210910 0.4316914 1.5472562
## 0.0003162278 0.50 3.952195 0.4223649 1.7480802
## 0.0003162278 0.55 4.412785 0.4152619 1.8761488
## 0.0003162278 0.60 4.832496 0.4088814 1.9950247
## 0.0003162278 0.65 5.110886 0.4039388 2.0762884
## 0.0003162278 0.70 5.316443 0.4007911 2.1362045
## 0.0003162278 0.75 5.539625 0.3993951 2.1984922
## 0.0003162278 0.80 5.773400 0.3979851 2.2634352
## 0.0003162278 0.85 5.993556 0.3965382 2.3242132
## 0.0003162278 0.90 6.214806 0.3953390 2.3842126
## 0.0003162278 0.95 6.435653 0.3951438 2.4434252
## 0.0003162278 1.00 6.627148 0.3960871 2.4938713
## 0.0004641589 0.05 1.261087 0.6085782 1.0189820
## 0.0004641589 0.10 1.123669 0.6390425 0.9000095
## 0.0004641589 0.15 1.164993 0.6133115 0.9298008
## 0.0004641589 0.20 1.189075 0.6096881 0.9377882
## 0.0004641589 0.25 1.427461 0.5497882 1.0335619
## 0.0004641589 0.30 1.843010 0.4927518 1.1528130
## 0.0004641589 0.35 2.227449 0.4661581 1.2686915
## 0.0004641589 0.40 2.485938 0.4499742 1.3462404
## 0.0004641589 0.45 2.829363 0.4376523 1.4468181
## 0.0004641589 0.50 3.742314 0.4273524 1.6878918
## 0.0004641589 0.55 4.174946 0.4207975 1.8081983
## 0.0004641589 0.60 4.594340 0.4141781 1.9261794
## 0.0004641589 0.65 4.975399 0.4090406 2.0348872
## 0.0004641589 0.70 5.181959 0.4046936 2.0950356
## 0.0004641589 0.75 5.376541 0.4015630 2.1509527
## 0.0004641589 0.80 5.600711 0.4000369 2.2133656
## 0.0004641589 0.85 5.816207 0.3984793 2.2731981
## 0.0004641589 0.90 6.034677 0.3966788 2.3331875
## 0.0004641589 0.95 6.238300 0.3954088 2.3886154
## 0.0004641589 1.00 6.406618 0.3943766 2.4349194
## 0.0006812921 0.05 1.281693 0.6087269 1.0353114
## 0.0006812921 0.10 1.127002 0.6374669 0.8996686
## 0.0006812921 0.15 1.155688 0.6166920 0.9261218
## 0.0006812921 0.20 1.217676 0.6003011 0.9474664
## 0.0006812921 0.25 1.315994 0.5734656 0.9948136
## 0.0006812921 0.30 1.688673 0.5086936 1.1077105
## 0.0006812921 0.35 2.053796 0.4773739 1.2185612
## 0.0006812921 0.40 2.370642 0.4580996 1.3121358
## 0.0006812921 0.45 2.616587 0.4452304 1.3865277
## 0.0006812921 0.50 3.436090 0.4323490 1.6052334
## 0.0006812921 0.55 3.926544 0.4255385 1.7384778
## 0.0006812921 0.60 4.298585 0.4197740 1.8424947
## 0.0006812921 0.65 4.697001 0.4136272 1.9551790
## 0.0006812921 0.70 5.009771 0.4093401 2.0448762
## 0.0006812921 0.75 5.182474 0.4056273 2.0945144
## 0.0006812921 0.80 5.380687 0.4022837 2.1512275
## 0.0006812921 0.85 5.581299 0.4005099 2.2070248
## 0.0006812921 0.90 5.794796 0.3984975 2.2660614
## 0.0006812921 0.95 5.984991 0.3966966 2.3185091
## 0.0006812921 1.00 6.143082 0.3950393 2.3627282
## 0.0010000000 0.05 1.307274 0.6081179 1.0540774
## 0.0010000000 0.10 1.134096 0.6349536 0.9034576
## 0.0010000000 0.15 1.147587 0.6205941 0.9240831
## 0.0010000000 0.20 1.201005 0.6031614 0.9424712
## 0.0010000000 0.25 1.233183 0.5885640 0.9613114
## 0.0010000000 0.30 1.503934 0.5360578 1.0549407
## 0.0010000000 0.35 1.908809 0.4891862 1.1733133
## 0.0010000000 0.40 2.234790 0.4676660 1.2713060
## 0.0010000000 0.45 2.460303 0.4536412 1.3400658
## 0.0010000000 0.50 2.885898 0.4405867 1.4599312
## 0.0010000000 0.55 3.628130 0.4302802 1.6572277
## 0.0010000000 0.60 4.008087 0.4246165 1.7607962
## 0.0010000000 0.65 4.348573 0.4193416 1.8570767
## 0.0010000000 0.70 4.718019 0.4134599 1.9620255
## 0.0010000000 0.75 4.958149 0.4096779 2.0314933
## 0.0010000000 0.80 5.120290 0.4063857 2.0775435
## 0.0010000000 0.85 5.301122 0.4032548 2.1296013
## 0.0010000000 0.90 5.493665 0.4008964 2.1831544
## 0.0010000000 0.95 5.676431 0.3986480 2.2336972
## 0.0010000000 1.00 5.834378 0.3967462 2.2775728
## 0.0014677993 0.05 1.336930 0.6059884 1.0771207
## 0.0014677993 0.10 1.147535 0.6300074 0.9126731
## 0.0014677993 0.15 1.138124 0.6263365 0.9180509
## 0.0014677993 0.20 1.188752 0.6050468 0.9382396
## 0.0014677993 0.25 1.157149 0.6203220 0.9286790
## 0.0014677993 0.30 1.367047 0.5635909 1.0111239
## 0.0014677993 0.35 1.760235 0.5030525 1.1275270
## 0.0014677993 0.40 2.060531 0.4785165 1.2200719
## 0.0014677993 0.45 2.323230 0.4620950 1.2989880
## 0.0014677993 0.50 2.544062 0.4504286 1.3654961
## 0.0014677993 0.55 3.182997 0.4376965 1.5377266
## 0.0014677993 0.60 3.686233 0.4298476 1.6733191
## 0.0014677993 0.65 4.013812 0.4245110 1.7630776
## 0.0014677993 0.70 4.327575 0.4193539 1.8521821
## 0.0014677993 0.75 4.642173 0.4140941 1.9420619
## 0.0014677993 0.80 4.855948 0.4103833 2.0038625
## 0.0014677993 0.85 4.992817 0.4069993 2.0431492
## 0.0014677993 0.90 5.168439 0.4040451 2.0934017
## 0.0014677993 0.95 5.338836 0.4013673 2.1411903
## 0.0014677993 1.00 5.487433 0.3990975 2.1830212
## 0.0021544347 0.05 1.370232 0.6016072 1.1033356
## 0.0021544347 0.10 1.165647 0.6237541 0.9300735
## 0.0021544347 0.15 1.128098 0.6331985 0.9099633
## 0.0021544347 0.20 1.176118 0.6082328 0.9332465
## 0.0021544347 0.25 1.202817 0.6032622 0.9434320
## 0.0021544347 0.30 1.296384 0.5726402 0.9841014
## 0.0021544347 0.35 1.534713 0.5302471 1.0633884
## 0.0021544347 0.40 1.894491 0.4911468 1.1687439
## 0.0021544347 0.45 2.171975 0.4718793 1.2523988
## 0.0021544347 0.50 2.379829 0.4584789 1.3168504
## 0.0021544347 0.55 2.627098 0.4479158 1.3892756
## 0.0021544347 0.60 3.306595 0.4365146 1.5701412
## 0.0021544347 0.65 3.662866 0.4301082 1.6679081
## 0.0021544347 0.70 3.959955 0.4249446 1.7493977
## 0.0021544347 0.75 4.234595 0.4200120 1.8273679
## 0.0021544347 0.80 4.503245 0.4150764 1.9045378
## 0.0021544347 0.85 4.696623 0.4110910 1.9604966
## 0.0021544347 0.90 4.837812 0.4075508 2.0006868
## 0.0021544347 0.95 4.981946 0.4047469 2.0422107
## 0.0021544347 1.00 5.116349 0.4021495 2.0809789
## 0.0031622777 0.05 1.404961 0.5952974 1.1321970
## 0.0031622777 0.10 1.187779 0.6160621 0.9519746
## 0.0031622777 0.15 1.124638 0.6357857 0.9044624
## 0.0031622777 0.20 1.161004 0.6139027 0.9286379
## 0.0031622777 0.25 1.202845 0.6014257 0.9418581
## 0.0031622777 0.30 1.195762 0.6016910 0.9473830
## 0.0031622777 0.35 1.370611 0.5608942 1.0107014
## 0.0031622777 0.40 1.715585 0.5074123 1.1145893
## 0.0031622777 0.45 1.982261 0.4837592 1.1959785
## 0.0031622777 0.50 2.207395 0.4683017 1.2651389
## 0.0031622777 0.55 2.421678 0.4561834 1.3294288
## 0.0031622777 0.60 2.745568 0.4463501 1.4201122
## 0.0031622777 0.65 3.303331 0.4370008 1.5694361
## 0.0031622777 0.70 3.588889 0.4312461 1.6489964
## 0.0031622777 0.75 3.853152 0.4260513 1.7218294
## 0.0031622777 0.80 4.098154 0.4213014 1.7906936
## 0.0031622777 0.85 4.307681 0.4166796 1.8513403
## 0.0031622777 0.90 4.501612 0.4119583 1.9079742
## 0.0031622777 0.95 4.619864 0.4089147 1.9411705
## 0.0031622777 1.00 4.739317 0.4061408 1.9756685
## 0.0046415888 0.05 1.440468 0.5867300 1.1609584
## 0.0046415888 0.10 1.212117 0.6083304 0.9798743
## 0.0046415888 0.15 1.130121 0.6344791 0.9029952
## 0.0046415888 0.20 1.145037 0.6224401 0.9210046
## 0.0046415888 0.25 1.187633 0.6041232 0.9361156
## 0.0046415888 0.30 1.193602 0.6052406 0.9406589
## 0.0046415888 0.35 1.318973 0.5650618 0.9942009
## 0.0046415888 0.40 1.459425 0.5429218 1.0424589
## 0.0046415888 0.45 1.817526 0.4974982 1.1452100
## 0.0046415888 0.50 2.041558 0.4792819 1.2137051
## 0.0046415888 0.55 2.247275 0.4660972 1.2771440
## 0.0046415888 0.60 2.439037 0.4557937 1.3345667
## 0.0046415888 0.65 2.779148 0.4463093 1.4283488
## 0.0046415888 0.70 3.222853 0.4386778 1.5481363
## 0.0046415888 0.75 3.481173 0.4332276 1.6202025
## 0.0046415888 0.80 3.714615 0.4280398 1.6852075
## 0.0046415888 0.85 3.929905 0.4231614 1.7458404
## 0.0046415888 0.90 4.104130 0.4184473 1.7966817
## 0.0046415888 0.95 4.254873 0.4143737 1.8404374
## 0.0046415888 1.00 4.373898 0.4113009 1.8738763
## 0.0068129207 0.05 1.474064 0.5789141 1.1878568
## 0.0068129207 0.10 1.236672 0.6067533 1.0024735
## 0.0068129207 0.15 1.143126 0.6301670 0.9130552
## 0.0068129207 0.20 1.131566 0.6314688 0.9112303
## 0.0068129207 0.25 1.168478 0.6107905 0.9306833
## 0.0068129207 0.30 1.205254 0.6001458 0.9416007
## 0.0068129207 0.35 1.221083 0.5945777 0.9535330
## 0.0068129207 0.40 1.382060 0.5532733 1.0156574
## 0.0068129207 0.45 1.578796 0.5230626 1.0795386
## 0.0068129207 0.50 1.884260 0.4916370 1.1657571
## 0.0068129207 0.55 2.079741 0.4762018 1.2255957
## 0.0068129207 0.60 2.273992 0.4653025 1.2847309
## 0.0068129207 0.65 2.443131 0.4564024 1.3357450
## 0.0068129207 0.70 2.764270 0.4475863 1.4240967
## 0.0068129207 0.75 3.119008 0.4409924 1.5208579
## 0.0068129207 0.80 3.362813 0.4357531 1.5885725
## 0.0068129207 0.85 3.564214 0.4305254 1.6454117
## 0.0068129207 0.90 3.744293 0.4255372 1.6967347
## 0.0068129207 0.95 3.899093 0.4210867 1.7418821
## 0.0068129207 1.00 4.032804 0.4177116 1.7806410
## 0.0100000000 0.05 1.505265 0.5705768 1.2139121
## 0.0100000000 0.10 1.266933 0.6056718 1.0231165
## 0.0100000000 0.15 1.168416 0.6209151 0.9320595
## 0.0100000000 0.20 1.129635 0.6328435 0.9077632
## 0.0100000000 0.25 1.153427 0.6182822 0.9250164
## 0.0100000000 0.30 1.188288 0.6034599 0.9359787
## 0.0100000000 0.35 1.220116 0.5967331 0.9474529
## 0.0100000000 0.40 1.318990 0.5637728 0.9929971
## 0.0100000000 0.45 1.445662 0.5447754 1.0390620
## 0.0100000000 0.50 1.673143 0.5122192 1.1087586
## 0.0100000000 0.55 1.936603 0.4883477 1.1822598
## 0.0100000000 0.60 2.115010 0.4753447 1.2361329
## 0.0100000000 0.65 2.298547 0.4656101 1.2912642
## 0.0100000000 0.70 2.447155 0.4580558 1.3367381
## 0.0100000000 0.75 2.712134 0.4505034 1.4103972
## 0.0100000000 0.80 2.999601 0.4444034 1.4896002
## 0.0100000000 0.85 3.237364 0.4389740 1.5556988
## 0.0100000000 0.90 3.422917 0.4336991 1.6087835
## 0.0100000000 0.95 3.584445 0.4291336 1.6552388
## 0.0100000000 1.00 3.722405 0.4252350 1.6948360
## 0.0146779927 0.05 1.535451 0.5620304 1.2401129
## 0.0146779927 0.10 1.303199 0.6040350 1.0503075
## 0.0146779927 0.15 1.197241 0.6102881 0.9567412
## 0.0146779927 0.20 1.137649 0.6303964 0.9119417
## 0.0146779927 0.25 1.139151 0.6269886 0.9164829
## 0.0146779927 0.30 1.171109 0.6094635 0.9326383
## 0.0146779927 0.35 1.202685 0.5989755 0.9398210
## 0.0146779927 0.40 1.267788 0.5834510 0.9652585
## 0.0146779927 0.45 1.400033 0.5491907 1.0197708
## 0.0146779927 0.50 1.522594 0.5356085 1.0654042
## 0.0146779927 0.55 1.735470 0.5078293 1.1288549
## 0.0146779927 0.60 1.975024 0.4874380 1.1944996
## 0.0146779927 0.65 2.154286 0.4757904 1.2467162
## 0.0146779927 0.70 2.304436 0.4674026 1.2923398
## 0.0146779927 0.75 2.465939 0.4602466 1.3409753
## 0.0146779927 0.80 2.694767 0.4536329 1.4057422
## 0.0146779927 0.85 2.915852 0.4481500 1.4678702
## 0.0146779927 0.90 3.119378 0.4429750 1.5248746
## 0.0146779927 0.95 3.299409 0.4377867 1.5759133
## 0.0146779927 1.00 3.443916 0.4335340 1.6174509
## 0.0215443469 0.05 1.563293 0.5528020 1.2638000
## 0.0215443469 0.10 1.344401 0.6004597 1.0834252
## 0.0215443469 0.15 1.221068 0.6053702 0.9842889
## 0.0215443469 0.20 1.157699 0.6227582 0.9243210
## 0.0215443469 0.25 1.134577 0.6298205 0.9131010
## 0.0215443469 0.30 1.155775 0.6170244 0.9265459
## 0.0215443469 0.35 1.184604 0.6045599 0.9365370
## 0.0215443469 0.40 1.225067 0.5928658 0.9474625
## 0.0215443469 0.45 1.345903 0.5609572 0.9966271
## 0.0215443469 0.50 1.470178 0.5403426 1.0412527
## 0.0215443469 0.55 1.600650 0.5272727 1.0905880
## 0.0215443469 0.60 1.797710 0.5035319 1.1491733
## 0.0215443469 0.65 2.017396 0.4868430 1.2090220
## 0.0215443469 0.70 2.177907 0.4767624 1.2542771
## 0.0215443469 0.75 2.316461 0.4692377 1.2953161
## 0.0215443469 0.80 2.488779 0.4627515 1.3448784
## 0.0215443469 0.85 2.681514 0.4568569 1.4007050
## 0.0215443469 0.90 2.868321 0.4517280 1.4536692
## 0.0215443469 0.95 3.038324 0.4467716 1.5019374
## 0.0215443469 1.00 3.195395 0.4421923 1.5467725
## 0.0316227766 0.05 1.588739 0.5423444 1.2854412
## 0.0316227766 0.10 1.385369 0.5953357 1.1164756
## 0.0316227766 0.15 1.246723 0.6051276 1.0061874
## 0.0316227766 0.20 1.185831 0.6118613 0.9454854
## 0.0316227766 0.25 1.143049 0.6261014 0.9186401
## 0.0316227766 0.30 1.143618 0.6249266 0.9199002
## 0.0316227766 0.35 1.171148 0.6099075 0.9340062
## 0.0316227766 0.40 1.198654 0.6001559 0.9407270
## 0.0316227766 0.45 1.311391 0.5770032 0.9733055
## 0.0316227766 0.50 1.420382 0.5475765 1.0214277
## 0.0316227766 0.55 1.535499 0.5350259 1.0603739
## 0.0316227766 0.60 1.662120 0.5213415 1.1104086
## 0.0316227766 0.65 1.855503 0.5015567 1.1674636
## 0.0316227766 0.70 2.060835 0.4873443 1.2243337
## 0.0316227766 0.75 2.195574 0.4784772 1.2615212
## 0.0316227766 0.80 2.331588 0.4713736 1.3012672
## 0.0316227766 0.85 2.509187 0.4651801 1.3515697
## 0.0316227766 0.90 2.674131 0.4599048 1.3989055
## 0.0316227766 0.95 2.831068 0.4553268 1.4432331
## 0.0316227766 1.00 2.973072 0.4508809 1.4840442
## 0.0464158883 0.05 1.611632 0.5309059 1.3044164
## 0.0464158883 0.10 1.423575 0.5895608 1.1472426
## 0.0464158883 0.15 1.276940 0.6058016 1.0286919
## 0.0464158883 0.20 1.209197 0.6049652 0.9693286
## 0.0464158883 0.25 1.162890 0.6174244 0.9320109
## 0.0464158883 0.30 1.143445 0.6243240 0.9204647
## 0.0464158883 0.35 1.160245 0.6157401 0.9292250
## 0.0464158883 0.40 1.185184 0.6046207 0.9399566
## 0.0464158883 0.45 1.232267 0.5912267 0.9509805
## 0.0464158883 0.50 1.369922 0.5630096 0.9970738
## 0.0464158883 0.55 1.481891 0.5400363 1.0408619
## 0.0464158883 0.60 1.590975 0.5315824 1.0799847
## 0.0464158883 0.65 1.715838 0.5167363 1.1275289
## 0.0464158883 0.70 1.892013 0.5008766 1.1793458
## 0.0464158883 0.75 2.072236 0.4887266 1.2306127
## 0.0464158883 0.80 2.197335 0.4807201 1.2666206
## 0.0464158883 0.85 2.357585 0.4740049 1.3104636
## 0.0464158883 0.90 2.507179 0.4687802 1.3525811
## 0.0464158883 0.95 2.649057 0.4638283 1.3941042
## 0.0464158883 1.00 2.772117 0.4594633 1.4301562
## 0.0681292069 0.05 1.631128 0.5190381 1.3204229
## 0.0681292069 0.10 1.457310 0.5834218 1.1748313
## 0.0681292069 0.15 1.311313 0.6047727 1.0561665
## 0.0681292069 0.20 1.229183 0.6038432 0.9918742
## 0.0681292069 0.25 1.188228 0.6071181 0.9525000
## 0.0681292069 0.30 1.155764 0.6175540 0.9279759
## 0.0681292069 0.35 1.152348 0.6202132 0.9252219
## 0.0681292069 0.40 1.174352 0.6100173 0.9361877
## 0.0681292069 0.45 1.196760 0.6011317 0.9444277
## 0.0681292069 0.50 1.298769 0.5792360 0.9708692
## 0.0681292069 0.55 1.406366 0.5549177 1.0118624
## 0.0681292069 0.60 1.518197 0.5393399 1.0513934
## 0.0681292069 0.65 1.629960 0.5297564 1.0937695
## 0.0681292069 0.70 1.756748 0.5125897 1.1404414
## 0.0681292069 0.75 1.905258 0.5002103 1.1847206
## 0.0681292069 0.80 2.060563 0.4897539 1.2308170
## 0.0681292069 0.85 2.189924 0.4826309 1.2684846
## 0.0681292069 0.90 2.338299 0.4764315 1.3098212
## 0.0681292069 0.95 2.471186 0.4719395 1.3467334
## 0.0681292069 1.00 2.587782 0.4679744 1.3803261
## 0.1000000000 0.05 1.645964 0.5109364 1.3333671
## 0.1000000000 0.10 1.484293 0.5785023 1.1976417
## 0.1000000000 0.15 1.343738 0.6023878 1.0832944
## 0.1000000000 0.20 1.252230 0.6036400 1.0099199
## 0.1000000000 0.25 1.205839 0.6018246 0.9718727
## 0.1000000000 0.30 1.176162 0.6076903 0.9422578
## 0.1000000000 0.35 1.160372 0.6140498 0.9298181
## 0.1000000000 0.40 1.165646 0.6155270 0.9293363
## 0.1000000000 0.45 1.189108 0.6054055 0.9419029
## 0.1000000000 0.50 1.259432 0.5856497 0.9624213
## 0.1000000000 0.55 1.345413 0.5696491 0.9897656
## 0.1000000000 0.60 1.449320 0.5493717 1.0279881
## 0.1000000000 0.65 1.547000 0.5397846 1.0600239
## 0.1000000000 0.70 1.667307 0.5294023 1.1078486
## 0.1000000000 0.75 1.792070 0.5124870 1.1517126
## 0.1000000000 0.80 1.931381 0.5005557 1.1931494
## 0.1000000000 0.85 2.044137 0.4923821 1.2290822
## 0.1000000000 0.90 2.174279 0.4856289 1.2671823
## 0.1000000000 0.95 2.306005 0.4802241 1.3047760
## 0.1000000000 1.00 2.417157 0.4765075 1.3362797
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were fraction = 0.1 and lambda = 1e-04.
best_params <- enet_chem$bestTune
best_cv_rmse <- min(enet_chem$results$RMSE)
best_cv_r2 <- enet_chem$results$Rsquared[which.min(enet_chem$results$RMSE)]
cat("Best lambda (fraction):", best_params$fraction, "\n")## Best lambda (fraction): 0.1## Best lambda (lambda): 1e-04## CV RMSE: 1.1193## CV R²: 0.6373The optimal Elastic Net model achieves a resampled cross-validated RMSE of approximately 1.119 and R² of 0.637.
(d) Test Set Performance
chem_pred <- predict(enet_chem, newdata = X_test2)
test_rmse2 <- RMSE(chem_pred, y_test2)
test_r2_2 <- 1 - sum((y_test2 - chem_pred)^2) / sum((y_test2 - mean(y_test2))^2)
cat("Test RMSE:", round(test_rmse2, 4), "\n")## Test RMSE: 1.149## Test R²: 0.5739tibble(Observed = y_test2, Predicted = chem_pred) %>%
ggplot(aes(Observed, Predicted)) +
geom_point(color = "#5C8A2C", alpha = 0.75, size = 2.5) +
geom_abline(slope = 1, intercept = 0, linetype = "dashed", color = "firebrick") +
labs(
title = "Elastic Net – Observed vs. Predicted Yield (Test Set)",
subtitle = paste0("Test R² = ", round(test_r2_2, 3),
" | CV R² = ", round(best_cv_r2, 3)),
x = "Observed Yield (%)",
y = "Predicted Yield (%)"
) +
theme_minimal(base_size = 13)
| Metric | CV (Training) | Test Set |
|---|---|---|
| RMSE | 1.119 | 1.149 |
| R² | 0.637 | 0.574 |
The test set performance is somewhat different from the resampled CV estimate, suggesting the model generalizes reasonably well.
(e) Variable Importance
imp <- varImp(enet_chem, scale = TRUE)
plot(imp, top = 20, main = "Top 20 Most Important Predictors (Elastic Net)")
imp_df <- imp$importance %>%
rownames_to_column("Predictor") %>%
arrange(desc(Overall)) %>%
head(20)
imp_df# Biological predictors are labeled BiologicalMaterial in this dataset
bio_top <- imp_df %>% filter(str_detect(Predictor, "BiologicalMaterial"))
proc_top <- imp_df %>% filter(str_detect(Predictor, "ManufacturingProcess"))
cat("Biological predictors in top 20: ", nrow(bio_top), "\n")## Biological predictors in top 20: 8## Manufacturing process predictors in top 20: 12Interpretation: The top 20 predictors include both biological and process variables, but manufacturing process predictors dominate the list. This is meaningful: process variables can be actively controlled, making them the most actionable targets for yield improvement.
(f) Relationships Between Top Predictors and Yield
We visualize the top 6 predictors against yield to understand directionality and form.
top6 <- imp_df$Predictor[1:6]
# Combine training data with yield for plotting
plot_df <- bind_cols(
as.data.frame(X_train2),
Yield = y_train2
) %>%
dplyr::select(all_of(top6), Yield) %>%
pivot_longer(-Yield, names_to = "Predictor", values_to = "Value")
ggplot(plot_df, aes(Value, Yield)) +
geom_point(alpha = 0.45, color = "#2C5C8F", size = 1.8) +
geom_smooth(method = "loess", se = TRUE, color = "firebrick", linewidth = 1) +
facet_wrap(~ Predictor, scales = "free_x", ncol = 2) +
labs(
title = "Top 6 Predictors vs. Yield (Training Set)",
subtitle = "LOESS smooth shown in red",
x = "Predictor Value",
y = "Yield (%)"
) +
theme_minimal(base_size = 12) +
theme(strip.text = element_text(face = "bold"))
How this information can improve yield:
Positive linear relationships — If a process variable (e.g., a temperature or pressure setting) shows a clear positive slope with yield, operators can target higher values of that variable in future runs, subject to feasibility constraints.
Negative relationships — Variables with inverse relationships to yield may represent impurities or inefficiencies; reducing them is a lever for improvement.
Non-linear patterns — Curved LOESS fits (e.g., an inverted-U shape) suggest optimal operating windows. Running the process near the peak of such a curve would maximize yield.
Biological material predictors — Although these cannot be changed in the process itself, they can serve as incoming quality controls. Batches with unfavorable biological predictor values could be flagged before processing, avoiding low-yield runs.
Financial impact — Since a 1% improvement in yield generates ~$100,000 per batch, even modest optimization (e.g., +1–2% yield by targeting favorable process settings) would represent substantial revenue gains.
Session Info
## R version 4.5.2 (2025-10-31)
## Platform: aarch64-apple-darwin20
## Running under: macOS Tahoe 26.3.1
##
## Matrix products: default
## BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.1
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## time zone: America/Chicago
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] lubridate_1.9.5 forcats_1.0.1
## [3] stringr_1.6.0 dplyr_1.2.0
## [5] purrr_1.2.1 readr_2.1.6
## [7] tidyr_1.3.2 tibble_3.3.1
## [9] tidyverse_2.0.0 MASS_7.3-65
## [11] elasticnet_1.3 lars_1.3
## [13] pls_2.9-0 caret_7.0-1
## [15] lattice_0.22-7 ggplot2_4.0.2
## [17] AppliedPredictiveModeling_1.1-7
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.2.1 timeDate_4052.112 farver_2.1.2
## [4] S7_0.2.1 fastmap_1.2.0 RANN_2.6.2
## [7] pROC_1.19.0.1 digest_0.6.39 rpart_4.1.24
## [10] timechange_0.4.0 lifecycle_1.0.5 cluster_2.1.8.1
## [13] survival_3.8-3 magrittr_2.0.4 compiler_4.5.2
## [16] rlang_1.1.7 sass_0.4.10 tools_4.5.2
## [19] plotrix_3.8-14 yaml_2.3.12 data.table_1.18.2.1
## [22] knitr_1.51 labeling_0.4.3 plyr_1.8.9
## [25] RColorBrewer_1.1-3 withr_3.0.2 nnet_7.3-20
## [28] grid_4.5.2 stats4_4.5.2 future_1.69.0
## [31] globals_0.19.0 scales_1.4.0 iterators_1.0.14
## [34] cli_3.6.5 ellipse_0.5.0 rmarkdown_2.30
## [37] generics_0.1.4 otel_0.2.0 rstudioapi_0.18.0
## [40] future.apply_1.20.2 tzdb_0.5.0 reshape2_1.4.5
## [43] cachem_1.1.0 splines_4.5.2 parallel_4.5.2
## [46] vctrs_0.7.1 hardhat_1.4.2 Matrix_1.7-4
## [49] jsonlite_2.0.0 hms_1.1.4 listenv_0.10.0
## [52] foreach_1.5.2 gower_1.0.2 jquerylib_0.1.4
## [55] recipes_1.3.1 glue_1.8.0 parallelly_1.46.1
## [58] codetools_0.2-20 stringi_1.8.7 gtable_0.3.6
## [61] rpart.plot_3.1.4 CORElearn_1.57.3.1 pillar_1.11.1
## [64] htmltools_0.5.9 ipred_0.9-15 lava_1.8.2
## [67] R6_2.6.1 evaluate_1.0.5 bslib_0.10.0
## [70] class_7.3-23 Rcpp_1.1.1 nlme_3.1-168
## [73] prodlim_2025.04.28 mgcv_1.9-3 xfun_0.56
## [76] pkgconfig_2.0.3 ModelMetrics_1.2.2.2