Kuhn and Johnson - Chapter 7 Exercises 7.2 and 7.5
Sachi Kapoor
April 26, 2026
Exercise 7.2
This exercise uses the Friedman simulation data. The response is generated from a nonlinear equation where only the first five predictors are truly informative. The remaining predictors are noise variables. The goal is to tune several nonlinear regression models, compare their performance, and determine whether MARS selects the truly informative predictors.
Simulate the training and test data
set.seed(200)
trainingData <- mlbench.friedman1(200, sd = 1)
trainingData$x <- as.data.frame(trainingData$x)
set.seed(201)
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- as.data.frame(testData$x)
str(trainingData$x)## 'data.frame': 200 obs. of 10 variables:
## $ V1 : num 0.534 0.584 0.59 0.691 0.667 ...
## $ V2 : num 0.648 0.438 0.588 0.226 0.819 ...
## $ V3 : num 0.8508 0.6727 0.4097 0.0334 0.7168 ...
## $ V4 : num 0.1816 0.6692 0.3381 0.0669 0.8032 ...
## $ V5 : num 0.929 0.1638 0.8941 0.6374 0.0831 ...
## $ V6 : num 0.3618 0.4531 0.0268 0.525 0.2234 ...
## $ V7 : num 0.827 0.649 0.179 0.513 0.664 ...
## $ V8 : num 0.421 0.845 0.35 0.797 0.904 ...
## $ V9 : num 0.5911 0.9282 0.0176 0.6899 0.397 ...
## $ V10: num 0.589 0.758 0.444 0.445 0.55 ...summary(trainingData$y)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3.556 10.756 14.556 14.416 17.970 28.382K-nearest neighbors model
ctrl_72 <- trainControl(method = "repeatedcv", number = 10, repeats = 3)
set.seed(921)
knnModel <- train(
x = trainingData$x,
y = trainingData$y,
method = "knn",
preProcess = c("center", "scale"),
tuneLength = 10,
trControl = ctrl_72
)
knnPred <- predict(knnModel, newdata = testData$x)
knnTest <- postResample(pred = knnPred, obs = testData$y)
knnModel## k-Nearest Neighbors
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold, repeated 3 times)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 3.151963 0.6214372 2.600597
## 7 3.087287 0.6516076 2.509247
## 9 3.058080 0.6843952 2.475974
## 11 3.034943 0.7013575 2.477236
## 13 3.020351 0.7279731 2.448917
## 15 3.063449 0.7304283 2.484342
## 17 3.061518 0.7450912 2.487711
## 19 3.093208 0.7421557 2.514214
## 21 3.118999 0.7450008 2.540937
## 23 3.139436 0.7514295 2.569148
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 13.knnTest## RMSE Rsquared MAE
## 3.2034139 0.6716079 2.5707670MARS model
marsGrid <- expand.grid(
degree = 1:2,
nprune = seq(2, 20, by = 2)
)
set.seed(921)
marsModel <- train(
x = trainingData$x,
y = trainingData$y,
method = "earth",
preProcess = c("center", "scale"),
tuneGrid = marsGrid,
trControl = ctrl_72
)
marsPred <- predict(marsModel, newdata = testData$x)
marsTest <- postResample(pred = marsPred, obs = testData$y)
marsModel## Multivariate Adaptive Regression Spline
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold, repeated 3 times)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 4.438050 0.2197911 3.6881529
## 1 4 2.672008 0.7236734 2.1709494
## 1 6 2.275852 0.8052266 1.7833757
## 1 8 1.727202 0.8865255 1.3546175
## 1 10 1.660787 0.8954520 1.3140280
## 1 12 1.641088 0.8982687 1.2816690
## 1 14 1.653936 0.8959369 1.2960813
## 1 16 1.655315 0.8957826 1.2964709
## 1 18 1.655315 0.8957826 1.2964709
## 1 20 1.655315 0.8957826 1.2964709
## 2 2 4.460945 0.2135576 3.7138066
## 2 4 2.694101 0.7205425 2.1805270
## 2 6 2.265618 0.8055884 1.7972836
## 2 8 1.715140 0.8864248 1.3453901
## 2 10 1.453653 0.9167060 1.1554820
## 2 12 1.328831 0.9321965 1.0509783
## 2 14 1.275209 0.9378036 1.0163297
## 2 16 1.268621 0.9383750 0.9983817
## 2 18 1.275822 0.9374554 1.0064832
## 2 20 1.275822 0.9374554 1.0064832
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 16 and degree = 2.marsTest## RMSE Rsquared MAE
## 1.2331551 0.9402676 0.9785410plot(marsModel, main = "MARS Tuning Results")
marsImportance <- varImp(marsModel)$importance %>%
tibble::rownames_to_column("Predictor") %>%
arrange(desc(Overall))
kable(marsImportance, digits = 3, caption = "MARS Variable Importance")| Predictor | Overall |
|---|---|
| V1 | 100.000 |
| V4 | 75.253 |
| V2 | 48.759 |
| V5 | 15.540 |
| V3 | 0.000 |
# Refit the final MARS model directly so the selected terms are easy to inspect.
finalMarsFit <- earth(
x = trainingData$x,
y = trainingData$y,
degree = marsModel$bestTune$degree,
nprune = marsModel$bestTune$nprune
)
summary(finalMarsFit)## Call: earth(x=trainingData$x, y=trainingData$y,
## degree=marsModel$bestTune$degree, nprune=marsModel$bestTune$nprune)
##
## coefficients
## (Intercept) 20.378441
## h(0.621722-V1) -15.512132
## h(V1-0.621722) 9.177132
## h(0.601063-V2) -17.940676
## h(V2-0.601063) 10.064356
## h(V3-0.281766) 11.590022
## h(0.447442-V3) 14.641640
## h(V3-0.447442) -12.924806
## h(V3-0.606015) 13.416764
## h(0.734892-V4) -10.074386
## h(V4-0.734892) 9.687149
## h(0.850094-V5) -5.385762
## h(0.218266-V1) * h(V2-0.601063) -55.372637
## h(V1-0.218266) * h(V2-0.601063) -27.542831
## h(V1-0.621722) * h(V2-0.295997) -26.527403
## h(0.649253-V1) * h(0.601063-V2) 26.129827
##
## Selected 16 of 18 terms, and 5 of 10 predictors (nprune=16)
## Termination condition: Reached nk 21
## Importance: V1, V4, V2, V5, V3, V6-unused, V7-unused, V8-unused, V9-unused, ...
## Number of terms at each degree of interaction: 1 11 4
## GCV 1.61518 RSS 210.6377 GRSq 0.934423 RSq 0.9568093plotmo(finalMarsFit, caption = "MARS partial dependence plots")## plotmo grid: V1 V2 V3 V4 V5 V6 V7
## 0.5139349 0.5106664 0.537307 0.4445841 0.5343299 0.4975981 0.4688035
## V8 V9 V10
## 0.497961 0.5288716 0.5359218
Radial basis function support vector machine
set.seed(921)
svmRModel <- train(
x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
preProcess = c("center", "scale"),
tuneLength = 8,
trControl = ctrl_72
)
svmPred <- predict(svmRModel, newdata = testData$x)
svmTest <- postResample(pred = svmPred, obs = testData$y)
svmRModel## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold, repeated 3 times)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.477496 0.8146722 1.972834
## 0.50 2.208826 0.8309997 1.745928
## 1.00 2.028275 0.8509798 1.598201
## 2.00 1.934791 0.8627124 1.514249
## 4.00 1.865448 0.8707502 1.460540
## 8.00 1.830309 0.8760225 1.445085
## 16.00 1.828418 0.8763048 1.448645
## 32.00 1.828338 0.8763089 1.448505
##
## Tuning parameter 'sigma' was held constant at a value of 0.06254979
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06254979 and C = 32.svmTest## RMSE Rsquared MAE
## 2.0820580 0.8285375 1.5849967plot(svmRModel, main = "Radial SVM Tuning Results")
Exercise 7.2 model comparison
results_72 <- rbind(
KNN = knnTest,
MARS = marsTest,
SVM_Radial = svmTest
) %>%
as.data.frame() %>%
tibble::rownames_to_column("Model") %>%
arrange(RMSE)
kable(results_72, digits = 3, caption = "Exercise 7.2 Test Set Performance")| Model | RMSE | Rsquared | MAE |
|---|---|---|---|
| MARS | 1.233 | 0.940 | 0.979 |
| SVM_Radial | 2.082 | 0.829 | 1.585 |
| KNN | 3.203 | 0.672 | 2.571 |
Exercise 7.2 answer
Based on the test set RMSE, the best model in my run is the model with the lowest RMSE in the table above. MARS and radial SVM both handle nonlinear relationships much better than K-nearest neighbors for this simulation. KNN performs weaker because the response is created from a structured nonlinear formula rather than from simple local neighborhoods in the predictor space.
The MARS variable importance table and final model summary show
whether the informative variables are selected. Since the
data-generating equation uses X1 through X5, a
strong model should mainly emphasize those predictors. In my run, the
most important predictors are concentrated among X1 through
X5, while the noise variables have little or no importance.
This means MARS is successfully detecting the true signal in the
simulated data.
Exercise 7.5
This exercise uses the chemical manufacturing process data. The goal is to use the same general preprocessing, splitting, and imputation approach as the previous linear modeling exercise, then train nonlinear regression models and compare performance.
The questions are:
- Which nonlinear regression model gives the best resampling and test set performance?
- Which predictors are most important in the best nonlinear model? Do the biological or process variables dominate?
- For predictors that are uniquely important to the best nonlinear model, what do the predictor-response plots suggest?
Load and split the chemical manufacturing data
data(ChemicalManufacturingProcess)
predictors <- ChemicalManufacturingProcess %>% select(-Yield)
yield <- ChemicalManufacturingProcess$Yield
set.seed(517)
trainingRows <- createDataPartition(yield, p = 0.70, list = FALSE)
trainPredictors <- predictors[trainingRows, ]
trainYield <- yield[trainingRows]
testPredictors <- predictors[-trainingRows, ]
testYield <- yield[-trainingRows]
dim(trainPredictors)## [1] 124 57dim(testPredictors)## [1] 52 57summary(trainYield)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 36.12 38.79 39.97 40.24 41.46 46.34Preprocess the predictors
The preprocessing steps below use median imputation, centering, scaling, removal of near-zero variance predictors, and removal of highly correlated predictors. I used median imputation instead of KNN imputation because KNN imputation can fail when the data contain NA, NaN, or infinite values.
# Convert to data frames.
trainPredictors <- as.data.frame(trainPredictors)
testPredictors <- as.data.frame(testPredictors)
# Replace Inf/-Inf with NA before preprocessing.
trainPredictors[] <- lapply(trainPredictors, function(x) {
x[is.infinite(x)] <- NA
x
})
testPredictors[] <- lapply(testPredictors, function(x) {
x[is.infinite(x)] <- NA
x
})
# Check missing values before preprocessing.
sum(is.na(trainPredictors))## [1] 88sum(is.na(testPredictors))## [1] 18# Use median imputation instead of knnImpute to avoid RANN::nn2 NA/NaN/Inf errors.
pp <- preProcess(
trainPredictors,
method = c("medianImpute", "center", "scale")
)
ppTrainPredictors <- predict(pp, trainPredictors)
ppTestPredictors <- predict(pp, testPredictors)
# Remove near-zero variance predictors.
nzv <- nearZeroVar(ppTrainPredictors)
if(length(nzv) > 0) {
ppTrainPredictors <- ppTrainPredictors[, -nzv, drop = FALSE]
ppTestPredictors <- ppTestPredictors[, -nzv, drop = FALSE]
}
# Remove highly correlated predictors.
corrMatrix <- cor(ppTrainPredictors, use = "pairwise.complete.obs")
highCorr <- findCorrelation(corrMatrix, cutoff = 0.75)
if(length(highCorr) > 0) {
ppTrainPredictors <- ppTrainPredictors[, -highCorr, drop = FALSE]
ppTestPredictors <- ppTestPredictors[, -highCorr, drop = FALSE]
}
# Final safety checks. All four should return 0.
sum(is.na(ppTrainPredictors))## [1] 0sum(is.na(ppTestPredictors))## [1] 0sum(is.infinite(as.matrix(ppTrainPredictors)))## [1] 0sum(is.infinite(as.matrix(ppTestPredictors)))## [1] 0dim(ppTrainPredictors)## [1] 124 35dim(ppTestPredictors)## [1] 52 35Train nonlinear models
ctrl_75 <- trainControl(method = "boot", number = 10)
set.seed(614)
marsChemGrid <- expand.grid(
degree = 1:2,
nprune = 2:12
)
marsChem <- train(
x = ppTrainPredictors,
y = trainYield,
method = "earth",
tuneGrid = marsChemGrid,
trControl = ctrl_75,
metric = "RMSE"
)
set.seed(614)
svmPolyGrid <- expand.grid(
degree = c(1, 2),
scale = c(0.25, 0.5, 1),
C = c(0.01, 0.05, 0.1, 0.5, 1)
)
svmPolyChem <- train(
x = ppTrainPredictors,
y = trainYield,
method = "svmPoly",
tuneGrid = svmPolyGrid,
trControl = ctrl_75,
metric = "RMSE"
)
set.seed(614)
svmRadialChem <- train(
x = ppTrainPredictors,
y = trainYield,
method = "svmRadial",
tuneLength = 8,
trControl = ctrl_75,
metric = "RMSE"
)
set.seed(614)
knnChem <- train(
x = ppTrainPredictors,
y = trainYield,
method = "knn",
tuneLength = 15,
trControl = ctrl_75,
metric = "RMSE"
)Resampling results
resampling_75 <- resamples(list(
MARS = marsChem,
SVM_Polynomial = svmPolyChem,
SVM_Radial = svmRadialChem,
KNN = knnChem
))
summary(resampling_75)##
## Call:
## summary.resamples(object = resampling_75)
##
## Models: MARS, SVM_Polynomial, SVM_Radial, KNN
## Number of resamples: 10
##
## MAE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## MARS 1.0381315 1.147816 1.226280 1.234798 1.305731 1.465944 0
## SVM_Polynomial 1.0424477 1.186413 1.286848 1.297622 1.449228 1.517574 0
## SVM_Radial 0.8368409 1.057719 1.097129 1.087872 1.141829 1.393940 0
## KNN 1.0070982 1.120923 1.187553 1.227485 1.354083 1.493270 0
##
## RMSE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## MARS 1.255149 1.474677 1.509305 1.539584 1.608809 1.881310 0
## SVM_Polynomial 1.270247 1.508996 1.691908 1.694125 1.840637 2.149853 0
## SVM_Radial 1.063622 1.319829 1.384302 1.377508 1.466748 1.689076 0
## KNN 1.290180 1.412057 1.567382 1.551075 1.724506 1.781058 0
##
## Rsquared
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## MARS 0.18823838 0.2819786 0.3659302 0.3596825 0.4144200 0.5220145
## SVM_Polynomial 0.09783666 0.1741580 0.2349950 0.2494173 0.2886982 0.5197313
## SVM_Radial 0.34920734 0.4021429 0.4575125 0.4562445 0.5052691 0.5543181
## KNN 0.16660638 0.2504763 0.2826843 0.3104218 0.3474161 0.5392228
## NA's
## MARS 0
## SVM_Polynomial 0
## SVM_Radial 0
## KNN 0bwplot(resampling_75, metric = "RMSE", main = "Resampled RMSE by Model")
Test set performance
model_list_75 <- list(
MARS = marsChem,
SVM_Polynomial = svmPolyChem,
SVM_Radial = svmRadialChem,
KNN = knnChem
)
test_results_75 <- lapply(names(model_list_75), function(model_name) {
preds <- predict(model_list_75[[model_name]], newdata = ppTestPredictors)
out <- postResample(pred = preds, obs = testYield)
data.frame(
Model = model_name,
RMSE = unname(out["RMSE"]),
Rsquared = unname(out["Rsquared"]),
MAE = unname(out["MAE"])
)
}) %>%
bind_rows() %>%
arrange(RMSE)
kable(test_results_75, digits = 3, caption = "Exercise 7.5 Test Set Performance")| Model | RMSE | Rsquared | MAE |
|---|---|---|---|
| SVM_Radial | 1.207 | 0.594 | 1.005 |
| MARS | 1.365 | 0.480 | 1.149 |
| SVM_Polynomial | 1.569 | 0.328 | 1.320 |
| KNN | 1.593 | 0.290 | 1.273 |
Best nonlinear model and variable importance
best_model_name <- test_results_75$Model[1]
best_model <- model_list_75[[best_model_name]]
best_model_name## [1] "SVM_Radial"best_model## Support Vector Machines with Radial Basis Function Kernel
##
## 124 samples
## 35 predictor
##
## No pre-processing
## Resampling: Bootstrapped (10 reps)
## Summary of sample sizes: 124, 124, 124, 124, 124, 124, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.535773 0.3933996 1.221425
## 0.50 1.452947 0.4179408 1.145201
## 1.00 1.413221 0.4302557 1.116163
## 2.00 1.406295 0.4352690 1.108370
## 4.00 1.396352 0.4455407 1.100935
## 8.00 1.380521 0.4557876 1.089338
## 16.00 1.377508 0.4562445 1.087872
## 32.00 1.377508 0.4562445 1.087872
##
## Tuning parameter 'sigma' was held constant at a value of 0.02058139
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.02058139 and C = 16.best_imp <- varImp(best_model)$importance %>%
tibble::rownames_to_column("Predictor") %>%
arrange(desc(Overall))
top10_best_imp <- head(best_imp, 10)
kable(top10_best_imp, digits = 3, caption = paste("Top 10 Predictors for", best_model_name))| Predictor | Overall |
|---|---|
| ManufacturingProcess17 | 100.000 |
| BiologicalMaterial03 | 91.296 |
| ManufacturingProcess36 | 90.292 |
| BiologicalMaterial11 | 75.886 |
| ManufacturingProcess06 | 73.847 |
| ManufacturingProcess33 | 64.054 |
| ManufacturingProcess02 | 61.388 |
| ManufacturingProcess04 | 48.458 |
| BiologicalMaterial09 | 47.686 |
| ManufacturingProcess12 | 41.442 |
top10_best_imp <- top10_best_imp %>%
mutate(
Predictor_Type = case_when(
grepl("ManufacturingProcess", Predictor) ~ "Process variable",
grepl("BiologicalMaterial", Predictor) ~ "Biological variable",
TRUE ~ "Other"
)
)
kable(top10_best_imp, digits = 3, caption = "Top 10 Predictors Classified by Type")| Predictor | Overall | Predictor_Type |
|---|---|---|
| ManufacturingProcess17 | 100.000 | Process variable |
| BiologicalMaterial03 | 91.296 | Biological variable |
| ManufacturingProcess36 | 90.292 | Process variable |
| BiologicalMaterial11 | 75.886 | Biological variable |
| ManufacturingProcess06 | 73.847 | Process variable |
| ManufacturingProcess33 | 64.054 | Process variable |
| ManufacturingProcess02 | 61.388 | Process variable |
| ManufacturingProcess04 | 48.458 | Process variable |
| BiologicalMaterial09 | 47.686 | Biological variable |
| ManufacturingProcess12 | 41.442 | Process variable |
Predictor-response plots for top predictors
top_predictors <- head(top10_best_imp$Predictor, 4)
plot_data <- ppTrainPredictors %>%
select(all_of(top_predictors)) %>%
mutate(Yield = trainYield) %>%
tidyr::pivot_longer(
cols = all_of(top_predictors),
names_to = "Predictor",
values_to = "Value"
)
ggplot(plot_data, aes(x = Value, y = Yield)) +
geom_point(alpha = 0.55) +
geom_smooth(method = "loess", se = FALSE) +
facet_wrap(~ Predictor, scales = "free_x") +
labs(
title = "Relationships Between Top Predictors and Yield",
x = "Preprocessed predictor value",
y = "Yield"
) +
theme_minimal()
#{r clean-memory-after-7-2, include=FALSE} rm(trainingData, testData, knnModel, marsModel, nnetModel, svmRadialModel) gc()
Exercise 7.5 answer
The best nonlinear model is identified by the lowest test set RMSE in the test performance table. The resampling plot is also useful because it shows whether the same model performed well consistently during resampling. If one model has the lowest test RMSE but unstable resampling results, I would be cautious about calling it clearly superior.
The variable importance table shows which predictors drive the best
nonlinear model. In my run, the top predictors should be reviewed by
their names: predictors beginning with ManufacturingProcess
are process variables, while predictors beginning with
BiologicalMaterial are biological variables. If most of the
top 10 are manufacturing process variables, then process variables
dominate the nonlinear model. If most are biological variables, then
biological variables dominate.
The predictor-response plots help show whether the nonlinear model is finding curved or threshold-like relationships. If the smooth lines are mostly straight, that suggests a roughly linear relationship between those predictors and yield. If the smooth lines bend or flatten, that suggests the nonlinear model is capturing patterns that a simple linear model may miss.
Overall, I would choose the nonlinear model with the strongest combination of low test RMSE, strong resampling performance, and interpretable variable importance. If the nonlinear models do not clearly outperform the earlier linear model from Exercise 6.3, then the chemical manufacturing data may be mostly explained by an approximately linear structure after preprocessing.
Session Information
sessionInfo()## R version 4.5.3 (2026-03-11)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.4 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8
## [4] LC_COLLATE=C.UTF-8 LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8
## [7] LC_PAPER=C.UTF-8 LC_NAME=C LC_ADDRESS=C
## [10] LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
##
## time zone: UTC
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] knitr_1.51 dplyr_1.2.1
## [3] mlbench_2.1-8 kernlab_0.9-33
## [5] earth_5.3.5 plotmo_3.7.0
## [7] plotrix_3.8-14 Formula_1.2-5
## [9] caret_7.0-1 lattice_0.22-9
## [11] ggplot2_4.0.3 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_0.2.2 fastmap_1.2.0 pROC_1.19.0.1
## [7] digest_0.6.39 rpart_4.1.24 timechange_0.4.0
## [10] lifecycle_1.0.5 cluster_2.1.8.2 survival_3.8-6
## [13] magrittr_2.0.5 compiler_4.5.3 rlang_1.2.0
## [16] sass_0.4.10 tools_4.5.3 yaml_2.3.12
## [19] data.table_1.18.2.1 labeling_0.4.3 plyr_1.8.9
## [22] RColorBrewer_1.1-3 withr_3.0.2 purrr_1.2.2
## [25] nnet_7.3-20 grid_4.5.3 stats4_4.5.3
## [28] future_1.70.0 globals_0.19.1 scales_1.4.0
## [31] iterators_1.0.14 MASS_7.3-65 cli_3.6.6
## [34] ellipse_0.5.0 rmarkdown_2.31 generics_0.1.4
## [37] otel_0.2.0 rstudioapi_0.18.0 future.apply_1.20.2
## [40] reshape2_1.4.5 cachem_1.1.0 stringr_1.6.0
## [43] splines_4.5.3 parallel_4.5.3 vctrs_0.7.3
## [46] hardhat_1.4.3 Matrix_1.7-4 jsonlite_2.0.0
## [49] listenv_0.10.1 foreach_1.5.2 tidyr_1.3.2
## [52] gower_1.0.2 jquerylib_0.1.4 recipes_1.3.2
## [55] glue_1.8.1 parallelly_1.47.0 codetools_0.2-20
## [58] lubridate_1.9.5 stringi_1.8.7 gtable_0.3.6
## [61] rpart.plot_3.1.4 tibble_3.3.1 CORElearn_1.57.3.1
## [64] pillar_1.11.1 htmltools_0.5.9 ipred_0.9-15
## [67] lava_1.9.0 R6_2.6.1 evaluate_1.0.5
## [70] bslib_0.10.0 class_7.3-23 Rcpp_1.1.1-1
## [73] nlme_3.1-168 prodlim_2026.03.11 mgcv_1.9-4
## [76] xfun_0.57 pkgconfig_2.0.3 ModelMetrics_1.2.2.2