Data_624_HW_8_YR
2025-05-03
library(AppliedPredictiveModeling)
library(caret)## Loading required package: ggplot2## Loading required package: latticelibrary(mlbench)
library(earth)## Loading required package: Formula## Loading required package: plotmo## Loading required package: plotrixlibrary(kernlab)##
## Attaching package: 'kernlab'## The following object is masked from 'package:ggplot2':
##
## alphalibrary(ggplot2)7.2
Friedman (1991) introduced several benchmark data sets create by sim- ulation. One of these simulations used the following nonlinear equation to create data: … where the x values are random variables uniformly distributed between [0, 1] (there are also 5 other non-informative variables also created in the simula- tion). The package mlbench contains a function called mlbench.friedman1 that simulates these data:
set.seed(200)
trainingData <- mlbench.friedman1(200, sd = 1)
## We convert the 'x' data from a matrix to a data frame
## One reason is that this will give the columns names.
trainingData$x <- data.frame(trainingData$x)
# Look at the data using
featurePlot(trainingData$x, trainingData$y)
## or other methods.
## This creates a list with a vector 'y' and a matrix
## of predictors 'x'. Also simulate a large test set to
## estimate the true error rate with good precision:
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)
#KNN Model
knnModel <- train(x = trainingData$x,
y = trainingData$y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModel## k-Nearest Neighbors
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 3.466085 0.5121775 2.816838
## 7 3.349428 0.5452823 2.727410
## 9 3.264276 0.5785990 2.660026
## 11 3.214216 0.6024244 2.603767
## 13 3.196510 0.6176570 2.591935
## 15 3.184173 0.6305506 2.577482
## 17 3.183130 0.6425367 2.567787
## 19 3.198752 0.6483184 2.592683
## 21 3.188993 0.6611428 2.588787
## 23 3.200458 0.6638353 2.604529
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 17.knnPred <- predict(knnModel, newdata = testData$x)
## The function 'postResample' can be used to get the test set
## perforamnce values
postResample(pred = knnPred, obs = testData$y)## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
# MARS Model
set.seed(0505)
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
marsFit <- train(x = trainingData$x,
y = trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsFit## Multivariate Adaptive Regression Spline
##
## 200 samples
## 10 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 4.473707 0.2059112 3.735380
## 1 3 3.674517 0.4563730 2.961494
## 1 4 2.693987 0.7219221 2.161466
## 1 5 2.354295 0.7839794 1.883063
## 1 6 2.261965 0.7996703 1.796800
## 1 7 1.842051 0.8681038 1.415254
## 1 8 1.667184 0.8936275 1.303338
## 1 9 1.658109 0.8960439 1.301385
## 1 10 1.642522 0.8967562 1.301455
## 1 11 1.638873 0.8981547 1.273918
## 1 12 1.627332 0.9007205 1.269979
## 1 13 1.641691 0.8994554 1.281886
## 1 14 1.668962 0.8957737 1.308823
## 1 15 1.668986 0.8956228 1.308573
## 1 16 1.668986 0.8956228 1.308573
## 1 17 1.668986 0.8956228 1.308573
## 1 18 1.668986 0.8956228 1.308573
## 1 19 1.668986 0.8956228 1.308573
## 1 20 1.668986 0.8956228 1.308573
## 1 21 1.668986 0.8956228 1.308573
## 1 22 1.668986 0.8956228 1.308573
## 1 23 1.668986 0.8956228 1.308573
## 1 24 1.668986 0.8956228 1.308573
## 1 25 1.668986 0.8956228 1.308573
## 1 26 1.668986 0.8956228 1.308573
## 1 27 1.668986 0.8956228 1.308573
## 1 28 1.668986 0.8956228 1.308573
## 1 29 1.668986 0.8956228 1.308573
## 1 30 1.668986 0.8956228 1.308573
## 1 31 1.668986 0.8956228 1.308573
## 1 32 1.668986 0.8956228 1.308573
## 1 33 1.668986 0.8956228 1.308573
## 1 34 1.668986 0.8956228 1.308573
## 1 35 1.668986 0.8956228 1.308573
## 1 36 1.668986 0.8956228 1.308573
## 1 37 1.668986 0.8956228 1.308573
## 1 38 1.668986 0.8956228 1.308573
## 2 2 4.473707 0.2059112 3.735380
## 2 3 3.674517 0.4563730 2.961494
## 2 4 2.644768 0.7301597 2.124856
## 2 5 2.258397 0.8004069 1.792321
## 2 6 2.206230 0.8065141 1.724864
## 2 7 1.792779 0.8767625 1.417441
## 2 8 1.626351 0.8975182 1.255933
## 2 9 1.417336 0.9210696 1.124743
## 2 10 1.364662 0.9281539 1.112445
## 2 11 1.289844 0.9363794 1.034030
## 2 12 1.307324 0.9350859 1.034410
## 2 13 1.345946 0.9316704 1.081343
## 2 14 1.321258 0.9325492 1.064430
## 2 15 1.326130 0.9317542 1.063100
## 2 16 1.330557 0.9314304 1.058284
## 2 17 1.336075 0.9308362 1.056546
## 2 18 1.336075 0.9308362 1.056546
## 2 19 1.336075 0.9308362 1.056546
## 2 20 1.336075 0.9308362 1.056546
## 2 21 1.336075 0.9308362 1.056546
## 2 22 1.336075 0.9308362 1.056546
## 2 23 1.336075 0.9308362 1.056546
## 2 24 1.336075 0.9308362 1.056546
## 2 25 1.336075 0.9308362 1.056546
## 2 26 1.336075 0.9308362 1.056546
## 2 27 1.336075 0.9308362 1.056546
## 2 28 1.336075 0.9308362 1.056546
## 2 29 1.336075 0.9308362 1.056546
## 2 30 1.336075 0.9308362 1.056546
## 2 31 1.336075 0.9308362 1.056546
## 2 32 1.336075 0.9308362 1.056546
## 2 33 1.336075 0.9308362 1.056546
## 2 34 1.336075 0.9308362 1.056546
## 2 35 1.336075 0.9308362 1.056546
## 2 36 1.336075 0.9308362 1.056546
## 2 37 1.336075 0.9308362 1.056546
## 2 38 1.336075 0.9308362 1.056546
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 11 and degree = 2.#The final values used for the model were nprune = 11 and degree = 2.
marsFit$bestTune## nprune degree
## 47 11 2marsPred <- predict(marsFit, newdata = testData$x)
marsResults <- postResample(pred = marsPred, obs = testData$y)
marsResults## RMSE Rsquared MAE
## 1.2803060 0.9335241 1.0168673#SVM Model
set.seed(0506)
svmFit <- train(x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 10,
trControl = trainControl(method = "cv"))
svmFit## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.475858 0.8121084 1.981112
## 0.50 2.212206 0.8276713 1.753422
## 1.00 2.048716 0.8481672 1.609954
## 2.00 1.906683 0.8678043 1.490096
## 4.00 1.824943 0.8763489 1.414680
## 8.00 1.805692 0.8800417 1.400559
## 16.00 1.803490 0.8814764 1.407560
## 32.00 1.804262 0.8813492 1.408806
## 64.00 1.804262 0.8813492 1.408806
## 128.00 1.804262 0.8813492 1.408806
##
## Tuning parameter 'sigma' was held constant at a value of 0.05936526
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.05936526 and C = 16.#The final values used for the model were sigma = 0.05936526 and C = 16.
svmFit$bestTune## sigma C
## 7 0.05936526 16svmPred <- predict(svmFit, newdata = testData$x)
svmResults <- postResample(pred = svmPred, obs = testData$y)
svmResults## RMSE Rsquared MAE
## 2.063427 0.827336 1.567688#comparing all three models
data.frame(
Model = c("KNN", "MARS", "SVM"),
rbind(
postResample(pred = knnPred, obs = testData$y),
postResample(pred = marsPred, obs = testData$y),
postResample(pred = svmPred, obs = testData$y)))## Model RMSE Rsquared MAE
## 1 KNN 3.204059 0.6819919 2.568346
## 2 MARS 1.280306 0.9335241 1.016867
## 3 SVM 2.063427 0.8273360 1.567688The performance comparison of three models shows that the MARS model has the best predictive accuracy, with the lowest RMSE (1.28), highest R-squared (0.93), and lowest MAE (1.02), indicating strong fit and minimal prediction error.
7.5
Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
- Which nonlinear regression model gives the optimal resampling and test set performance?
data(ChemicalManufacturingProcess)
# pre-processing from 6.3 exercise
set.seed(0506)
cmp <- preProcess(ChemicalManufacturingProcess, method = "knnImpute")
cmp_imputed <- predict(cmp, ChemicalManufacturingProcess)
any(is.na(cmp_imputed))## [1] FALSE#removing near zero values
cmp_filtered <- cmp_imputed[,-nearZeroVar(cmp_imputed)]
set.seed(0507)
cmp_train_index <- createDataPartition(cmp_filtered$Yield, p= 0.8, list = FALSE)
cmp_train <- cmp_filtered[cmp_train_index,]
cmp_test <- cmp_filtered[-cmp_train_index,]
#MARS model
cmp_mars <- train(cmp_train[, !names(cmp_train) %in% "Yield"],
cmp_train$Yield,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
cmp_mars## Multivariate Adaptive Regression Spline
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 129, 128, 130, 129, 130, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 0.7752911 0.4275023 0.6120243
## 1 3 0.6032026 0.6463356 0.4839294
## 1 4 0.6265973 0.6185799 0.4896057
## 1 5 0.6118502 0.6378310 0.4824863
## 1 6 0.6393335 0.6083814 0.5094254
## 1 7 0.6373448 0.6209967 0.5059538
## 1 8 0.6052538 0.6584140 0.4823153
## 1 9 0.6097183 0.6517113 0.4872536
## 1 10 0.6330484 0.6296726 0.5018900
## 1 11 0.6418055 0.6240124 0.5115936
## 1 12 0.6387257 0.6177895 0.5145750
## 1 13 0.6247542 0.6358544 0.5033395
## 1 14 0.6296166 0.6242255 0.4995423
## 1 15 0.6355888 0.6210653 0.5009699
## 1 16 0.6369560 0.6193788 0.5015336
## 1 17 0.6434263 0.6202956 0.5119013
## 1 18 0.6257724 0.6377410 0.5051716
## 1 19 0.6255045 0.6374628 0.5042739
## 1 20 0.6271235 0.6372024 0.5051993
## 1 21 0.6271235 0.6372024 0.5051993
## 1 22 0.6271235 0.6372024 0.5051993
## 1 23 0.6271235 0.6372024 0.5051993
## 1 24 0.6271235 0.6372024 0.5051993
## 1 25 0.6271235 0.6372024 0.5051993
## 1 26 0.6271235 0.6372024 0.5051993
## 1 27 0.6271235 0.6372024 0.5051993
## 1 28 0.6271235 0.6372024 0.5051993
## 1 29 0.6271235 0.6372024 0.5051993
## 1 30 0.6271235 0.6372024 0.5051993
## 1 31 0.6271235 0.6372024 0.5051993
## 1 32 0.6271235 0.6372024 0.5051993
## 1 33 0.6271235 0.6372024 0.5051993
## 1 34 0.6271235 0.6372024 0.5051993
## 1 35 0.6271235 0.6372024 0.5051993
## 1 36 0.6271235 0.6372024 0.5051993
## 1 37 0.6271235 0.6372024 0.5051993
## 1 38 0.6271235 0.6372024 0.5051993
## 2 2 0.7752911 0.4275023 0.6120243
## 2 3 0.6326674 0.5975588 0.4993006
## 2 4 0.6257613 0.6279919 0.4927481
## 2 5 0.6805299 0.5581148 0.5380409
## 2 6 0.6796221 0.5643818 0.5440488
## 2 7 0.7037744 0.5509892 0.5476012
## 2 8 0.7064711 0.5546050 0.5399432
## 2 9 0.6927406 0.5684102 0.5352198
## 2 10 0.6937045 0.5922264 0.5114920
## 2 11 0.7224928 0.5715142 0.5307765
## 2 12 0.7665474 0.5649394 0.5503282
## 2 13 0.7237082 0.6089367 0.5307078
## 2 14 0.7719478 0.5759897 0.5492151
## 2 15 0.7414732 0.6028284 0.5395357
## 2 16 0.9468814 0.5641198 0.6214396
## 2 17 0.7852803 0.6028986 0.5560268
## 2 18 0.7681858 0.6158080 0.5442563
## 2 19 0.9819668 0.5849192 0.6281134
## 2 20 0.9596390 0.5861546 0.6179385
## 2 21 0.9526725 0.5848057 0.6195384
## 2 22 0.9790833 0.5798822 0.6333510
## 2 23 0.9773756 0.5825638 0.6344239
## 2 24 0.9749866 0.5839369 0.6381853
## 2 25 0.9540998 0.5862773 0.6319302
## 2 26 0.9623307 0.5832487 0.6395288
## 2 27 0.9687205 0.5803439 0.6454061
## 2 28 0.9687205 0.5803439 0.6454061
## 2 29 0.9687205 0.5803439 0.6454061
## 2 30 0.9710568 0.5803154 0.6441091
## 2 31 0.9710568 0.5803154 0.6441091
## 2 32 0.9710568 0.5803154 0.6441091
## 2 33 0.9710568 0.5803154 0.6441091
## 2 34 0.9710568 0.5803154 0.6441091
## 2 35 0.9710568 0.5803154 0.6441091
## 2 36 0.9710568 0.5803154 0.6441091
## 2 37 0.9710568 0.5803154 0.6441091
## 2 38 0.9710568 0.5803154 0.6441091
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 3 and degree = 1.#The final values used for the model were nprune = 3 and degree = 1.
cmp_marsPred <- predict(cmp_mars, newdata = cmp_test)
cmp_marsResults <- postResample(pred = cmp_marsPred, obs = cmp_test$Yield)
cmp_marsResults## RMSE Rsquared MAE
## 0.7191385 0.4663119 0.5946564# KNN Model
set.seed(0507)
cmp_knnFit <- train(Yield ~ .,
data = cmp_train,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
cmp_knnFit## k-Nearest Neighbors
##
## 144 samples
## 56 predictor
##
## Pre-processing: centered (56), scaled (56)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 144, 144, 144, 144, 144, 144, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 0.7984837 0.4037482 0.6275656
## 7 0.7858486 0.4259869 0.6253203
## 9 0.7813229 0.4430184 0.6210572
## 11 0.7894727 0.4322128 0.6315689
## 13 0.7985669 0.4192702 0.6396295
## 15 0.7990100 0.4285618 0.6404208
## 17 0.8022355 0.4291068 0.6446410
## 19 0.8054929 0.4302893 0.6463274
## 21 0.8075964 0.4349964 0.6463214
## 23 0.8140798 0.4314819 0.6526266
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 9.#The final value used for the model was k = 19.
cmp_knnPred <- predict(cmp_knnFit, newdata = cmp_test)
cmp_knnResults <- postResample(pred = cmp_knnPred, obs = cmp_test$Yield)
cmp_knnResults## RMSE Rsquared MAE
## 0.7102753 0.5300573 0.5627296#SVM Model
set.seed(0507)
cmp_svmFit <- train(Yield ~ .,
data = cmp_train,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 10,
trControl = trainControl(method = "cv"))
cmp_svmFit## Support Vector Machines with Radial Basis Function Kernel
##
## 144 samples
## 56 predictor
##
## Pre-processing: centered (56), scaled (56)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 130, 129, 131, 130, 129, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 0.7591297 0.5331625 0.6239359
## 0.50 0.7032320 0.5790092 0.5749790
## 1.00 0.6510864 0.6223945 0.5253027
## 2.00 0.6178926 0.6448386 0.4907204
## 4.00 0.6140758 0.6398688 0.4757480
## 8.00 0.6124280 0.6411986 0.4822256
## 16.00 0.6070621 0.6496302 0.4798367
## 32.00 0.6070621 0.6496302 0.4798367
## 64.00 0.6070621 0.6496302 0.4798367
## 128.00 0.6070621 0.6496302 0.4798367
##
## Tuning parameter 'sigma' was held constant at a value of 0.01360243
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01360243 and C = 16.#The final values used for the model were sigma = 0.01360243 and C = 16.
cmp_svmPred <- predict(cmp_svmFit, newdata = cmp_test)
cmp_svmResults <- postResample(pred = cmp_svmPred, obs = cmp_test$Yield)
cmp_svmResults## RMSE Rsquared MAE
## 0.6202813 0.6358902 0.5351300#comparing all three models
cmp_models <- data.frame(
Model = c("KNN", "MARS", "SVM"),
rbind(
postResample(pred = cmp_knnPred, obs = cmp_test$Yield),
postResample(pred = cmp_marsPred, obs = cmp_test$Yield),
postResample(pred = cmp_svmPred, obs = cmp_test$Yield)))
cmp_models## Model RMSE Rsquared MAE
## 1 KNN 0.7102753 0.5300573 0.5627296
## 2 MARS 0.7191385 0.4663119 0.5946564
## 3 SVM 0.6202813 0.6358902 0.5351300Among the nonlinear regression models evaluated, the Support Vector Machine (SVM) model demonstrated the best performance, achieving the lowest RMSE (0.62), highest R-squared (0.64), and lowest MAE (0.54), indicating the most accurate predictions.
- Which predictors are most important in the optimal nonlinear regres- sion model? Do either the biological or process variables dominate the list? How do the top ten important predictors compare to the top ten predictors from the optimal linear model?
cmp_predictors <- varImp(cmp_svmFit, scale = FALSE)
plot(cmp_predictors, top = 10)
The most important predictors were mainly process variables. The
top predictors included ManufacturingProcess13, ManufacturingProcess32,
ManufacturingProcess09, and ManufacturingProcess17, indicating a strong
influence of manufacturing process factors on yield. A few biological
variables also appeared in the top ten, including BiologicalMaterial06,
BiologicalMaterial03, and BiologicalMaterial12, but overall the list was
dominated by process-related features. These were also present in the
optimal linear model.
- Explore the relationshipsbetween the top predictors and the response for the predictors that are unique to the optimal nonlinear regression model.
Do these plots reveal intuition about the biological or process predictors and their relationship with yield?
unique_predictors <- c("ManufacturingProcess13",
"ManufacturingProcess32",
"ManufacturingProcess09")
# Loop over predictors and create scatterplots with LOESS smooth line
for (pred in unique_predictors) {
p <- ggplot(cmp_train, aes_string(x = pred, y = "Yield")) +
geom_point(alpha = 0.5) +
geom_smooth(method = "loess", se = FALSE, color = "blue", linewidth = 1) +
labs(title = paste("Yield vs.", pred),
x = pred,
y = "Yield") +
theme_minimal()
print(p)
}## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation idioms with `aes()`.
## ℹ See also `vignette("ggplot2-in-packages")` for more information.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
There is a clear non-linear relationship for all three top
predictors. ManufacturingProcess13 had a strong inverse nonlinear
relationship. ManufacturingProcess32 peaked at moderate values and
dropped at the extremes. ManufacturingProcess09 showed a positive
relationship with a noticeable increase after a certain point. These
trends suggest nonlinear process effects that the linear model
missed.