7.2. Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data:
y = 10 sin(πx1x2) + 20(x3 − 0.5)2 + 10x4 + 5x5 + N(0, σ2)where the x values are random variables uniformly distributed between [0, 1] (there are also 5 other non-informative variables also created in the simulation). The package mlbench contains a function called mlbench.friedman1 that simulates these data:
library(caret)
library(mlbench)
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)
## 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
Tune several models on these data. For example:
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.The RMSE is 3.183130, the optimal R^2 0.6425367
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 2.5683461The RMSE is 3.2040595, the optimal R^2 0.6819919
X4 is the most important
Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
Neural Networks
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1), .size = c(1:10),.bag = FALSE)
set.seed(100)
nnetmodel <- train(x = trainingData$x,
y = trainingData$y,
method = "avNNet",
tuneGrid = nnetGrid,
trControl = trainControl(method = "cv"),
## Automatically standardize data prior to modeling
## and prediction
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 10 * (ncol(trainingData$x) + 1) + 5 + 1,
maxit = 500)
nnetmodel## Model Averaged Neural Network
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 2.392711 0.7610354 1.897330
## 0.00 2 2.410532 0.7567109 1.907478
## 0.00 3 2.043957 0.8224281 1.630751
## 0.00 4 2.289347 0.8130639 1.749187
## 0.00 5 2.445600 0.7709399 1.824446
## 0.00 6 2.898295 0.7388800 2.052725
## 0.00 7 3.351563 0.6644147 2.460366
## 0.00 8 6.513566 0.4418645 3.563297
## 0.00 9 4.484215 0.5644107 2.877950
## 0.00 10 NaN NaN NaN
## 0.01 1 2.385381 0.7602926 1.887906
## 0.01 2 2.425125 0.7510903 1.935991
## 0.01 3 2.151209 0.8016018 1.701951
## 0.01 4 2.091925 0.8154383 1.676653
## 0.01 5 2.169742 0.7999255 1.738715
## 0.01 6 2.262032 0.8056619 1.817195
## 0.01 7 2.318301 0.7861811 1.856908
## 0.01 8 2.413847 0.7772629 1.938009
## 0.01 9 2.317190 0.7847500 1.857641
## 0.01 10 NaN NaN NaN
## 0.10 1 2.393965 0.7596431 1.894191
## 0.10 2 2.423612 0.7525959 1.935872
## 0.10 3 2.169914 0.7982380 1.726854
## 0.10 4 2.059080 0.8224160 1.648610
## 0.10 5 1.975656 0.8394000 1.578979
## 0.10 6 2.152198 0.8098015 1.693056
## 0.10 7 2.161512 0.8163011 1.693526
## 0.10 8 2.273716 0.7922525 1.822713
## 0.10 9 2.315333 0.7811273 1.785409
## 0.10 10 NaN NaN NaN
##
## Tuning parameter 'bag' was held constant at a value of FALSE
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 5, decay = 0.1 and bag = FALSE.size = 5, decay = 0.1, RMSE of 1.975656, R^2 of 0.8394000
nnetpredict <- predict(nnetmodel, newdata = testData$x)
postResample(pred = nnetpredict, obs = testData$y)## RMSE Rsquared MAE
## 2.1113956 0.8277556 1.5739011The RMSE is 2.1113956, and the R^2 is 0.8277556, the resample is better.
X4 is more important
Multivariate Adaptive Regression Splines
# Define the candidate models to test
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
# Fix the seed so that the results can be reproduced
set.seed(100)
marsmodel <- train(x = trainingData$x,
y = trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"),
preProc = c("center", "scale"),
tuneLength = 10)
marsmodel## Multivariate Adaptive Regression Spline
##
## 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:
##
## degree nprune RMSE Rsquared MAE
## 1 2 4.327937 0.2544880 3.6004742
## 1 3 3.572450 0.4912720 2.8958113
## 1 4 2.596841 0.7183600 2.1063410
## 1 5 2.370161 0.7659777 1.9186686
## 1 6 2.276141 0.7881481 1.8100006
## 1 7 1.766728 0.8751831 1.3902146
## 1 8 1.780946 0.8723243 1.4013449
## 1 9 1.665091 0.8819775 1.3255147
## 1 10 1.663804 0.8821283 1.3276573
## 1 11 1.657738 0.8822967 1.3317299
## 1 12 1.653784 0.8827903 1.3315041
## 1 13 1.648496 0.8823663 1.3164065
## 1 14 1.639073 0.8841742 1.3128329
## 1 15 1.639073 0.8841742 1.3128329
## 1 16 1.639073 0.8841742 1.3128329
## 1 17 1.639073 0.8841742 1.3128329
## 1 18 1.639073 0.8841742 1.3128329
## 1 19 1.639073 0.8841742 1.3128329
## 1 20 1.639073 0.8841742 1.3128329
## 1 21 1.639073 0.8841742 1.3128329
## 1 22 1.639073 0.8841742 1.3128329
## 1 23 1.639073 0.8841742 1.3128329
## 1 24 1.639073 0.8841742 1.3128329
## 1 25 1.639073 0.8841742 1.3128329
## 1 26 1.639073 0.8841742 1.3128329
## 1 27 1.639073 0.8841742 1.3128329
## 1 28 1.639073 0.8841742 1.3128329
## 1 29 1.639073 0.8841742 1.3128329
## 1 30 1.639073 0.8841742 1.3128329
## 1 31 1.639073 0.8841742 1.3128329
## 1 32 1.639073 0.8841742 1.3128329
## 1 33 1.639073 0.8841742 1.3128329
## 1 34 1.639073 0.8841742 1.3128329
## 1 35 1.639073 0.8841742 1.3128329
## 1 36 1.639073 0.8841742 1.3128329
## 1 37 1.639073 0.8841742 1.3128329
## 1 38 1.639073 0.8841742 1.3128329
## 2 2 4.327937 0.2544880 3.6004742
## 2 3 3.572450 0.4912720 2.8958113
## 2 4 2.661826 0.7070510 2.1734709
## 2 5 2.404015 0.7578971 1.9753867
## 2 6 2.243927 0.7914805 1.7830717
## 2 7 1.856336 0.8605482 1.4356822
## 2 8 1.754607 0.8763186 1.3968406
## 2 9 1.653859 0.8870129 1.2813884
## 2 10 1.434159 0.9166537 1.1339203
## 2 11 1.320482 0.9289120 1.0347278
## 2 12 1.317547 0.9306879 1.0359899
## 2 13 1.296910 0.9306902 1.0146112
## 2 14 1.221407 0.9395223 0.9631486
## 2 15 1.230516 0.9390469 0.9761484
## 2 16 1.236911 0.9387407 0.9745362
## 2 17 1.236911 0.9387407 0.9745362
## 2 18 1.236911 0.9387407 0.9745362
## 2 19 1.236911 0.9387407 0.9745362
## 2 20 1.236911 0.9387407 0.9745362
## 2 21 1.236911 0.9387407 0.9745362
## 2 22 1.236911 0.9387407 0.9745362
## 2 23 1.236911 0.9387407 0.9745362
## 2 24 1.236911 0.9387407 0.9745362
## 2 25 1.236911 0.9387407 0.9745362
## 2 26 1.236911 0.9387407 0.9745362
## 2 27 1.236911 0.9387407 0.9745362
## 2 28 1.236911 0.9387407 0.9745362
## 2 29 1.236911 0.9387407 0.9745362
## 2 30 1.236911 0.9387407 0.9745362
## 2 31 1.236911 0.9387407 0.9745362
## 2 32 1.236911 0.9387407 0.9745362
## 2 33 1.236911 0.9387407 0.9745362
## 2 34 1.236911 0.9387407 0.9745362
## 2 35 1.236911 0.9387407 0.9745362
## 2 36 1.236911 0.9387407 0.9745362
## 2 37 1.236911 0.9387407 0.9745362
## 2 38 1.236911 0.9387407 0.9745362
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 14 and degree = 2.nprune = 14 and degree = 2, the optimal RMSE is 1.221407 and the R^2 0.9395223
marspredict <- predict(marsmodel, newdata = testData$x)
postResample(pred = marspredict, obs = testData$y)## RMSE Rsquared MAE
## 1.2779993 0.9338365 1.0147070The RMSE is 1.2779993 and the R^2 0.9338365, the resample is better
X1 is the important value
Support Vector Machines
svmmodel <- train(x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmmodel## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.490737 0.8009120 1.982118
## 0.50 2.246868 0.8153042 1.774454
## 1.00 2.051872 0.8400992 1.614368
## 2.00 1.949707 0.8534618 1.524201
## 4.00 1.886125 0.8610205 1.465373
## 8.00 1.849240 0.8654699 1.436630
## 16.00 1.834604 0.8673639 1.429807
## 32.00 1.833221 0.8675754 1.428687
## 64.00 1.833221 0.8675754 1.428687
## 128.00 1.833221 0.8675754 1.428687
## 256.00 1.833221 0.8675754 1.428687
## 512.00 1.833221 0.8675754 1.428687
## 1024.00 1.833221 0.8675754 1.428687
## 2048.00 1.833221 0.8675754 1.428687
##
## Tuning parameter 'sigma' was held constant at a value of 0.06315483
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06315483 and C = 32.sigma = 0.06315483 and C = 32, the optimal RMSE of 1.833221 and the R^2 of 0.8675754
svmpredict <- predict(svmmodel, newdata = testData$x)
postResample(pred = svmpredict, obs = testData$y)## RMSE Rsquared MAE
## 2.0741473 0.8255848 1.5755185The RMSE is 2.0741473, the optimal R^2 0.8255848.
library(kableExtra)
results<-rbind(
"KNN" = postResample(pred = knnPred, obs = testData$y),
"NNET" = postResample(pred = nnetpredict, obs = testData$y),
"MARS" = postResample(pred = marspredict, obs = testData$y),
"SVM" = postResample(pred = svmpredict, obs = testData$y)
)
results %>%
kable() %>%
kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| KNN | 3.204060 | 0.6819919 | 2.568346 |
| NNET | 2.111396 | 0.8277556 | 1.573901 |
| MARS | 1.277999 | 0.9338365 | 1.014707 |
| SVM | 2.074147 | 0.8255848 | 1.575519 |
The MARS Model performed the best with a RMSE of 1.277999, R^2 of 0.9338365.
7.5. Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
library(AppliedPredictiveModeling)
data("ChemicalManufacturingProcess")
library(tidyverse)
preP <- preProcess(ChemicalManufacturingProcess,
method = c( "knnImpute", "center", "scale"))
df <- predict(preP, ChemicalManufacturingProcess)
## Restore the response variable values to original
df$Yield = ChemicalManufacturingProcess$Yield
## Split the data into a training and a test set
trainRows <- createDataPartition(df$Yield, p = .80, list = FALSE)
df.train <- df[trainRows, ]
df.test <- df[-trainRows, ]
colYield <- which(colnames(df) == "Yield")
trainingX <- df.train[, -colYield]
trainingY <- df.train$Yield
testingX <- df.test[, -colYield]
testingY <- df.test$Yield- Which nonlinear regression model gives the optimal resampling and testset performance?
KNN Model
knnmodelcmp <- train(x = trainingX,
y = trainingY,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnmodelcmp## k-Nearest Neighbors
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 144, 144, 144, 144, 144, 144, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.407356 0.4212720 1.112452
## 7 1.424206 0.4010071 1.136401
## 9 1.411586 0.4117033 1.129366
## 11 1.406807 0.4191247 1.133421
## 13 1.403986 0.4248304 1.132403
## 15 1.416444 0.4191244 1.145144
## 17 1.417817 0.4229133 1.145318
## 19 1.428014 0.4193412 1.155316
## 21 1.439530 0.4129175 1.160116
## 23 1.453014 0.4042676 1.175374
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 13.knncmppredict <- predict(knnmodelcmp, newdata = testingX)
postResample(pred = knncmppredict, obs = testingY)## RMSE Rsquared MAE
## 1.2788738 0.6050203 1.0462036Neural Networks
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1), .size = c(1:10),.bag = FALSE)
nnetmodelcmp <- train(x = trainingX,
y = trainingY,
method = "avNNet",
tuneGrid = nnetGrid,
trControl = trainControl(method = "cv"),
## Automatically standardize data prior to modeling
## and prediction
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 5 * (ncol(trainingData$x) + 1) + 5 + 1,
maxit = 500)
nnetmodelcmp## Model Averaged Neural Network
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 128, 131, 129, 129, 131, 129, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 1.584571 0.3130882 1.318742
## 0.00 2 NaN NaN NaN
## 0.00 3 NaN NaN NaN
## 0.00 4 NaN NaN NaN
## 0.00 5 NaN NaN NaN
## 0.00 6 NaN NaN NaN
## 0.00 7 NaN NaN NaN
## 0.00 8 NaN NaN NaN
## 0.00 9 NaN NaN NaN
## 0.00 10 NaN NaN NaN
## 0.01 1 1.556010 0.3951200 1.209399
## 0.01 2 NaN NaN NaN
## 0.01 3 NaN NaN NaN
## 0.01 4 NaN NaN NaN
## 0.01 5 NaN NaN NaN
## 0.01 6 NaN NaN NaN
## 0.01 7 NaN NaN NaN
## 0.01 8 NaN NaN NaN
## 0.01 9 NaN NaN NaN
## 0.01 10 NaN NaN NaN
## 0.10 1 1.416094 0.4921079 1.132066
## 0.10 2 NaN NaN NaN
## 0.10 3 NaN NaN NaN
## 0.10 4 NaN NaN NaN
## 0.10 5 NaN NaN NaN
## 0.10 6 NaN NaN NaN
## 0.10 7 NaN NaN NaN
## 0.10 8 NaN NaN NaN
## 0.10 9 NaN NaN NaN
## 0.10 10 NaN NaN NaN
##
## Tuning parameter 'bag' was held constant at a value of FALSE
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 1, decay = 0.1 and bag = FALSE.nnetcmppredictcmp <- predict(nnetmodelcmp, newdata = testingX)
postResample(pred = nnetcmppredictcmp, obs = testingY)## RMSE Rsquared MAE
## 1.5994384 0.4379603 1.2752048Multivariate Adaptive Regression Splines
# Define the candidate models to test
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
# Fix the seed so that the results can be reproduced
set.seed(100)
marsmodelcmp <- train(x = trainingX,
y = trainingY,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"),
preProc = c("center", "scale"),
tuneLength = 10)
marsmodelcmp## Multivariate Adaptive Regression Spline
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 130, 130, 130, 130, 130, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 1.444061 0.4013748 1.1603587
## 1 3 1.175249 0.5894302 0.9428001
## 1 4 1.160211 0.6175903 0.9544192
## 1 5 1.252364 0.5833326 1.0365835
## 1 6 1.166502 0.6234591 0.9586637
## 1 7 1.166563 0.6274583 0.9398742
## 1 8 1.188837 0.5913858 0.9746283
## 1 9 1.208097 0.5815979 0.9870916
## 1 10 1.253156 0.5632424 1.0138561
## 1 11 1.218883 0.5798604 0.9773617
## 1 12 1.205409 0.5906900 0.9582228
## 1 13 1.167951 0.6147467 0.9358355
## 1 14 1.171180 0.6111584 0.9431921
## 1 15 1.172861 0.6069284 0.9546878
## 1 16 1.178332 0.6041202 0.9623805
## 1 17 1.184181 0.6003157 0.9647359
## 1 18 1.184181 0.6003157 0.9647359
## 1 19 1.188177 0.5949445 0.9595038
## 1 20 1.205153 0.5862832 0.9716499
## 1 21 1.210471 0.5834874 0.9765732
## 1 22 1.203311 0.5868768 0.9759124
## 1 23 1.202759 0.5866147 0.9770236
## 1 24 1.197362 0.5900525 0.9706005
## 1 25 1.204300 0.5887813 0.9779134
## 1 26 1.204062 0.5885443 0.9777640
## 1 27 1.204062 0.5885443 0.9777640
## 1 28 1.204062 0.5885443 0.9777640
## 1 29 1.204062 0.5885443 0.9777640
## 1 30 1.204062 0.5885443 0.9777640
## 1 31 1.204062 0.5885443 0.9777640
## 1 32 1.204062 0.5885443 0.9777640
## 1 33 1.204062 0.5885443 0.9777640
## 1 34 1.204062 0.5885443 0.9777640
## 1 35 1.204062 0.5885443 0.9777640
## 1 36 1.204062 0.5885443 0.9777640
## 1 37 1.204062 0.5885443 0.9777640
## 1 38 1.204062 0.5885443 0.9777640
## 2 2 1.444061 0.4013748 1.1603587
## 2 3 1.185007 0.5792362 0.9527514
## 2 4 1.152807 0.6069138 0.9256951
## 2 5 1.202202 0.5910730 0.9576904
## 2 6 1.177337 0.6101047 0.9237307
## 2 7 1.195214 0.6003549 0.9526190
## 2 8 1.182864 0.6039753 0.9322286
## 2 9 1.200815 0.5660698 0.9443230
## 2 10 1.193127 0.5873743 0.9398984
## 2 11 1.224756 0.5986929 0.9710711
## 2 12 1.245054 0.5701015 0.9942423
## 2 13 1.197486 0.5992033 0.9702788
## 2 14 1.356648 0.5265972 1.0741137
## 2 15 1.424230 0.4999087 1.0887157
## 2 16 1.463153 0.4680344 1.1104824
## 2 17 1.476454 0.4767288 1.0899221
## 2 18 1.501645 0.4633398 1.1117723
## 2 19 1.511346 0.4563785 1.1178177
## 2 20 1.510317 0.4549507 1.1167499
## 2 21 1.509077 0.4533001 1.1173055
## 2 22 1.507403 0.4562626 1.1204202
## 2 23 1.515518 0.4537028 1.1217657
## 2 24 1.526465 0.4458162 1.1132488
## 2 25 1.527829 0.4453549 1.1127053
## 2 26 1.534977 0.4433749 1.1136937
## 2 27 1.534977 0.4433749 1.1136937
## 2 28 1.534977 0.4433749 1.1136937
## 2 29 1.534977 0.4433749 1.1136937
## 2 30 1.534977 0.4433749 1.1136937
## 2 31 1.534977 0.4433749 1.1136937
## 2 32 1.534977 0.4433749 1.1136937
## 2 33 1.534977 0.4433749 1.1136937
## 2 34 1.534977 0.4433749 1.1136937
## 2 35 1.534977 0.4433749 1.1136937
## 2 36 1.534977 0.4433749 1.1136937
## 2 37 1.534977 0.4433749 1.1136937
## 2 38 1.534977 0.4433749 1.1136937
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 4 and degree = 2.marspredictcmp <- predict(marsmodelcmp, newdata = testingX)
postResample(pred = marspredictcmp, obs = testingY)## RMSE Rsquared MAE
## 2.1493604 0.1937177 1.5058004Support Vector Machines
svmmodelcmp <- train(x = trainingX,
y = trainingY,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmmodelcmp## Support Vector Machines with Radial Basis Function Kernel
##
## 144 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 128, 129, 130, 131, 130, 128, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.2997257 0.5920603 1.0581932
## 0.50 1.1465549 0.6535431 0.9292275
## 1.00 1.0502310 0.6850454 0.8501671
## 2.00 1.0068258 0.7008040 0.8080636
## 4.00 0.9823643 0.7137321 0.7960622
## 8.00 0.9738763 0.7172855 0.7876829
## 16.00 0.9744713 0.7168080 0.7868434
## 32.00 0.9744713 0.7168080 0.7868434
## 64.00 0.9744713 0.7168080 0.7868434
## 128.00 0.9744713 0.7168080 0.7868434
## 256.00 0.9744713 0.7168080 0.7868434
## 512.00 0.9744713 0.7168080 0.7868434
## 1024.00 0.9744713 0.7168080 0.7868434
## 2048.00 0.9744713 0.7168080 0.7868434
##
## Tuning parameter 'sigma' was held constant at a value of 0.01440193
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01440193 and C = 8.svmpredictcmp <- predict(svmmodelcmp, newdata = testingX)
postResample(pred = svmpredictcmp, obs = testingY)## RMSE Rsquared MAE
## 1.5199166 0.4181622 1.2680693Resampling Models
resample<-rbind(
"KNN" = postResample(pred = predict(knnmodelcmp), obs = trainingY),
"N NET" = postResample(pred = predict(nnetmodelcmp), obs = trainingY),
"MARS" = postResample(pred = predict(marsmodelcmp), obs = trainingY),
"SVM" = postResample(pred = predict(svmmodelcmp), obs = trainingY)
)
resample %>%
kable() %>%
kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| KNN | 1.2480907 | 0.5713744 | 1.0147753 |
| N NET | 0.7500020 | 0.8347106 | 0.6072691 |
| MARS | 1.0720993 | 0.6521974 | 0.8412929 |
| SVM | 0.1753772 | 0.9920798 | 0.1677212 |
The SVM Model for the resample.
Testset Models
predictcmp<-rbind(
"KNN" = postResample(pred = knncmppredict, obs = testingY),
"N NET" = postResample(pred = nnetcmppredictcmp, obs = testingY),
"MARS" = postResample(pred = marspredictcmp, obs = testingY),
"SVM" = postResample(pred = svmpredictcmp, obs = testingY)
)
predictcmp %>%
kable() %>%
kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| KNN | 1.278874 | 0.6050203 | 1.046204 |
| N NET | 1.599438 | 0.4379603 | 1.275205 |
| MARS | 2.149360 | 0.1937177 | 1.505800 |
| SVM | 1.519917 | 0.4181622 | 1.268069 |
The best model is the KNN for the test set.
- Which predictors are most important in the optimal nonlinear regression 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?
## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess09 97.95
## ManufacturingProcess13 95.63
## BiologicalMaterial06 90.79
## BiologicalMaterial03 87.58
## ManufacturingProcess17 82.67
## ManufacturingProcess06 82.18
## BiologicalMaterial12 80.61
## ManufacturingProcess31 75.49
## ManufacturingProcess36 72.65
## ManufacturingProcess11 71.96
## BiologicalMaterial02 58.48
## BiologicalMaterial11 58.39
## ManufacturingProcess18 53.40
## BiologicalMaterial09 52.76
## ManufacturingProcess25 49.90
## BiologicalMaterial04 42.53
## ManufacturingProcess29 41.18
## ManufacturingProcess30 40.60
## BiologicalMaterial08 40.31ManufacturingProcess32 was the most important for the KNN. ManufacturingProcess dominates the list.
## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess09 97.95
## ManufacturingProcess13 95.63
## BiologicalMaterial06 90.79
## BiologicalMaterial03 87.58
## ManufacturingProcess17 82.67
## ManufacturingProcess06 82.18
## BiologicalMaterial12 80.61
## ManufacturingProcess31 75.49
## ManufacturingProcess36 72.65
## ManufacturingProcess11 71.96
## BiologicalMaterial02 58.48
## BiologicalMaterial11 58.39
## ManufacturingProcess18 53.40
## BiologicalMaterial09 52.76
## ManufacturingProcess25 49.90
## BiologicalMaterial04 42.53
## ManufacturingProcess29 41.18
## ManufacturingProcess30 40.60
## BiologicalMaterial08 40.31ManufacturingProcess32 was the most important for the NNet model. ManufacturingProcess dominates the list
## earth variable importance
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess09 49.81
## ManufacturingProcess17 0.00
## BiologicalMaterial04 0.00ManufacturingProcess32 was the most important in the MARS model. ManufacturingProcess dominates the list
## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess09 97.95
## ManufacturingProcess13 95.63
## BiologicalMaterial06 90.79
## BiologicalMaterial03 87.58
## ManufacturingProcess17 82.67
## ManufacturingProcess06 82.18
## BiologicalMaterial12 80.61
## ManufacturingProcess31 75.49
## ManufacturingProcess36 72.65
## ManufacturingProcess11 71.96
## BiologicalMaterial02 58.48
## BiologicalMaterial11 58.39
## ManufacturingProcess18 53.40
## BiologicalMaterial09 52.76
## ManufacturingProcess25 49.90
## BiologicalMaterial04 42.53
## ManufacturingProcess29 41.18
## ManufacturingProcess30 40.60
## BiologicalMaterial08 40.31ManufacturingProcess32 was the most important for the SVM Model. ManufacturingProcess dominates the list
For all of the models ManufacturingProcess32 was more important, while in all of the Models ManufacturingProcess09 was second.
- Explore the relationships between 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?
mimp <- varImp(knnmodelcmp)$importance
imp10 <- head(rownames(mimp)[order(-mimp$Overall)], 10)
as.data.frame(imp10)## imp10
## 1 ManufacturingProcess32
## 2 ManufacturingProcess09
## 3 ManufacturingProcess13
## 4 BiologicalMaterial06
## 5 BiologicalMaterial03
## 6 ManufacturingProcess17
## 7 ManufacturingProcess06
## 8 BiologicalMaterial12
## 9 ManufacturingProcess31
## 10 ManufacturingProcess36#featurePlot(trainX[,top10Vars], trainY)
X <- df[,imp10]
Y <- df[,colYield]
## Shorten the variable names for readability
colnames(X) <- gsub("(Process|Material)", "", colnames(X))
featurePlot(X, Y)
This shows the relationship of the Top 10 most important predictors and how it responds to them. The optimal model was the KNN these plots show that they are mostly linear. Looking at the top two you can see that there are some outliers. Seeing the relationship of the plots allows one to decide how to change the data in order to accomadate what they want to showcase.