HW8
XiaoFei Mei
2025-11-09
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=10sin(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(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)
ctrl = trainControl(method='cv', number = 10)Tune several models on these data. KNN was provided, I also tried NN, MARS, SVM to compare:
library(caret)
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 2.5683461#NN
#check correlations above 0.75
cutoff <- findCorrelation(cor(trainingData$x), cutoff = .75)
# resulted no variable.
# tuning grid
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1),
.size = c(1:10))
# 10-fold cross-validation
ctrl <- trainControl(method = "cv", number = 10)
set.seed(200)
# tune
nnetTune <- train(trainingData$x, trainingData$y,
method = "nnet",
tuneGrid = nnetGrid,
trControl = ctrl,
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 10 * (ncol(trainingData$x) + 1) + 10 + 1,
maxit = 500)
nnetTune## 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 2.611167 0.7034326 2.057976
## 0.00 2 2.598785 0.7234665 2.050280
## 0.00 3 2.297113 0.7838555 1.861644
## 0.00 4 2.368843 0.7648841 1.865206
## 0.00 5 2.645231 0.7432411 2.085563
## 0.00 6 2.755188 0.7166304 2.232277
## 0.00 7 5.890779 0.4576737 3.525435
## 0.00 8 4.864418 0.5024423 3.226141
## 0.00 9 3.560161 0.6246193 2.840954
## 0.00 10 7.918946 0.4288706 4.285780
## 0.01 1 2.381033 0.7641663 1.871496
## 0.01 2 2.584223 0.7329824 2.039030
## 0.01 3 2.149663 0.8043669 1.700742
## 0.01 4 2.581742 0.7251690 2.060772
## 0.01 5 2.552497 0.7241392 2.087565
## 0.01 6 2.723551 0.7268775 2.152613
## 0.01 7 3.135450 0.6400705 2.466251
## 0.01 8 3.169068 0.6323455 2.496601
## 0.01 9 3.283663 0.6212971 2.589913
## 0.01 10 3.201798 0.6355282 2.580713
## 0.10 1 2.392301 0.7614537 1.873847
## 0.10 2 2.516063 0.7411240 1.959889
## 0.10 3 2.300698 0.7631105 1.799187
## 0.10 4 2.420099 0.7688275 1.973457
## 0.10 5 2.376897 0.7815448 1.884554
## 0.10 6 2.391501 0.7705770 1.893266
## 0.10 7 2.790691 0.7118671 2.266089
## 0.10 8 2.992085 0.6806902 2.370561
## 0.10 9 3.152233 0.6543316 2.480312
## 0.10 10 3.203597 0.6701593 2.600855
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 3 and decay = 0.01.nnPred <- predict(nnetTune, testData$x)
postResample(nnPred, testData$y)## RMSE Rsquared MAE
## 2.2831109 0.7952037 1.7216052MARS
#MARS
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
set.seed(200)
marsTuned <- train(trainingData$x, trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = ctrl)
marsTuned## 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 4.188280 0.3042527 3.460689
## 1 3 3.551182 0.4999832 2.837116
## 1 4 2.653143 0.7167280 2.128222
## 1 5 2.405769 0.7562160 1.948161
## 1 6 2.295006 0.7754603 1.853199
## 1 7 1.771950 0.8611767 1.391357
## 1 8 1.647182 0.8774867 1.299564
## 1 9 1.609816 0.8837307 1.299705
## 1 10 1.635035 0.8798236 1.309436
## 1 11 1.571915 0.8896147 1.260711
## 1 12 1.571561 0.8898750 1.253077
## 1 13 1.567577 0.8906927 1.250795
## 1 14 1.571673 0.8909652 1.245508
## 1 15 1.571673 0.8909652 1.245508
## 1 16 1.571673 0.8909652 1.245508
## 1 17 1.571673 0.8909652 1.245508
## 1 18 1.571673 0.8909652 1.245508
## 1 19 1.571673 0.8909652 1.245508
## 1 20 1.571673 0.8909652 1.245508
## 1 21 1.571673 0.8909652 1.245508
## 1 22 1.571673 0.8909652 1.245508
## 1 23 1.571673 0.8909652 1.245508
## 1 24 1.571673 0.8909652 1.245508
## 1 25 1.571673 0.8909652 1.245508
## 1 26 1.571673 0.8909652 1.245508
## 1 27 1.571673 0.8909652 1.245508
## 1 28 1.571673 0.8909652 1.245508
## 1 29 1.571673 0.8909652 1.245508
## 1 30 1.571673 0.8909652 1.245508
## 1 31 1.571673 0.8909652 1.245508
## 1 32 1.571673 0.8909652 1.245508
## 1 33 1.571673 0.8909652 1.245508
## 1 34 1.571673 0.8909652 1.245508
## 1 35 1.571673 0.8909652 1.245508
## 1 36 1.571673 0.8909652 1.245508
## 1 37 1.571673 0.8909652 1.245508
## 1 38 1.571673 0.8909652 1.245508
## 2 2 4.188280 0.3042527 3.460689
## 2 3 3.551182 0.4999832 2.837116
## 2 4 2.615256 0.7216809 2.128763
## 2 5 2.344223 0.7683855 1.890080
## 2 6 2.275048 0.7762472 1.807779
## 2 7 1.841464 0.8418935 1.457945
## 2 8 1.641647 0.8839822 1.288520
## 2 9 1.535119 0.9002991 1.214772
## 2 10 1.473254 0.9101555 1.158761
## 2 11 1.379476 0.9207735 1.080991
## 2 12 1.285380 0.9283193 1.033426
## 2 13 1.267261 0.9328905 1.014726
## 2 14 1.261797 0.9327541 1.009821
## 2 15 1.266663 0.9320714 1.005751
## 2 16 1.270858 0.9322465 1.009757
## 2 17 1.263778 0.9327687 1.007653
## 2 18 1.263778 0.9327687 1.007653
## 2 19 1.263778 0.9327687 1.007653
## 2 20 1.263778 0.9327687 1.007653
## 2 21 1.263778 0.9327687 1.007653
## 2 22 1.263778 0.9327687 1.007653
## 2 23 1.263778 0.9327687 1.007653
## 2 24 1.263778 0.9327687 1.007653
## 2 25 1.263778 0.9327687 1.007653
## 2 26 1.263778 0.9327687 1.007653
## 2 27 1.263778 0.9327687 1.007653
## 2 28 1.263778 0.9327687 1.007653
## 2 29 1.263778 0.9327687 1.007653
## 2 30 1.263778 0.9327687 1.007653
## 2 31 1.263778 0.9327687 1.007653
## 2 32 1.263778 0.9327687 1.007653
## 2 33 1.263778 0.9327687 1.007653
## 2 34 1.263778 0.9327687 1.007653
## 2 35 1.263778 0.9327687 1.007653
## 2 36 1.263778 0.9327687 1.007653
## 2 37 1.263778 0.9327687 1.007653
## 2 38 1.263778 0.9327687 1.007653
##
## 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.Check predictors importance. Based on the result below, predictors X1,4,2,5,3 (top 5 from the list) will be used for prediction.
varImp(marsTuned)## earth variable importance
##
## Overall
## X1 100.00
## X4 75.24
## X2 48.74
## X5 15.53
## X3 0.00marsTuned$results[32,1:5]## degree nprune RMSE Rsquared MAE
## 53 2 17 1.263778 0.9327687 1.007653mars_pred <- predict(marsTuned, newdata = testData$x)
postResample(obs = testData$y, pred=mars_pred[,1])## RMSE Rsquared MAE
## 1.1722635 0.9448890 0.9324923#SVM
set.seed(100)
svmRTuned <- train(trainingData$x, trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = ctrl)
svmRTuned## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.530787 0.7922715 2.013175
## 0.50 2.259539 0.8064569 1.789962
## 1.00 2.099789 0.8274242 1.656154
## 2.00 2.002943 0.8412934 1.583791
## 4.00 1.943618 0.8504425 1.546586
## 8.00 1.918711 0.8547582 1.532981
## 16.00 1.920651 0.8536189 1.536116
## 32.00 1.920651 0.8536189 1.536116
## 64.00 1.920651 0.8536189 1.536116
## 128.00 1.920651 0.8536189 1.536116
## 256.00 1.920651 0.8536189 1.536116
## 512.00 1.920651 0.8536189 1.536116
## 1024.00 1.920651 0.8536189 1.536116
## 2048.00 1.920651 0.8536189 1.536116
##
## Tuning parameter 'sigma' was held constant at a value of 0.06509124
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06509124 and C = 8.svm_pred <- predict(svmRTuned, newdata = testData$x)
postResample(obs = testData$y, pred=svm_pred)## RMSE Rsquared MAE
## 2.0631908 0.8275736 1.56622132b) Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
In the above, we can see that the best model to use would be the MARS model, with RMSE,Rsquared and MAE score to be 1.1589948, 0.9460418 and 0.9250230. It has relatvely low RMSE and mAE score and a high Rsquared number. The MARS model does select informative predictors (X1-X5). In order of importance (most to less important): X1, X4, X2, X5, X3 with score as follow: X1-100.00000, X4-75.23592,X2-48.72974 ,X5-15.51884, and X3-0.
7.5. Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
- Which nonlinear regression model gives the optimal resampling and test set performance?
After data imputation, data splitting, and pre-processing stepsm, I
will be using KNN, NN, MARS and SVM model and compare them all.
Result showing below indicating the best model are SVM, with the lowest
RMSE and MAE value 1.14 and 0.75. plus the highest rsquared value of
0.75.
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)
# imputation
miss <- preProcess(ChemicalManufacturingProcess, method = "bagImpute")
Chemical <- predict(miss, ChemicalManufacturingProcess)
# filtering low frequencies
Chemical <- Chemical[, -nearZeroVar(Chemical)]
set.seed(624)
# index for training
index <- createDataPartition(Chemical$Yield, p = .8, list = FALSE)
# train
train_x <- Chemical[index, -1]
train_y <- Chemical[index, 1]
# test
test_x <- Chemical[-index, -1]
test_y <- Chemical[-index, 1]KNN:
#KNN
knnModel <- train(train_x, train_y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModel## 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 1.471125 0.3330992 1.161484
## 7 1.447346 0.3519621 1.150975
## 9 1.439505 0.3614781 1.153856
## 11 1.440067 0.3597565 1.157491
## 13 1.446347 0.3556436 1.165135
## 15 1.437409 0.3693582 1.165991
## 17 1.448196 0.3618152 1.176400
## 19 1.452990 0.3601724 1.182114
## 21 1.456702 0.3606783 1.183356
## 23 1.457503 0.3658775 1.185981
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 15.knnPred <- predict(knnModel, test_x)
## The function 'postResample' can be used to get the test set
## perforamnce values
postResample(pred = knnPred, test_y)## RMSE Rsquared MAE
## 1.5262067 0.6187302 1.1800625NN:
#NN
# remove predictors to ensure maximum abs pairwise corr between predictors < 0.75
tooHigh <- findCorrelation(cor(train_x), cutoff = .75)
# removing 21 variables
train_x_nnet <- train_x[, -tooHigh]
test_x_nnet <- test_x[, -tooHigh]
# create a tuning grid
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1),
.size = c(1:10))
# 10-fold cross-validation to make reasonable estimates
ctrl <- trainControl(method = "cv", number = 10)
set.seed(100)
# tune
nnetTune <- train(train_x_nnet, train_y,
method = "nnet",
tuneGrid = nnetGrid,
trControl = ctrl,
preProc = c("center", "scale"),
linout = TRUE,
trace = FALSE,
MaxNWts = 10 * (ncol(train_x_nnet) + 1) + 10 + 1,
maxit = 500)
nnetTune## Neural Network
##
## 144 samples
## 35 predictor
##
## Pre-processing: centered (35), scaled (35)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 130, 130, 130, 130, 130, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 1.653183 0.2181442 1.345706
## 0.00 2 2.534424 0.2369899 1.836159
## 0.00 3 3.171155 0.2925081 2.287581
## 0.00 4 3.712864 0.1207665 2.918384
## 0.00 5 3.431782 0.1500205 2.741843
## 0.00 6 4.519146 0.1324394 3.247006
## 0.00 7 4.572852 0.1511347 3.586819
## 0.00 8 4.897815 0.1777553 3.195740
## 0.00 9 6.323278 0.1664817 4.360992
## 0.00 10 8.667370 0.1152399 5.988899
## 0.01 1 1.667606 0.3154025 1.388167
## 0.01 2 2.265838 0.1993149 1.714076
## 0.01 3 2.332248 0.2440199 1.842895
## 0.01 4 3.014612 0.1547884 2.259713
## 0.01 5 2.548550 0.2082747 1.952267
## 0.01 6 2.616978 0.2014794 2.005287
## 0.01 7 2.701263 0.1879326 2.110111
## 0.01 8 2.918798 0.1889725 2.273816
## 0.01 9 2.608475 0.2538457 2.126104
## 0.01 10 3.326500 0.2346732 2.486645
## 0.10 1 1.618516 0.3543468 1.325739
## 0.10 2 1.852789 0.3901490 1.390430
## 0.10 3 2.908159 0.1839385 2.024600
## 0.10 4 2.656395 0.2030849 1.958837
## 0.10 5 2.943660 0.2220188 2.101576
## 0.10 6 2.548716 0.2985445 1.890078
## 0.10 7 2.174937 0.3069096 1.702733
## 0.10 8 2.795824 0.2115189 1.979132
## 0.10 9 2.282975 0.2389936 1.862757
## 0.10 10 2.656864 0.2044295 1.990448
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 1 and decay = 0.1.nnPred <- predict(nnetTune, test_x_nnet)
postResample(nnPred, test_y)## RMSE Rsquared MAE
## 1.5064579 0.5140357 1.1159762MARS:
#MARS
# create a tuning grid
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
set.seed(100)
# tune
marsTune <- train(train_x, train_y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsTune## Multivariate Adaptive Regression Spline
##
## 144 samples
## 56 predictor
##
## No pre-processing
## 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 1.382295 0.4386629 1.1032611
## 1 3 1.240867 0.5448952 0.9985512
## 1 4 1.259935 0.5341424 1.0107010
## 1 5 1.245790 0.5272274 1.0113559
## 1 6 1.269935 0.5136793 1.0204522
## 1 7 1.310209 0.5055710 1.0295204
## 1 8 1.288293 0.5221112 1.0036609
## 1 9 1.293021 0.5193283 1.0156268
## 1 10 1.286486 0.5258144 1.0107051
## 1 11 1.350612 0.5108572 1.0494019
## 1 12 1.354690 0.5164837 1.0502417
## 1 13 1.371710 0.5124198 1.0535178
## 1 14 1.386234 0.5064731 1.0729218
## 1 15 1.377159 0.5169364 1.0708723
## 1 16 1.377159 0.5169364 1.0708723
## 1 17 1.377159 0.5169364 1.0708723
## 1 18 1.377159 0.5169364 1.0708723
## 1 19 1.377159 0.5169364 1.0708723
## 1 20 1.377159 0.5169364 1.0708723
## 1 21 1.377159 0.5169364 1.0708723
## 1 22 1.377159 0.5169364 1.0708723
## 1 23 1.377159 0.5169364 1.0708723
## 1 24 1.377159 0.5169364 1.0708723
## 1 25 1.377159 0.5169364 1.0708723
## 1 26 1.377159 0.5169364 1.0708723
## 1 27 1.377159 0.5169364 1.0708723
## 1 28 1.377159 0.5169364 1.0708723
## 1 29 1.377159 0.5169364 1.0708723
## 1 30 1.377159 0.5169364 1.0708723
## 1 31 1.377159 0.5169364 1.0708723
## 1 32 1.377159 0.5169364 1.0708723
## 1 33 1.377159 0.5169364 1.0708723
## 1 34 1.377159 0.5169364 1.0708723
## 1 35 1.377159 0.5169364 1.0708723
## 1 36 1.377159 0.5169364 1.0708723
## 1 37 1.377159 0.5169364 1.0708723
## 1 38 1.377159 0.5169364 1.0708723
## 2 2 1.382295 0.4386629 1.1032611
## 2 3 1.237952 0.5375297 1.0083290
## 2 4 1.253568 0.5221886 1.0335088
## 2 5 1.204199 0.5507043 0.9713244
## 2 6 1.241877 0.5180123 1.0022903
## 2 7 1.228535 0.5360710 0.9772064
## 2 8 1.236188 0.5297973 0.9891217
## 2 9 1.224202 0.5377333 0.9943605
## 2 10 1.196350 0.5532418 0.9855648
## 2 11 1.217007 0.5502910 1.0105749
## 2 12 1.236600 0.5473328 1.0021900
## 2 13 1.227170 0.5587354 0.9909744
## 2 14 1.263470 0.5599646 1.0158323
## 2 15 1.230580 0.5620079 1.0103784
## 2 16 1.241609 0.5506318 0.9964320
## 2 17 1.233933 0.5689345 0.9858733
## 2 18 1.241566 0.5806316 1.0029570
## 2 19 1.236775 0.5859195 0.9987440
## 2 20 1.317821 0.5266260 1.0648319
## 2 21 1.388138 0.5126592 1.1035179
## 2 22 1.402762 0.5068048 1.1134955
## 2 23 1.396884 0.5054997 1.1196368
## 2 24 1.380184 0.5113281 1.1059875
## 2 25 1.380184 0.5113281 1.1059875
## 2 26 1.386388 0.5070473 1.1174699
## 2 27 1.380683 0.5101973 1.1123044
## 2 28 1.361918 0.5211907 1.0932094
## 2 29 1.366147 0.5191619 1.0957169
## 2 30 1.366147 0.5191619 1.0957169
## 2 31 1.366147 0.5191619 1.0957169
## 2 32 1.360840 0.5200205 1.0921339
## 2 33 1.360840 0.5200205 1.0921339
## 2 34 1.360840 0.5200205 1.0921339
## 2 35 1.360840 0.5200205 1.0921339
## 2 36 1.360840 0.5200205 1.0921339
## 2 37 1.360840 0.5200205 1.0921339
## 2 38 1.360840 0.5200205 1.0921339
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 10 and degree = 2.marsPred <- predict(marsTune, test_x)
postResample(marsPred, test_y)## RMSE Rsquared MAE
## 1.3464789 0.6138875 0.9826902SVM:
#SVM
set.seed(100)
# tune
svmRTune <- train(train_x, train_y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmRTune## 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, 130, 130, 130, 130, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.413177 0.4630126 1.1760898
## 0.50 1.314625 0.5018046 1.0947625
## 1.00 1.217731 0.5647210 1.0095889
## 2.00 1.164634 0.5994161 0.9630243
## 4.00 1.124391 0.6199423 0.9192936
## 8.00 1.119796 0.6170091 0.9287431
## 16.00 1.118734 0.6174115 0.9308110
## 32.00 1.118734 0.6174115 0.9308110
## 64.00 1.118734 0.6174115 0.9308110
## 128.00 1.118734 0.6174115 0.9308110
## 256.00 1.118734 0.6174115 0.9308110
## 512.00 1.118734 0.6174115 0.9308110
## 1024.00 1.118734 0.6174115 0.9308110
## 2048.00 1.118734 0.6174115 0.9308110
##
## Tuning parameter 'sigma' was held constant at a value of 0.0139359
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.0139359 and C = 16.svmRPred <- predict(svmRTune, test_x)
postResample(svmRPred, test_y)## RMSE Rsquared MAE
## 1.1412463 0.7513994 0.8006586compare all:
rbind(knn = postResample(knnPred, test_y),
nn = postResample(nnPred, test_y),
mars = postResample(marsPred, test_y),
svmR = postResample(svmRPred, test_y))## RMSE Rsquared MAE
## knn 1.526207 0.6187302 1.1800625
## nn 1.506458 0.5140357 1.1159762
## mars 1.346479 0.6138875 0.9826902
## svmR 1.141246 0.7513994 0.8006586- 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?
Here is the list of predictors with importantce from top: ManufacturingProcess32,36,09,13,31 nad 06, along with biologicalMaterial06,03,02 and 12 are the top 10. From the comparison below, seems top ten important predictors are the same as the top ten predictors from the optimal linear model(used LARS).
varImp(svmRTune)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## BiologicalMaterial06 89.32
## BiologicalMaterial03 77.48
## ManufacturingProcess36 76.64
## ManufacturingProcess09 73.90
## ManufacturingProcess13 73.24
## ManufacturingProcess31 67.06
## BiologicalMaterial02 66.92
## BiologicalMaterial12 64.94
## ManufacturingProcess06 59.23
## ManufacturingProcess17 53.07
## BiologicalMaterial11 49.11
## BiologicalMaterial04 48.27
## ManufacturingProcess11 45.42
## ManufacturingProcess29 45.31
## ManufacturingProcess33 44.62
## BiologicalMaterial01 40.70
## BiologicalMaterial08 38.19
## ManufacturingProcess30 35.52
## BiologicalMaterial09 29.60#from linear model:
set.seed(100)
larsTune <- train(train_x, train_y,
method = "lars",
metric = "Rsquared",
tuneLength = 20,
trControl = ctrl,
preProc = c("center", "scale"))
lars_predict <- predict(larsTune, test_x)
plot(varImp(svmRTune), top = 10,
main = "Nonlinear: Top 10 Important Predictors")
plot(varImp(larsTune), top = 10,
main = "Linear: Top 10 Important Predictors")
- 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?
From the plot below, we can see among the top predictors, ManufacturingProcess32 has the highest positive correlation with Yield but other than that most other manufactureingProcess tends to negatively correlated with yield whereas biological materials tiends to positively corelated with yield, with BiologicalMaterial 06 and 03 from the top of the list.
library("corrplot")## corrplot 0.95 loadedtop10 <- varImp(svmRTune)$importance |>
arrange(-Overall) |>
head(10)
Chemical |>
select(c("Yield", row.names(top10))) |>
cor() |>
corrplot()