DATA 624 Homework 8
Orli Khaimova
4/24/2022
7.2
Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data:
\[y = 10 \sin(\pi x_1 x_2) + 20(x_3 - 0.5)^2 + 10x_4 +5x_5 +N(0,\sigma^2)\]
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)
## 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)Tune several models on these data. For example:
KNN
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 2.5683461NN
# remove predictors to ensure maximum abs pairwise corr between predictors < 0.75
tooHigh <- findCorrelation(cor(trainingData$x), cutoff = .75)
# returns an empty variable
# create a tuning grid
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1),
.size = c(1:10))
# 10-fold cross-validation to make reasonable estimates
ctrl <- trainControl(method = "cv", number = 10)
set.seed(100)
# tune
nnetTune <- train(trainingData$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 2.540546 0.7254252 2.008197
## 0.00 2 2.655140 0.7062546 2.133980
## 0.00 3 2.309494 0.7716862 1.814365
## 0.00 4 2.268793 0.8065021 1.790856
## 0.00 5 2.489348 0.7556147 1.938090
## 0.00 6 3.445815 0.6172907 2.291949
## 0.00 7 5.262992 0.5126718 3.145832
## 0.00 8 5.095620 0.4496682 3.298365
## 0.00 9 6.723562 0.5042550 4.089071
## 0.00 10 3.529390 0.6008587 2.765906
## 0.01 1 2.385136 0.7603460 1.887587
## 0.01 2 2.583767 0.7260485 2.018814
## 0.01 3 2.282621 0.7815267 1.812073
## 0.01 4 2.274770 0.7901402 1.842674
## 0.01 5 2.653199 0.7237241 2.117160
## 0.01 6 2.753791 0.7153838 2.190768
## 0.01 7 2.799525 0.7123252 2.209083
## 0.01 8 3.342508 0.6050453 2.630355
## 0.01 9 3.806778 0.6109666 2.881245
## 0.01 10 3.470039 0.5982097 2.841658
## 0.10 1 2.394058 0.7596252 1.894319
## 0.10 2 2.618767 0.7152952 2.117662
## 0.10 3 2.580239 0.7353788 2.005527
## 0.10 4 2.442308 0.7777448 1.907659
## 0.10 5 2.617403 0.7227439 2.059628
## 0.10 6 2.543814 0.7549811 2.060699
## 0.10 7 2.811804 0.6959408 2.149668
## 0.10 8 2.900555 0.6937553 2.336332
## 0.10 9 3.101137 0.6579733 2.481149
## 0.10 10 2.973902 0.6608165 2.360836
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 4 and decay = 0.nnPred <- predict(nnetTune, testData$x)
postResample(nnPred, testData$y)## RMSE Rsquared MAE
## 2.4700280 0.7762282 1.9078363MARS
# create a tuning grid
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38)
set.seed(100)
# tune
marsTune <- train(trainingData$x, trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsTune## 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 4.327937 0.2544880 3.600474
## 1 3 3.572450 0.4912720 2.895811
## 1 4 2.596841 0.7183600 2.106341
## 1 5 2.370161 0.7659777 1.918669
## 1 6 2.276141 0.7881481 1.810001
## 1 7 1.766728 0.8751831 1.390215
## 1 8 1.780946 0.8723243 1.401345
## 1 9 1.665091 0.8819775 1.325515
## 1 10 1.663804 0.8821283 1.327657
## 1 11 1.657738 0.8822967 1.331730
## 1 12 1.653784 0.8827903 1.331504
## 1 13 1.648496 0.8823663 1.316407
## 1 14 1.639073 0.8841742 1.312833
## 1 15 1.639073 0.8841742 1.312833
## 1 16 1.639073 0.8841742 1.312833
## 1 17 1.639073 0.8841742 1.312833
## 1 18 1.639073 0.8841742 1.312833
## 1 19 1.639073 0.8841742 1.312833
## 1 20 1.639073 0.8841742 1.312833
## 1 21 1.639073 0.8841742 1.312833
## 1 22 1.639073 0.8841742 1.312833
## 1 23 1.639073 0.8841742 1.312833
## 1 24 1.639073 0.8841742 1.312833
## 1 25 1.639073 0.8841742 1.312833
## 1 26 1.639073 0.8841742 1.312833
## 1 27 1.639073 0.8841742 1.312833
## 1 28 1.639073 0.8841742 1.312833
## 1 29 1.639073 0.8841742 1.312833
## 1 30 1.639073 0.8841742 1.312833
## 1 31 1.639073 0.8841742 1.312833
## 1 32 1.639073 0.8841742 1.312833
## 1 33 1.639073 0.8841742 1.312833
## 1 34 1.639073 0.8841742 1.312833
## 1 35 1.639073 0.8841742 1.312833
## 1 36 1.639073 0.8841742 1.312833
## 1 37 1.639073 0.8841742 1.312833
## 1 38 1.639073 0.8841742 1.312833
## 2 2 4.327937 0.2544880 3.600474
## 2 3 3.572450 0.4912720 2.895811
## 2 4 2.661826 0.7070510 2.173471
## 2 5 2.404015 0.7578971 1.975387
## 2 6 2.243927 0.7914805 1.783072
## 2 7 1.856336 0.8605482 1.435682
## 2 8 1.754607 0.8763186 1.396841
## 2 9 1.603578 0.8938666 1.261361
## 2 10 1.492421 0.9084998 1.168700
## 2 11 1.317350 0.9292504 1.033926
## 2 12 1.304327 0.9320133 1.019108
## 2 13 1.277510 0.9323681 1.002927
## 2 14 1.269626 0.9350024 1.003346
## 2 15 1.266217 0.9359400 1.013893
## 2 16 1.268470 0.9354868 1.011414
## 2 17 1.268470 0.9354868 1.011414
## 2 18 1.268470 0.9354868 1.011414
## 2 19 1.268470 0.9354868 1.011414
## 2 20 1.268470 0.9354868 1.011414
## 2 21 1.268470 0.9354868 1.011414
## 2 22 1.268470 0.9354868 1.011414
## 2 23 1.268470 0.9354868 1.011414
## 2 24 1.268470 0.9354868 1.011414
## 2 25 1.268470 0.9354868 1.011414
## 2 26 1.268470 0.9354868 1.011414
## 2 27 1.268470 0.9354868 1.011414
## 2 28 1.268470 0.9354868 1.011414
## 2 29 1.268470 0.9354868 1.011414
## 2 30 1.268470 0.9354868 1.011414
## 2 31 1.268470 0.9354868 1.011414
## 2 32 1.268470 0.9354868 1.011414
## 2 33 1.268470 0.9354868 1.011414
## 2 34 1.268470 0.9354868 1.011414
## 2 35 1.268470 0.9354868 1.011414
## 2 36 1.268470 0.9354868 1.011414
## 2 37 1.268470 0.9354868 1.011414
## 2 38 1.268470 0.9354868 1.011414
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 15 and degree = 2.marsPred <- predict(marsTune, testData$x)
postResample(marsPred, testData$y)## RMSE Rsquared MAE
## 1.1589948 0.9460418 0.9250230SVM
set.seed(100)
# tune
svmRTune <- train(trainingData$x, trainingData$y,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmRTune## 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.svmRPred <- predict(svmRTune, testData$x)
postResample(svmRPred, testData$y)## RMSE Rsquared MAE
## 2.0631908 0.8275736 1.5662213Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
MARS appears to give the best performance since it has the lowest RMSE and MAE and highest \(R^2\). The second best would be SVM radial.
varImp(marsTune)## earth variable importance
##
## Overall
## X1 100.00
## X4 75.24
## X2 48.73
## X5 15.52
## X3 0.00plot(varImp(marsTune))
MARS does select the informative variables, X1 - X5, however, X3 seems to be insignificant, as it has an importance of about 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.
# load the data
library(AppliedPredictiveModeling)## Warning: package 'AppliedPredictiveModeling' was built under R version 4.0.5data(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]- Which nonlinear regression model gives the optimal resampling and test set performance?
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
# 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 0.2926531 2.287708
## 0.00 4 3.711481 0.1209223 2.917563
## 0.00 5 3.431171 0.1500184 2.741639
## 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 3.002585 0.1568784 2.241891
## 0.01 5 2.559003 0.2072487 1.958315
## 0.01 6 2.615888 0.2014615 2.003966
## 0.01 7 2.704167 0.1873030 2.115345
## 0.01 8 2.884852 0.1905781 2.225486
## 0.01 9 2.664823 0.2242711 2.139448
## 0.01 10 3.327161 0.2317687 2.496622
## 0.10 1 1.618516 0.3543468 1.325739
## 0.10 2 1.852789 0.3901490 1.390430
## 0.10 3 2.907373 0.1839412 2.024455
## 0.10 4 2.664748 0.1941929 1.965063
## 0.10 5 2.946770 0.2249056 2.098324
## 0.10 6 2.533430 0.2964670 1.884063
## 0.10 7 2.175251 0.3069320 1.702581
## 0.10 8 2.696990 0.1964820 1.966580
## 0.10 9 2.282723 0.2395294 1.862798
## 0.10 10 2.780285 0.1640235 2.034794
##
## 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
# 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
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 0.8006586rbind(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 0.8006586SVM gives the best performance with the radial method as it has the lowest RMSE and MAE and the highest \(R^2\).
- 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?
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 29.60plot(varImp(svmRTune), top = 20) 
The process variables dominate the list with a ratio of 11:9. This was the same for the optimal linear model from Homework 7.
#optimal linear method from homework 7
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)The top ten important predictors are the same as the top ten predictors from the optimal linear model, which was the LARS model.
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?
top10 <- varImp(svmRTune)$importance %>%
arrange(-Overall) %>%
head(10)
Chemical %>%
select(c("Yield", row.names(top10))) %>%
cor() %>%
corrplot()
train_x %>%
select(row.names(top10)) %>%
featurePlot(., train_y)
According to the correlation plot, ManufacturingProcess32 has the highest positive correlation with Yield. Two of the top ten variables are negatively correlated with Yield.
It seems that the biological predictors have a positive relationship with the yield, while the manufacturing processes vary with their relationship with the yield. For instance, ManufacturingProcess31 is mostly centered around a value and ManufacturingProcess36 has levels.