H8_624
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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(x_1x_2) + 20(x_3 − 0.5)^2 + 10x_4 + 5x_5 + 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:
set.seed(200)
# install.packages("mlbench")
library(mlbench)
library(caret)## Loading required package: lattice## Loading required package: ggplot2trainingData <- mlbench.friedman1(200, sd = 1)
trainingData$x <- data.frame(trainingData$x)
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)
featurePlot(trainingData$x, trainingData$y)
## Tune several models on these data. For example: # KNN
library(caret)
knnFit <- train(x = trainingData$x,
y = trainingData$y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 12)
knnFit## 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
## 25 3.209660 0.6723668 2.606953
## 27 3.230715 0.6719092 2.620592
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 17.knnPred <- predict(knnFit, newdata = testData$x)
## The function 'postResample' can be used to get the test set
## performance values
knn_model_stats <-postResample(pred = knnPred, obs = testData$y)
knn_model_stats## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461knn_pred_actual <- data.frame(testData$y,knnPred)
ggplot(knn_pred_actual,aes(x=testData$y,y=knnPred))+geom_bin2d(binwidth = c(0.25, 0.25))+scale_fill_gradientn(limits=c(0,30), breaks=seq(0, 40, by=10), colours=rainbow(10))
# SVM
library(caret)
svmFit <- train(x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
tuneLength=10,
preProc = c("center", "scale"))
svmPred <- predict(svmFit, newdata = testData$x)
svmFit## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.547516 0.7726615 2.006413
## 0.50 2.320477 0.7917364 1.820893
## 1.00 2.192587 0.8088843 1.722220
## 2.00 2.116056 0.8196188 1.654683
## 4.00 2.069331 0.8263439 1.617767
## 8.00 2.045729 0.8300195 1.603214
## 16.00 2.041974 0.8306705 1.601002
## 32.00 2.041974 0.8306705 1.601002
## 64.00 2.041974 0.8306705 1.601002
## 128.00 2.041974 0.8306705 1.601002
##
## Tuning parameter 'sigma' was held constant at a value of 0.06148849
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06148849 and C = 16.## The function 'postResample' can be used to get the test set
## performance values
svm_model_stats <- postResample(pred = svmPred, obs = testData$y)
svm_model_stats## RMSE Rsquared MAE
## 2.0700606 0.8262412 1.5723919svm_pred_actual <- data.frame(testData$y,svmPred)
ggplot(knn_pred_actual,aes(x=testData$y,y=svmPred))+geom_bin2d(binwidth = c(0.25, 0.25))+scale_fill_gradientn(limits=c(0,30), breaks=seq(0, 40, by=10), colours=rainbow(10))
Neural Network
library(caret)
nnetGrid <- expand.grid(.decay=c( 0.01), ## , 0, 0.1, 0.5, 0.9
.size=c(1), ##, 10, 15, 20
.bag=FALSE)
nnetFit <- train(x = trainingData$x,
y = trainingData$y,
method = "avNNet",
tuneGrid = nnetGrid,
preProc = c("center", "scale"),
trace=FALSE,
linout=TRUE,
maxit=500)## Warning: executing %dopar% sequentially: no parallel backend registerednnetFit## Model Averaged Neural Network
##
## 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:
##
## RMSE Rsquared MAE
## 2.543819 0.7414866 1.994862
##
## Tuning parameter 'size' was held constant at a value of 1
## Tuning
## parameter 'decay' was held constant at a value of 0.01
## Tuning parameter
## 'bag' was held constant at a value of FALSEnnetPred <- predict(nnetFit, newdata = testData$x)
nnet_model_stats <- postResample(pred = nnetPred, obs = testData$y)
nnet_model_stats## RMSE Rsquared MAE
## 2.6433128 0.7194467 2.0232365nnet_pred_actual <- data.frame(testData$y,nnetPred)
ggplot(knn_pred_actual,aes(x=testData$y,y=nnetPred))+geom_bin2d(binwidth = c(0.25, 0.25))+scale_fill_gradientn(limits=c(0,30), breaks=seq(0, 40, by=10), colours=rainbow(10))
MARS
library(caret)
marsGrid <- expand.grid(.degree=1:2,
.nprune=2:20)
marsFit <- train(x = trainingData$x,
y = trainingData$y,
method = "earth",
tuneGrid = marsGrid,
preProc = c("center", "scale"))## Loading required package: earth## Loading required package: Formula## Loading required package: plotmo## Loading required package: plotrix## Loading required package: TeachingDemosmarsPred <- predict(marsFit, newdata = testData$x)
varImp(marsFit)## earth variable importance
##
## Overall
## X1 100.00
## X4 75.31
## X2 48.86
## X5 15.61
## X3 0.00marsFit$finalModel## Selected 15 of 21 terms, and 5 of 10 predictors (nprune=15)
## Termination condition: Reached nk 21
## Importance: X1, X4, X2, X5, X3, X6-unused, X7-unused, X8-unused, X9-unused, ...
## Number of terms at each degree of interaction: 1 10 4
## GCV 1.726398 RSS 232.166 GRSq 0.9299075 RSq 0.952395## The function 'postResample' can be used to get the test set
## performance values
mars_model_stats <- postResample(pred = marsPred, obs = testData$y)
mars_pred_actual <- data.frame(testData$y,marsPred)
ggplot(knn_pred_actual,aes(x=testData$y,y=marsPred))+geom_bin2d(binwidth = c(0.25, 0.25))+scale_fill_gradientn(limits=c(0,30), breaks=seq(0, 40, by=10), colours=rainbow(10))
Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
Answer: MARS model is best with the highest R^2. It assigned the following importance: X1, X4, X2, X5, X3, X6-unused, X7-unused, X8-unused, X9-unused, X10-unused ; hence no importance has been assigned to the terms which were not relevant.
7.5. Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
(a) Which nonlinear regression model gives the optimal resampling and test set performance?
library(caret)
library(RANN)
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)
set.seed(200)
prepro <- preProcess(ChemicalManufacturingProcess,
method = c("knnImpute"))
newdata <- predict(prepro, ChemicalManufacturingProcess)
trainRow <- createDataPartition(ChemicalManufacturingProcess$Yield, p=0.8, list=FALSE)
colY <- which(colnames(ChemicalManufacturingProcess) == "Yield")
X.train <- newdata[,-colY][trainRow, ]
y.train <- ChemicalManufacturingProcess$Yield[trainRow]
X.test <- newdata[,-colY][-trainRow, ]
y.test <- ChemicalManufacturingProcess$Yield[-trainRow]KNN:
set.seed(200)
knnModel <- train(x = X.train,
y = y.train,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07knnModel## 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 1.365780 0.4528739 1.069655
## 7 1.378378 0.4413042 1.080146
## 9 1.375417 0.4373255 1.080401
## 11 1.380799 0.4355753 1.086821
## 13 1.386536 0.4292945 1.093778
## 15 1.392702 0.4240102 1.103727
## 17 1.400824 0.4188405 1.113934
## 19 1.403301 0.4181174 1.117597
## 21 1.410616 0.4141855 1.127149
## 23 1.421834 0.4068892 1.138288
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 5.Neural network:
nnetGrid <- expand.grid(.decay=c( 0.01), ## , 0, 0.1, 0.5, 0.9
.size=c(1), ##, 10, 15, 20
.bag=FALSE)
set.seed(1)
nnetModel <- train(x = X.train,
y = y.train,
method = "avNNet",
tuneGrid = nnetGrid,
preProc = c("center", "scale"),
trace=FALSE,
linout=TRUE,
maxit=500)## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07nnetModel## Model Averaged Neural Network
##
## 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:
##
## RMSE Rsquared MAE
## 1.767153 0.2912356 1.400779
##
## Tuning parameter 'size' was held constant at a value of 1
## Tuning
## parameter 'decay' was held constant at a value of 0.01
## Tuning parameter
## 'bag' was held constant at a value of FALSEMARS
marsGrid <- expand.grid(.degree=1:2,
.nprune=2:10)
set.seed(200)
marsModel <- train(x = X.train,
y = y.train,
method = "earth",
tuneGrid = marsGrid,
preProc = c("center", "scale"))## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07marsModel## Multivariate Adaptive Regression Spline
##
## 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:
##
## degree nprune RMSE Rsquared MAE
## 1 2 1.411268 0.4098750 1.107151
## 1 3 1.307141 0.4905180 1.041817
## 1 4 1.293540 0.5052587 1.029050
## 1 5 1.347941 0.4793801 1.073904
## 1 6 1.340144 0.4906001 1.065237
## 1 7 1.353643 0.4856596 1.077448
## 1 8 1.375574 0.4804173 1.098254
## 1 9 1.416728 0.4559809 1.115563
## 1 10 1.436290 0.4458512 1.124370
## 2 2 1.411268 0.4098750 1.107151
## 2 3 1.350940 0.4625688 1.062403
## 2 4 1.319006 0.4921283 1.044610
## 2 5 1.382786 0.4831476 1.079984
## 2 6 1.475093 0.4640268 1.096458
## 2 7 1.449447 0.4568105 1.095087
## 2 8 1.795503 0.4321311 1.165932
## 2 9 1.764130 0.4390187 1.158521
## 2 10 1.808197 0.4213890 1.173868
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 4 and degree = 1.SVM
set.seed(200)
svm<- train(x = X.train,
y = y.train,
method = "svmRadial",
tuneLength=10,
preProc = c("center", "scale"))## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.svm## Support Vector Machines with Radial Basis Function Kernel
##
## 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:
##
## C RMSE Rsquared MAE
## 0.25 1.398759 0.4537202 1.1255148
## 0.50 1.302496 0.5047557 1.0358766
## 1.00 1.235115 0.5467465 0.9743106
## 2.00 1.186407 0.5784783 0.9385012
## 4.00 1.164512 0.5926128 0.9296788
## 8.00 1.157286 0.5969286 0.9258664
## 16.00 1.155801 0.5969221 0.9263493
## 32.00 1.155801 0.5969221 0.9263493
## 64.00 1.155801 0.5969221 0.9263493
## 128.00 1.155801 0.5969221 0.9263493
##
## Tuning parameter 'sigma' was held constant at a value of 0.01358238
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01358238 and C = 16.model_list <- resamples(list(KNN=knnModel, NNet=nnetModel, MARS=marsModel, SVM=svm))
summary(model_list)##
## Call:
## summary.resamples(object = model_list)
##
## Models: KNN, NNet, MARS, SVM
## Number of resamples: 25
##
## MAE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## KNN 0.8734492 0.9571554 1.0865124 1.0696545 1.1243159 1.352277 0
## NNet 0.9626489 1.2795109 1.3712333 1.4007793 1.5123413 1.918635 0
## MARS 0.8156142 0.9428349 1.0231779 1.0290501 1.1064248 1.285856 0
## SVM 0.7583403 0.8596967 0.9117069 0.9263493 0.9853586 1.093117 0
##
## RMSE
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## KNN 1.1367187 1.296736 1.347619 1.365780 1.448562 1.684804 0
## NNet 1.2162841 1.641295 1.773364 1.767153 1.877997 2.365619 0
## MARS 1.0404386 1.185886 1.281264 1.293540 1.398731 1.633982 0
## SVM 0.9692539 1.069662 1.135511 1.155801 1.214655 1.358024 0
##
## Rsquared
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## KNN 0.3054950 0.4084353 0.4456296 0.4528739 0.5260487 0.5634062 0
## NNet 0.1053802 0.2118994 0.2853491 0.2912356 0.3541902 0.6115916 0
## MARS 0.2658525 0.4538329 0.4983337 0.5052587 0.5865740 0.6711294 0
## SVM 0.4116634 0.5367938 0.5850332 0.5969221 0.6594510 0.7138705 0Answer: Looking at the mean R^2 across models, SVM appears to have the highest R^2.
(b) 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?
Features <- varImp(svm)
Features## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## BiologicalMaterial06 87.81
## ManufacturingProcess13 78.23
## BiologicalMaterial03 76.45
## BiologicalMaterial12 69.16
## ManufacturingProcess17 68.44
## ManufacturingProcess36 67.83
## ManufacturingProcess31 67.82
## ManufacturingProcess09 64.12
## ManufacturingProcess06 60.80
## ManufacturingProcess29 53.56
## BiologicalMaterial02 53.50
## ManufacturingProcess11 50.24
## BiologicalMaterial11 48.66
## ManufacturingProcess33 46.88
## ManufacturingProcess30 45.42
## BiologicalMaterial09 38.34
## BiologicalMaterial04 37.89
## BiologicalMaterial08 36.89
## ManufacturingProcess12 36.65