Data 624 Homework 8
Farhana Zahir
4/25/2021
Textbook: Max Kuhn and Kjell Johnson. Applied Predictive Modeling. Springer, New York, 2013.
#Load packages
library(tidyverse)
library(AppliedPredictiveModeling)
library(mlbench)
library(caret)
library(kableExtra)Exercise 7.2
Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data:
\[y = 10 sin(\pi x_1x_2) + 20(x_3 − 0.5)^2 + 10x_4 + 5x_5 + N(0, \sigma^2)\]
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:
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)## 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 Model
set.seed(522)
knn_model <- train(trainingData$x, trainingData$y,
method="knn", preProc = c("center", "scale"),
tuneLength = 10,
trControl = trainControl(method="cv"))
knn_model## k-Nearest Neighbors
##
## 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:
##
## k RMSE Rsquared MAE
## 5 3.228481 0.6036980 2.673649
## 7 3.178096 0.6423871 2.584947
## 9 3.104596 0.6860325 2.523957
## 11 3.076279 0.7050698 2.500496
## 13 3.143129 0.6954467 2.534185
## 15 3.150320 0.7049198 2.548329
## 17 3.166626 0.7156530 2.559601
## 19 3.114165 0.7416197 2.527780
## 21 3.187294 0.7253828 2.589612
## 23 3.205232 0.7244692 2.611535
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 11.knn_model$bestTune## k
## 4 11#Predict
knn_pred <- predict(knn_model, testData$x)
#Performance using postResample
knn_PR <- postResample(knn_pred, testData$y)
knn_PR## RMSE Rsquared MAE
## 3.0977783 0.6726827 2.4817458We will do SVM, Mars and Neural Network models
SVM Model
#SVM Model
set.seed(523)
svm_model <- train(trainingData$x, trainingData$y,
method='svmRadial', preProc = c('center', 'scale'),
tuneLength = 10,
trControl = trainControl(method="cv"))
svm_model## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 180, 180, 180, 180, 180, 180, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.497214 0.7962809 1.994198
## 0.50 2.253418 0.8142203 1.781069
## 1.00 2.074618 0.8356609 1.642539
## 2.00 1.977957 0.8483550 1.550495
## 4.00 1.908320 0.8566316 1.510192
## 8.00 1.894891 0.8598874 1.508390
## 16.00 1.897449 0.8606028 1.514757
## 32.00 1.896460 0.8608272 1.513806
## 64.00 1.896460 0.8608272 1.513806
## 128.00 1.896460 0.8608272 1.513806
##
## Tuning parameter 'sigma' was held constant at a value of 0.06321079
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06321079 and C = 8.svm_model$bestTune## sigma C
## 6 0.06321079 8#Predict
svm_pred <- predict(svm_model, testData$x)
#Performance using postResample
svm_PR <- postResample(svm_pred, testData$y)
svm_PR## RMSE Rsquared MAE
## 2.0116240 0.8341343 1.5603945Multivariate Adaptive Regression Spline (MARS)
#MARS
grid <- expand.grid(.degree = 1:2, .nprune = 2:38)
mars_model = train(x = trainingData$x,
y = trainingData$y,
method = 'earth',
tuneGrid = grid,
trControl = trainControl(method = 'cv',
number = 10))## Loading required package: earth## Loading required package: Formula## Loading required package: plotmo## Loading required package: plotrix## Loading required package: TeachingDemosmars_model## 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 4.450919 0.2143516 3.6734248
## 1 3 3.830815 0.4088022 3.0984800
## 1 4 2.675243 0.7165022 2.1280381
## 1 5 2.365007 0.7816522 1.8525112
## 1 6 2.289095 0.7924202 1.8212124
## 1 7 1.789719 0.8713738 1.4351452
## 1 8 1.694092 0.8901055 1.3234340
## 1 9 1.643706 0.8977664 1.2811407
## 1 10 1.646732 0.8968699 1.2752174
## 1 11 1.607833 0.9021749 1.2575943
## 1 12 1.579022 0.9040028 1.2270389
## 1 13 1.647705 0.8946842 1.2809962
## 1 14 1.643119 0.8958301 1.2696374
## 1 15 1.630169 0.8980775 1.2668929
## 1 16 1.630169 0.8980775 1.2668929
## 1 17 1.630169 0.8980775 1.2668929
## 1 18 1.630169 0.8980775 1.2668929
## 1 19 1.630169 0.8980775 1.2668929
## 1 20 1.630169 0.8980775 1.2668929
## 1 21 1.630169 0.8980775 1.2668929
## 1 22 1.630169 0.8980775 1.2668929
## 1 23 1.630169 0.8980775 1.2668929
## 1 24 1.630169 0.8980775 1.2668929
## 1 25 1.630169 0.8980775 1.2668929
## 1 26 1.630169 0.8980775 1.2668929
## 1 27 1.630169 0.8980775 1.2668929
## 1 28 1.630169 0.8980775 1.2668929
## 1 29 1.630169 0.8980775 1.2668929
## 1 30 1.630169 0.8980775 1.2668929
## 1 31 1.630169 0.8980775 1.2668929
## 1 32 1.630169 0.8980775 1.2668929
## 1 33 1.630169 0.8980775 1.2668929
## 1 34 1.630169 0.8980775 1.2668929
## 1 35 1.630169 0.8980775 1.2668929
## 1 36 1.630169 0.8980775 1.2668929
## 1 37 1.630169 0.8980775 1.2668929
## 1 38 1.630169 0.8980775 1.2668929
## 2 2 4.450919 0.2143516 3.6734248
## 2 3 3.830815 0.4088022 3.0984800
## 2 4 2.675243 0.7165022 2.1280381
## 2 5 2.362650 0.7821262 1.8469646
## 2 6 2.353512 0.7788763 1.8558599
## 2 7 1.836903 0.8667230 1.4606448
## 2 8 1.695449 0.8911044 1.2703273
## 2 9 1.480061 0.9148698 1.1322960
## 2 10 1.384976 0.9286363 1.0674459
## 2 11 1.348503 0.9316396 1.0331269
## 2 12 1.283040 0.9384212 0.9921473
## 2 13 1.271339 0.9389152 0.9850019
## 2 14 1.234497 0.9421385 0.9688186
## 2 15 1.252860 0.9394388 0.9817668
## 2 16 1.265204 0.9382027 0.9871744
## 2 17 1.256888 0.9392198 0.9792562
## 2 18 1.256888 0.9392198 0.9792562
## 2 19 1.256888 0.9392198 0.9792562
## 2 20 1.256888 0.9392198 0.9792562
## 2 21 1.256888 0.9392198 0.9792562
## 2 22 1.256888 0.9392198 0.9792562
## 2 23 1.256888 0.9392198 0.9792562
## 2 24 1.256888 0.9392198 0.9792562
## 2 25 1.256888 0.9392198 0.9792562
## 2 26 1.256888 0.9392198 0.9792562
## 2 27 1.256888 0.9392198 0.9792562
## 2 28 1.256888 0.9392198 0.9792562
## 2 29 1.256888 0.9392198 0.9792562
## 2 30 1.256888 0.9392198 0.9792562
## 2 31 1.256888 0.9392198 0.9792562
## 2 32 1.256888 0.9392198 0.9792562
## 2 33 1.256888 0.9392198 0.9792562
## 2 34 1.256888 0.9392198 0.9792562
## 2 35 1.256888 0.9392198 0.9792562
## 2 36 1.256888 0.9392198 0.9792562
## 2 37 1.256888 0.9392198 0.9792562
## 2 38 1.256888 0.9392198 0.9792562
##
## 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.mars_model$bestTune## nprune degree
## 50 14 2#Predict
mars_pred <- predict(mars_model, testData$x)
#Performance using postResample
mars_PR <- postResample(mars_pred, testData$y)
mars_PR## RMSE Rsquared MAE
## 1.1409847 0.9468881 0.9050118Neural network
set.seed(524)
#Neural network
neural_grid <- expand.grid(.decay=c(0, 0.01, 0.1, 0.5, 0.9),
.size=c(1, 10, 15, 20),
.bag=FALSE)
nnet_model <- train(x = trainingData$x,
y = trainingData$y,
method = "avNNet",
tuneGrid = neural_grid,
preProc = c("center", "scale"),
trace=FALSE,
linout=TRUE,
maxit=500)## Warning: executing %dopar% sequentially: no parallel backend registerednnet_model## Model Averaged Neural Network
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 2.549738 0.7509075 2.004835
## 0.00 10 2.866368 0.7013522 2.213792
## 0.00 15 2.621844 0.7313389 2.079292
## 0.00 20 2.531681 0.7514466 2.006906
## 0.01 1 2.510233 0.7547375 1.947588
## 0.01 10 2.715290 0.7178671 2.153340
## 0.01 15 2.382769 0.7771917 1.873671
## 0.01 20 2.242542 0.8020937 1.762573
## 0.10 1 2.497886 0.7559208 1.935479
## 0.10 10 2.467482 0.7646477 1.951694
## 0.10 15 2.227135 0.8046949 1.747027
## 0.10 20 2.169068 0.8144227 1.691132
## 0.50 1 2.535862 0.7491868 1.966092
## 0.50 10 2.253272 0.7999209 1.773413
## 0.50 15 2.165550 0.8144580 1.680169
## 0.50 20 2.166531 0.8144850 1.684386
## 0.90 1 2.565883 0.7440710 1.991535
## 0.90 10 2.216199 0.8057737 1.731018
## 0.90 15 2.166214 0.8145014 1.686501
## 0.90 20 2.165934 0.8146613 1.676838
##
## Tuning parameter 'bag' was held constant at a value of FALSE
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 15, decay = 0.5 and bag = FALSE.nnet_model$bestTune## size decay bag
## 15 15 0.5 FALSE#Predict
nnet_pred <- predict(nnet_model, testData$x)
#Performance using postResample
nnet_PR <- postResample(nnet_pred, testData$y)
nnet_PR## RMSE Rsquared MAE
## 1.9049449 0.8523567 1.4737367Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
comp<-data.frame(rbind(knn_PR,svm_PR, mars_PR, nnet_PR ))
comp %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
full_width = F)| RMSE | Rsquared | MAE | |
|---|---|---|---|
| knn_PR | 3.097778 | 0.6726827 | 2.4817458 |
| svm_PR | 2.011624 | 0.8341343 | 1.5603945 |
| mars_PR | 1.140985 | 0.9468881 | 0.9050118 |
| nnet_PR | 1.904945 | 0.8523567 | 1.4737367 |
From the table above, Mars has the highest \(r^2\) and explains the highest portion of the variability with X1-X5 informative predictors. The RMSE and MAE are also lowest for the Mars model.
varImp(mars_model)## earth variable importance
##
## Overall
## X1 100.00
## X4 75.24
## X2 48.74
## X5 15.53
## X3 0.00MARS did select the informative predictors with X1 as the most important variable and X3 as the least important variable.
Exercise 7.3
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.
data(ChemicalManufacturingProcess)The matrix process Predictors contains the 57 predictors (12 describing the input biological material and 45 describing the process predictors) for the 176 manufacturing runs. yield contains the percent yield for each run.
Impute missing values
pre_process <-preProcess(ChemicalManufacturingProcess[, -c(1)], method = "knnImpute")
chemical_imp <- predict(pre_process, ChemicalManufacturingProcess[, -c(1)])Remove pairs with correlation abobe 0.90
correlations <- cor(chemical_imp)
highCorr <- findCorrelation(correlations, cutoff = .9)
chemical_imp <- chemical_imp[, -highCorr]Remove near zero variance
near0 <- nearZeroVar(chemical_imp)
chemical_imp <- chemical_imp[,-near0]Split
set.seed(420)
trainRow <- createDataPartition(ChemicalManufacturingProcess$Yield, p=0.8, list=FALSE)
train_X <- chemical_imp[trainRow, ]
train_y <- ChemicalManufacturingProcess$Yield[trainRow]
test_X <- chemical_imp[-trainRow, ]
test_y <- ChemicalManufacturingProcess$Yield[-trainRow]kNN model
set.seed(421)
knn_model <- train(x = train_X,
y = train_y,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knn_model## k-Nearest Neighbors
##
## 144 samples
## 46 predictor
##
## Pre-processing: centered (46), scaled (46)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 144, 144, 144, 144, 144, 144, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.409536 0.3776307 1.137217
## 7 1.377913 0.3997948 1.111391
## 9 1.354392 0.4181553 1.096218
## 11 1.346958 0.4284230 1.101977
## 13 1.330118 0.4492008 1.092717
## 15 1.336551 0.4447446 1.103701
## 17 1.343605 0.4423617 1.107250
## 19 1.346562 0.4434431 1.109906
## 21 1.351739 0.4440656 1.112934
## 23 1.355510 0.4488470 1.117200
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 13.MARS
mars_grid <- expand.grid(.degree=1:2,
.nprune=2:10)
set.seed(1)
mars_model <- train(x = train_X,
y = train_y,
method = "earth",
tuneGrid = mars_grid,
preProc = c("center", "scale"))
mars_model## Multivariate Adaptive Regression Spline
##
## 144 samples
## 46 predictor
##
## Pre-processing: centered (46), scaled (46)
## 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 1.474240 0.3910106 1.172063
## 1 3 1.460509 0.4932852 1.069873
## 1 4 1.295495 0.5303977 1.018799
## 1 5 1.466921 0.5071752 1.060716
## 1 6 1.468971 0.5102403 1.061393
## 1 7 1.496935 0.5110456 1.058454
## 1 8 1.530576 0.5043556 1.070192
## 1 9 1.567819 0.4944717 1.091810
## 1 10 1.585640 0.4868699 1.103259
## 2 2 1.474240 0.3910106 1.172063
## 2 3 1.375036 0.4630789 1.084824
## 2 4 1.366646 0.4916712 1.074731
## 2 5 1.369570 0.4907674 1.072225
## 2 6 1.488398 0.4376590 1.132782
## 2 7 1.480439 0.4498945 1.124263
## 2 8 1.512448 0.4337287 1.144685
## 2 9 1.511464 0.4358332 1.139535
## 2 10 1.551652 0.4223434 1.160382
##
## 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(424)
svm_model <- train(x = train_X,
y = train_y,
method = "svmRadial",
tuneLength=10,
preProc = c("center", "scale"))
svm_model## Support Vector Machines with Radial Basis Function Kernel
##
## 144 samples
## 46 predictor
##
## Pre-processing: centered (46), scaled (46)
## 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 1.381268 0.4446768 1.1241875
## 0.50 1.315707 0.4698946 1.0646536
## 1.00 1.267987 0.4947014 1.0240494
## 2.00 1.244078 0.5110422 1.0075866
## 4.00 1.231350 0.5211678 0.9965781
## 8.00 1.222248 0.5273994 0.9877924
## 16.00 1.221956 0.5276342 0.9875321
## 32.00 1.221956 0.5276342 0.9875321
## 64.00 1.221956 0.5276342 0.9875321
## 128.00 1.221956 0.5276342 0.9875321
##
## Tuning parameter 'sigma' was held constant at a value of 0.01645132
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01645132 and C = 16.Neural Network
set.seed(425)
#Neural Network
neural_grid <- expand.grid(.decay=c(0, 0.01, 0.1),
.size=c(1, 10, 15, 20),
.bag=FALSE)
nnet_model <- train(x = train_X,
y = train_y,
method = "avNNet",
tuneGrid = neural_grid,
preProc = c("center", "scale"),
trace=FALSE,
linout=TRUE,
maxit=500)
nnet_model## Model Averaged Neural Network
##
## 144 samples
## 46 predictor
##
## Pre-processing: centered (46), scaled (46)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 144, 144, 144, 144, 144, 144, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 1.551880 0.32561407 1.267736
## 0.00 10 11.402856 0.06924055 6.467790
## 0.00 15 3.245216 0.13853026 2.196686
## 0.00 20 2.169981 0.22106320 1.637850
## 0.01 1 1.523509 0.40201287 1.203582
## 0.01 10 2.832644 0.20785194 2.025772
## 0.01 15 2.303925 0.24294869 1.639688
## 0.01 20 2.191011 0.22904556 1.557309
## 0.10 1 1.729906 0.36008637 1.272695
## 0.10 10 1.894456 0.27469643 1.436813
## 0.10 15 2.113697 0.26317723 1.409187
## 0.10 20 2.018267 0.27793686 1.394605
##
## 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 = 0.01 and bag = FALSE.(a) Which non-linear regression model gives the optimal resampling and test set performance?
#Predict
svm_pred <- predict(svm_model, test_X)
mars_pred <- predict(mars_model, test_X)
knn_pred <- predict(knn_model, test_X)
nnet_pred <- predict(nnet_model, test_X)
#Performance of test data
svm_rs <- postResample(svm_pred, test_y)
mars_rs <- postResample(mars_pred, test_y)
knn_rs <- postResample(knn_pred, test_y)
nnet_rs <- postResample(nnet_pred, test_y)
comp1 <- data.frame(rbind(svm_rs, mars_rs, knn_rs, nnet_rs))
comp1 %>% kable() %>% kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| svm_rs | 1.114729 | 0.7043145 | 0.9095641 |
| mars_rs | 1.243172 | 0.6172914 | 0.9682900 |
| knn_rs | 1.643258 | 0.3571572 | 1.2951116 |
| nnet_rs | 1.602106 | 0.3688266 | 1.2949973 |
It looks like the SVM model with an \(r^2\) of 70% and lowest RMSE and MAE is the optimal model.
(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?
(svm_imp = varImp(svm_model))## loess r-squared variable importance
##
## only 20 most important variables shown (out of 46)
##
## Overall
## ManufacturingProcess32 100.00
## BiologicalMaterial06 97.23
## ManufacturingProcess13 85.70
## BiologicalMaterial03 81.56
## ManufacturingProcess36 76.57
## ManufacturingProcess17 63.98
## ManufacturingProcess09 62.34
## ManufacturingProcess33 51.74
## BiologicalMaterial08 49.36
## BiologicalMaterial01 48.71
## BiologicalMaterial11 45.32
## ManufacturingProcess06 43.50
## ManufacturingProcess11 41.74
## ManufacturingProcess02 39.72
## BiologicalMaterial09 34.20
## ManufacturingProcess30 32.47
## ManufacturingProcess20 29.05
## ManufacturingProcess12 28.82
## ManufacturingProcess35 24.44
## ManufacturingProcess10 23.52model_pls <- train(x = train_X, y = train_y, method='pls', metric='RMSE',
tuneLength=20, trControl = trainControl(method='cv'))
(pls_imp = varImp(model_pls))## pls variable importance
##
## only 20 most important variables shown (out of 46)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess36 82.94
## ManufacturingProcess13 77.78
## ManufacturingProcess09 71.26
## BiologicalMaterial06 67.87
## ManufacturingProcess17 65.54
## ManufacturingProcess33 63.92
## BiologicalMaterial03 63.18
## BiologicalMaterial08 62.64
## BiologicalMaterial01 57.74
## BiologicalMaterial11 52.39
## ManufacturingProcess06 52.29
## ManufacturingProcess11 51.33
## ManufacturingProcess12 47.71
## ManufacturingProcess28 46.37
## ManufacturingProcess04 42.18
## ManufacturingProcess02 38.43
## BiologicalMaterial10 36.00
## ManufacturingProcess10 35.36
## ManufacturingProcess30 29.08p1<-plot(svm_imp, top=10, main='SVM')
p2<-plot(pls_imp, top=10, main='PLS')
gridExtra::grid.arrange(p1, p2, ncol = 2)
ManufacturingProcess32 dominates both the models. There are 2 biological processes in the top 5 compared to only 1 in the PLS model. Even though the rank of the variables have changed, the top ten list contains the same predictors.
(c) 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?
temp <- svm_imp$importance
temp$predictor <- row.names(temp)
temp <- temp[order(temp$Overall, decreasing = TRUE),]
temp_v = row.names(temp[1:10,])
par(mfrow = c(5,2))
for (i in 1:10){
x = ChemicalManufacturingProcess[,temp_v[i]]
y = ChemicalManufacturingProcess$Yield
plot(x, y, xlab = temp_v[i], ylab = 'Yield')
abline(lm(y~x))
}
The plots show linear relationships between the biological and process predictors and yield.