# Load libraries
library(caret)
library(mlbench)
library(earth)
library(kernlab)
library(nnet)
library(AppliedPredictiveModeling)
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.
Tune several models on these data.Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
Based on lowest RSME, highest Rsquared amd lowest MAE, the MARS model performed best to explain the variance in the non linear data. MARS was able to identify/select the most informative predictors X1 - X5 and their degrees of importance relative to X1.
Model | RMSE | R-squared | MAE |
---|---|---|---|
KNN | 3.2040595 | 0.6819919 | 2.5683461 |
MARS | 1.1589948 | 0.9460418 | 0.9250230 |
SVM | 2.0067929 | 0.8366234 | 1.5104528 |
NNET | 2.3663646 | 0.7831041 | 1.7857183 |
set.seed(200)
# Create training data
trainingData <- mlbench.friedman1(200, sd = 1)
# 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
featurePlot(trainingData$x, trainingData$y)
# Create test data
# 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
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.
# 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$x)
#The function ’postResample’ can be used to get the test set performance values
postResample(pred = knnPred, obs = testData$y)
## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461
# Set seed
set.seed(200)
# Generate training/test data
trainingData <- mlbench.friedman1(200, sd = 1)
trainingData$x <- data.frame(trainingData$x)
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)
# MARS tuning grid for degrees and terms
marsGrid <- expand.grid(.degree = 1:2,
.nprune = 2:38)
# Train MARS model with CV
marsTuned <- train(
x = trainingData$x,
y = trainingData$y,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
# View model results
head(marsTuned$results)
## degree nprune RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 2 4.418191 0.2220731 3.693914 0.5810075 0.13108125 0.5115431
## 38 2 2 4.418191 0.2220731 3.693914 0.5810075 0.13108125 0.5115431
## 2 1 3 3.587804 0.4951608 2.883269 0.4425236 0.11284760 0.4103648
## 39 2 3 3.587804 0.4951608 2.883269 0.4425236 0.11284760 0.4103648
## 3 1 4 2.649942 0.7213442 2.111172 0.4508144 0.07395655 0.3700952
## 40 2 4 2.649942 0.7213442 2.111172 0.4508144 0.07395655 0.3700952
## nprune degree
## 51 15 2
## 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 4.418191 0.2220731 3.6939136
## 1 3 3.587804 0.4951608 2.8832687
## 1 4 2.649942 0.7213442 2.1111721
## 1 5 2.291556 0.7964068 1.8323488
## 1 6 2.280375 0.8004019 1.7664797
## 1 7 1.784776 0.8754863 1.3753836
## 1 8 1.682249 0.8891793 1.2994890
## 1 9 1.646664 0.8954811 1.2539206
## 1 10 1.624818 0.8987479 1.2715170
## 1 11 1.610740 0.9019619 1.2576772
## 1 12 1.608672 0.8997193 1.2572225
## 1 13 1.634826 0.8969655 1.2749330
## 1 14 1.637957 0.8967907 1.2768206
## 1 15 1.637957 0.8967907 1.2768206
## 1 16 1.637957 0.8967907 1.2768206
## 1 17 1.637957 0.8967907 1.2768206
## 1 18 1.637957 0.8967907 1.2768206
## 1 19 1.637957 0.8967907 1.2768206
## 1 20 1.637957 0.8967907 1.2768206
## 1 21 1.637957 0.8967907 1.2768206
## 1 22 1.637957 0.8967907 1.2768206
## 1 23 1.637957 0.8967907 1.2768206
## 1 24 1.637957 0.8967907 1.2768206
## 1 25 1.637957 0.8967907 1.2768206
## 1 26 1.637957 0.8967907 1.2768206
## 1 27 1.637957 0.8967907 1.2768206
## 1 28 1.637957 0.8967907 1.2768206
## 1 29 1.637957 0.8967907 1.2768206
## 1 30 1.637957 0.8967907 1.2768206
## 1 31 1.637957 0.8967907 1.2768206
## 1 32 1.637957 0.8967907 1.2768206
## 1 33 1.637957 0.8967907 1.2768206
## 1 34 1.637957 0.8967907 1.2768206
## 1 35 1.637957 0.8967907 1.2768206
## 1 36 1.637957 0.8967907 1.2768206
## 1 37 1.637957 0.8967907 1.2768206
## 1 38 1.637957 0.8967907 1.2768206
## 2 2 4.418191 0.2220731 3.6939136
## 2 3 3.587804 0.4951608 2.8832687
## 2 4 2.649942 0.7213442 2.1111721
## 2 5 2.291556 0.7964068 1.8323488
## 2 6 2.280375 0.8004019 1.7664797
## 2 7 1.803484 0.8735581 1.3868197
## 2 8 1.654427 0.8922574 1.2862049
## 2 9 1.546581 0.9071585 1.2029829
## 2 10 1.521494 0.9109646 1.1652663
## 2 11 1.422808 0.9228462 1.1019263
## 2 12 1.355197 0.9296603 1.0571268
## 2 13 1.244245 0.9405589 0.9808320
## 2 14 1.262326 0.9389157 0.9857827
## 2 15 1.239616 0.9402979 0.9733206
## 2 16 1.255372 0.9388680 0.9884378
## 2 17 1.263593 0.9378911 0.9924776
## 2 18 1.263593 0.9378911 0.9924776
## 2 19 1.263593 0.9378911 0.9924776
## 2 20 1.263593 0.9378911 0.9924776
## 2 21 1.263593 0.9378911 0.9924776
## 2 22 1.263593 0.9378911 0.9924776
## 2 23 1.263593 0.9378911 0.9924776
## 2 24 1.263593 0.9378911 0.9924776
## 2 25 1.263593 0.9378911 0.9924776
## 2 26 1.263593 0.9378911 0.9924776
## 2 27 1.263593 0.9378911 0.9924776
## 2 28 1.263593 0.9378911 0.9924776
## 2 29 1.263593 0.9378911 0.9924776
## 2 30 1.263593 0.9378911 0.9924776
## 2 31 1.263593 0.9378911 0.9924776
## 2 32 1.263593 0.9378911 0.9924776
## 2 33 1.263593 0.9378911 0.9924776
## 2 34 1.263593 0.9378911 0.9924776
## 2 35 1.263593 0.9378911 0.9924776
## 2 36 1.263593 0.9378911 0.9924776
## 2 37 1.263593 0.9378911 0.9924776
## 2 38 1.263593 0.9378911 0.9924776
##
## 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.
# Predict test set
marsPred <- predict(marsTuned, newdata = testData$x)
# Test set performance metrics
postResample(pred = marsPred, obs = testData$y)
## RMSE Rsquared MAE
## 1.1589948 0.9460418 0.9250230
# Variable importance
varImp_mars <- varImp(marsTuned)
# Plot variable importance
plot(varImp_mars, main = "Variable Importance for MARS Model")
# Set seed
set.seed(200)
# Generate training/test data
trainingData <- mlbench.friedman1(200, sd = 1)
trainingData$x <- data.frame(trainingData$x)
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)
# SVM tuning grid
svmGrid <- expand.grid(
sigma = kernlab::sigest(as.matrix(trainingData$x))[1],
C = 2^seq(-2, 11, length = 14))
# Train SVM model with CV
svmTuned <- train(
x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
tuneGrid = svmGrid,
preProc = c("center", "scale"),
trControl = trainControl(method = "cv"),
metric = "RMSE")
# Model results
head(svmTuned$results)
## sigma C RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 0.03413237 0.25 2.571925 0.7743376 2.071077 0.2400015 0.08920597 0.2627740
## 2 0.03413237 0.50 2.330949 0.7891793 1.860996 0.3092339 0.08345690 0.2851714
## 3 0.03413237 1.00 2.190281 0.8041334 1.740172 0.3448926 0.07427752 0.3022471
## 4 0.03413237 2.00 2.033562 0.8267485 1.604770 0.3201126 0.06069353 0.3073858
## 5 0.03413237 4.00 1.921808 0.8440954 1.516867 0.2992432 0.05308139 0.3068404
## 6 0.03413237 8.00 1.830444 0.8581127 1.445103 0.2865026 0.05416399 0.2843860
## sigma C
## 6 0.03413237 8
# Test set predictions & metrics
svmPred <- predict(svmTuned, newdata = testData$x)
postResample(pred = svmPred, obs = testData$y)
## RMSE Rsquared MAE
## 2.0067929 0.8366234 1.5104528
# Variable importance
svmVarImp <- varImp(svmTuned)
# Variable importance
plot(svmVarImp, main = "Variable Importance for SVM Model")
## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 8
##
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.0341323677577858
##
## Number of Support Vectors : 147
##
## Objective Function Value : -155.5642
## Training error : 0.02686
## [1] 147
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.0341323677577858
# Set seed
set.seed(200)
# Generate training/test data
trainingData <- mlbench.friedman1(200, sd = 1)
trainingData$x <- data.frame(trainingData$x)
testData <- mlbench.friedman1(5000, sd = 1)
testData$x <- data.frame(testData$x)
# Neural net tuning grid
nnetGrid <- expand.grid(
size = 1:10,
decay = c(0, 0.01, 0.1))
# Train neural net with CV
nnetTuned <- train(
x = trainingData$x,
y = trainingData$y,
method = "nnet",
tuneGrid = nnetGrid,
preProc = c("center", "scale"),
trControl = trainControl(method = "cv"),
linout = TRUE,
trace = FALSE,
maxit = 500,
MaxNWts = 10 * (ncol(trainingData$x) + 1) + 10 + 1)
# Model results
head(nnetTuned$results)
## size decay RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.00 2.437191 0.7652028 1.911389 0.3778913 0.05749024 0.3244792
## 2 1 0.01 2.432988 0.7659100 1.905811 0.3816866 0.05623267 0.3274606
## 3 1 0.10 2.440477 0.7640614 1.909477 0.3993041 0.05547645 0.3385159
## 4 2 0.00 2.615029 0.7370369 2.113315 0.4106364 0.06701454 0.4449996
## 5 2 0.01 2.558425 0.7398896 2.055602 0.4925137 0.08833787 0.3978898
## 6 2 0.10 2.616198 0.7315751 2.102083 0.2575369 0.05807522 0.2347142
## size decay
## 7 3 0
# Test set predictions
nnetPred <- predict(nnetTuned, newdata = testData$x)
postResample(pred = nnetPred, obs = testData$y)
## RMSE Rsquared MAE
## 2.3663646 0.7831041 1.7857183
# Variable importance
nnetVarImp <- varImp(nnetTuned)
# Variable importance plot
plot(nnetVarImp, main = "Variable Importance for Neural Network Model")
Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
data("ChemicalManufacturingProcess")
# Separate predictors and outcome variables
predictors <- ChemicalManufacturingProcess[, !names(ChemicalManufacturingProcess) %in% "Yield"]
outcome <- ChemicalManufacturingProcess$Yield
# Create a KNN imputation
imputation_model <- preProcess(predictors, method = "knnImpute")
# Apply the imputation model to fill in missing values
predictors_imputed <- predict(imputation_model, newdata = predictors)
# Recombine the imputed predictor variables with the original outcome
data_imputed <- cbind(predictors_imputed, Yield = outcome)
Which nonlinear regression model gives the optimal resampling and test set performance?
Based on these performance metrics, MARS is the best performing model, with the lowest RMSE (1.13), highest R-squared (0.64), and lowest MAE (0.88). SVM is next with moderate performance as well. KNN and NNET show weaker performance, particularly KNN with the lowest R-squared of 0.42.
However, the MARS R-squared valued explaining roughly 64% of the variance only has moderate predictive power, I would not recommend it as a model.
Model | RMSE | R-squared | MAE |
---|---|---|---|
KNN | 1.4383785 | 0.4220841 | 1.2071970 |
MARS | 1.1285523 | 0.6370613 | 0.8805678 |
SVM | 1.2382642 | 0.5849938 | 0.9442940 |
NNET | 1.4275480 | 0.5169966 | 1.0926083 |
# Set seed
set.seed(200)
# Split data into training/test sets (using 75% for training)
trainIndex <- createDataPartition(data_imputed$Yield, p = 0.75, list = FALSE)
trainData <- data_imputed[trainIndex, ]
testData <- data_imputed[-trainIndex, ]
# Separate predictors and outcome for training data
x_train <- trainData[, !names(trainData) %in% "Yield"]
y_train <- trainData$Yield
# Separate predictors and outcome for test data
x_test <- testData[, !names(testData) %in% "Yield"]
y_test <- testData$Yield
# Train KNN model with CV
knnModel <- train(
x = x_train,
y = y_train,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10,
trControl = trainControl(method = "cv"))
# Model results
head(knnModel$results)
## k RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 5 1.359972 0.4721301 1.059979 0.3049140 0.1524457 0.2180544
## 2 7 1.348804 0.4750832 1.079011 0.2993717 0.1690338 0.2336064
## 3 9 1.333911 0.4919537 1.066791 0.2839310 0.1728645 0.1978271
## 4 11 1.337299 0.5005712 1.093707 0.2833658 0.1750383 0.1960178
## 5 13 1.341294 0.4967197 1.096107 0.2992423 0.1751587 0.2004236
## 6 15 1.374989 0.4665072 1.131406 0.2955441 0.1848535 0.1996309
## k
## 3 9
# Test set predictions & metrics
knnPred <- predict(knnModel, newdata = x_test)
postResample(pred = knnPred, obs = y_test)
## RMSE Rsquared MAE
## 1.4383785 0.4220841 1.2071970
## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess17 80.31
## BiologicalMaterial06 75.09
## ManufacturingProcess13 74.47
## BiologicalMaterial03 67.11
## ManufacturingProcess06 66.44
## ManufacturingProcess36 66.30
## BiologicalMaterial12 65.94
## ManufacturingProcess09 65.29
## BiologicalMaterial02 55.94
## ManufacturingProcess31 53.57
## ManufacturingProcess29 47.17
## ManufacturingProcess33 45.79
## BiologicalMaterial08 44.98
## ManufacturingProcess11 41.72
## ManufacturingProcess02 41.58
## BiologicalMaterial11 41.18
## BiologicalMaterial04 40.31
## BiologicalMaterial09 35.12
## BiologicalMaterial01 34.15
# Set seed
set.seed(200)
# Split data into training/test sets
trainIndex <- createDataPartition(data_imputed$Yield, p = 0.75, list = FALSE)
trainData <- data_imputed[trainIndex, ]
testData <- data_imputed[-trainIndex, ]
# Separate predictors and outcome for training data
x_train <- trainData[, !names(trainData) %in% "Yield"]
y_train <- trainData$Yield
# Separate predictors and outcome for test data
x_test <- testData[, !names(testData) %in% "Yield"]
y_test <- testData$Yield
# MARS tuning grid
marsGrid <- expand.grid(
.degree = 1:2,
.nprune = 2:38)
# Train MARS model with CV
marsTuned <- train(
x = x_train,
y = y_train,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
# Model results
head(marsTuned$results)
## degree nprune RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 2 1.373863 0.4641852 1.0914312 0.3101635 0.1490830 0.2047572
## 38 2 2 1.371014 0.4674806 1.0894104 0.3117095 0.1502947 0.2045887
## 2 1 3 1.213194 0.5946863 0.9923436 0.3130848 0.1314158 0.2698188
## 39 2 3 1.151606 0.6229947 0.9534037 0.1997636 0.1070254 0.1976835
## 3 1 4 1.150495 0.6372393 0.9417116 0.2546411 0.1345659 0.2198082
## 40 2 4 1.307310 0.5448013 1.0289397 0.3326252 0.1694827 0.2465038
## nprune degree
## 3 4 1
# Test set predictions & metrics
marsPred <- predict(marsTuned, newdata = x_test)
postResample(pred = marsPred, obs = y_test)
## RMSE Rsquared MAE
## 1.1285523 0.6370613 0.8805678
## earth variable importance
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess09 46.54
## ManufacturingProcess13 0.00
# Set seed
set.seed(200)
# Split data into training/test sets
trainIndex <- createDataPartition(data_imputed$Yield, p = 0.75, list = FALSE)
trainData <- data_imputed[trainIndex, ]
testData <- data_imputed[-trainIndex, ]
# Separate predictors and outcome for training data
x_train <- trainData[, !names(trainData) %in% "Yield"]
y_train <- trainData$Yield
# Separate predictors and outcome for test data
x_test <- testData[, !names(testData) %in% "Yield"]
y_test <- testData$Yield
# SVM tuning grid
svmGrid <- expand.grid(
sigma = kernlab::sigest(as.matrix(x_train))[1],
C = 2^seq(-2, 11, length = 14))
# Train SVM model with CV
svmTuned <- train(
x = x_train,
y = y_train,
method = "svmRadial",
tuneGrid = svmGrid,
preProc = c("center", "scale"),
trControl = trainControl(method = "cv"),
metric = "RMSE")
# Model results
head(svmTuned$results)
## sigma C RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 0.005113201 0.25 1.418520 0.4930477 1.1657529 0.2605062 0.11517217 0.1865589
## 2 0.005113201 0.50 1.308933 0.5279864 1.0802562 0.2223870 0.09848656 0.1781604
## 3 0.005113201 1.00 1.243171 0.5520036 1.0154208 0.1737705 0.10351787 0.1428009
## 4 0.005113201 2.00 1.212312 0.5663325 0.9661878 0.1539465 0.12856239 0.1375778
## 5 0.005113201 4.00 1.167104 0.5958077 0.9242702 0.1646867 0.15385490 0.1596887
## 6 0.005113201 8.00 1.173760 0.5949558 0.9337971 0.2042906 0.17586608 0.1817306
## sigma C
## 5 0.005113201 4
# Test set predictions & metrics
svmPred <- predict(svmTuned, newdata = x_test)
postResample(pred = svmPred, obs = y_test)
## RMSE Rsquared MAE
## 1.2382642 0.5849938 0.9442940
## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess17 80.31
## BiologicalMaterial06 75.09
## ManufacturingProcess13 74.47
## BiologicalMaterial03 67.11
## ManufacturingProcess06 66.44
## ManufacturingProcess36 66.30
## BiologicalMaterial12 65.94
## ManufacturingProcess09 65.29
## BiologicalMaterial02 55.94
## ManufacturingProcess31 53.57
## ManufacturingProcess29 47.17
## ManufacturingProcess33 45.79
## BiologicalMaterial08 44.98
## ManufacturingProcess11 41.72
## ManufacturingProcess02 41.58
## BiologicalMaterial11 41.18
## BiologicalMaterial04 40.31
## BiologicalMaterial09 35.12
## BiologicalMaterial01 34.15
## Support Vector Machine object of class "ksvm"
##
## SV type: eps-svr (regression)
## parameter : epsilon = 0.1 cost C = 4
##
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.00511320051817451
##
## Number of Support Vectors : 111
##
## Objective Function Value : -138.8385
## Training error : 0.154714
## [1] 111
## Gaussian Radial Basis kernel function.
## Hyperparameter : sigma = 0.00511320051817451
# Split data into training/test sets
trainIndex <- createDataPartition(data_imputed$Yield, p = 0.75, list = FALSE)
trainData <- data_imputed[trainIndex, ]
testData <- data_imputed[-trainIndex, ]
# Separate predictors and outcome for training data
chemx_train <- trainData[, !names(trainData) %in% "Yield"]
chemy_train <- trainData$Yield
# Separate predictors and outcome for test data
x_test <- testData[, !names(testData) %in% "Yield"]
y_test <- testData$Yield
# Neural net tuning grid
nnetGrid <- expand.grid(
size = 1:10,
decay = c(0, 0.001, 0.01, 0.1, 0.3, 0.5))
# Train neural net with CV
nnetTuned <- train(
x = chemx_train,
y = chemy_train,
method = "nnet",
tuneGrid = nnetGrid,
preProc = c("center", "scale", "nzv"),
trControl = trainControl(method = "cv"),
linout = TRUE,
trace = FALSE,
maxit = 500,
MaxNWts = 10 * (ncol(x_train) + 1) + 10 + 1)
# Model results
head(nnetTuned$results)
## size decay RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.000 1.755169 0.1768797 1.455565 0.3146626 0.1631902 0.1918572
## 2 1 0.001 1.856906 0.2455660 1.499637 0.3691828 0.2032322 0.2770922
## 3 1 0.010 1.644305 0.4661764 1.293050 0.4449435 0.1776262 0.2982960
## 4 1 0.100 1.409446 0.4911999 1.148571 0.3571027 0.2047787 0.3208863
## 5 1 0.300 2.042982 0.3583754 1.435583 1.2455139 0.2235291 0.6888214
## 6 1 0.500 2.403655 0.3376808 1.602566 1.5849201 0.2295900 0.7899551
## size decay
## 4 1 0.1
# Test set predictions & metrics
nnetPred <- predict(nnetTuned, newdata = x_test)
postResample(pred = nnetPred, obs = y_test)
## RMSE Rsquared MAE
## 1.4548170 0.4936753 1.1268198
## nnet variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess25 100.000
## ManufacturingProcess26 68.272
## ManufacturingProcess29 51.321
## ManufacturingProcess32 50.918
## ManufacturingProcess18 46.904
## ManufacturingProcess20 44.807
## ManufacturingProcess33 30.757
## BiologicalMaterial11 29.972
## ManufacturingProcess27 28.433
## BiologicalMaterial12 23.543
## ManufacturingProcess31 18.689
## ManufacturingProcess10 12.612
## BiologicalMaterial05 11.702
## BiologicalMaterial01 11.407
## ManufacturingProcess04 11.239
## ManufacturingProcess09 11.091
## ManufacturingProcess13 10.590
## ManufacturingProcess17 10.528
## ManufacturingProcess45 10.513
## BiologicalMaterial09 9.928
*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?
The MARS model was the best performing model. It selected only 3 manufacturing processes as the most important variables. In comparison to the linear PLS and Lasso models, all models identified ManufacturingProcess32 as the most important predictor. The other MARS variables ManufacturingProcess09 and ManufacturingProcess13 also appeared in the linear models but were interspersed between other manufacturing and biological predictors. The MARS model did not identify any biological variables.
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?
The MARS model identified three key manufacturing process variables (09, 13, and 32) as the most important predictors of yield; none of the biological materials were selected as significant predictors. The model achieved a moderate fit with an R-squared of 0.658. The model is additive suggesting that each manufacturing process variable independently affects the yield where their relationships with yield change.
The plots reveal distinct nonlinear relationships between the three manufacturing processes and yield. ManufacturingProcess13 shows a positive relationship with yield until it plateaus. ManufacturingProcess09 shows a negative relationship with yield until reaching a threshold. ManufacturingProcess32 demonstrates a positive impact on yield after crossing a threshold.
## Call: earth(x=data.frame[132,57], y=c(38,42.44,42.0...), keepxy=TRUE, degree=1,
## nprune=4)
##
## coefficients
## (Intercept) 39.718231
## h(0.995762-ManufacturingProcess09) -0.809311
## h(-1.18977-ManufacturingProcess13) 2.230866
## h(ManufacturingProcess32- -0.827442) 1.240221
##
## Selected 4 of 22 terms, and 3 of 57 predictors (nprune=4)
## Termination condition: RSq changed by less than 0.001 at 22 terms
## Importance: ManufacturingProcess32, ManufacturingProcess09, ...
## Number of terms at each degree of interaction: 1 3 (additive model)
## GCV 1.282191 RSS 151.7746 GRSq 0.6246959 RSq 0.6582876
# Plot MARS model significant predictors
plotmo(marsTuned$finalModel,
main="Manufacturing Process Effects",
degree1.names=c("Manufacturing Process 13",
"Manufacturing Process 09",
"Manufacturing Process 32"),
xlab=c("Process 13 Value", "Process 09 Value", "Process 32 Value"),
ylab="Response",
caption="",
col.response="darkgray",
lwd=1.5)
## plotmo grid: BiologicalMaterial01 BiologicalMaterial02 BiologicalMaterial03
## -0.156068 -0.06661163 -0.08122839
## BiologicalMaterial04 BiologicalMaterial05 BiologicalMaterial06
## -0.1404558 -0.02095088 -0.07218167
## BiologicalMaterial07 BiologicalMaterial08 BiologicalMaterial09
## -0.1313107 0.0003357286 -0.03626614
## BiologicalMaterial10 BiologicalMaterial11 BiologicalMaterial12
## -0.2180196 -0.1811404 -0.006316853
## ManufacturingProcess01 ManufacturingProcess02 ManufacturingProcess03
## 0.1056672 0.5214313 0.3319348
## ManufacturingProcess04 ManufacturingProcess05 ManufacturingProcess06
## 0.3424324 -0.09641047 -0.2229103
## ManufacturingProcess07 ManufacturingProcess08 ManufacturingProcess09
## -0.9580199 0.8941637 0.1292558
## ManufacturingProcess10 ManufacturingProcess11 ManufacturingProcess12
## -0.09005212 0.02020002 -0.4806937
## ManufacturingProcess13 ManufacturingProcess14 ManufacturingProcess15
## 0.09066017 0.1216248 -0.07580629
## ManufacturingProcess16 ManufacturingProcess17 ManufacturingProcess18
## 0.08160103 -0.1151653 0.06617593
## ManufacturingProcess19 ManufacturingProcess20 ManufacturingProcess21
## -0.1360039 0.07317798 -0.1744786
## ManufacturingProcess22 ManufacturingProcess23 ManufacturingProcess24
## -0.1218132 -0.01031118 -0.3162877
## ManufacturingProcess25 ManufacturingProcess26 ManufacturingProcess27
## 0.06379054 0.05464677 0.06777473
## ManufacturingProcess28 ManufacturingProcess29 ManufacturingProcess30
## 0.7064583 -0.066778 0.03954225
## ManufacturingProcess31 ManufacturingProcess32 ManufacturingProcess33
## 0.1107323 -0.08632349 0.1836771
## ManufacturingProcess34 ManufacturingProcess35 ManufacturingProcess36
## 0.1182687 0.03729394 -0.4269245
## ManufacturingProcess37 ManufacturingProcess38 ManufacturingProcess39
## -0.03063781 0.7174727 0.231727
## ManufacturingProcess40 ManufacturingProcess41 ManufacturingProcess42
## -0.4626528 -0.4405878 0.2027957
## ManufacturingProcess43 ManufacturingProcess44 ManufacturingProcess45
## -0.1865604 0.2946725 0.1522024