DATA624-HW8
Calvin Cox
11/15/2020
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(πx1x2) + 20(x3 − 0.5)2 + 10x4 + 5x5 + N(0, σ2) where the x values are random variables uniformly distributed between [0, 1] (there are also 5 other non-informative variables also created in the simulation). The package mlbench contains a function called mlbench.friedman1 that simulates these data:
library(mlbench)
library(caret)## Loading required package: lattice## Loading required package: ggplot2## Warning in as.POSIXlt.POSIXct(Sys.time()): unknown timezone 'zone/tz/2020d.1.0/
## zoneinfo/America/New_York'library(magrittr)
library(dplyr) ##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionset.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.
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 3.565620 0.4887976 2.886629
## 7 3.422420 0.5300524 2.752964
## 9 3.368072 0.5536927 2.715310
## 11 3.323010 0.5779056 2.669375
## 13 3.275835 0.6030846 2.628663
## 15 3.261864 0.6163510 2.621192
## 17 3.261973 0.6267032 2.616956
## 19 3.286299 0.6281075 2.640585
## 21 3.280950 0.6390386 2.643807
## 23 3.292397 0.6440392 2.656080
##
## 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, 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
## 3.1750657 0.6785946 2.5443169SVM
svmRModel <- train(x=trainingData$x, y=trainingData$y,
method="svmRadial",
preProcess=c("center", "scale"),
tuneLength=20)
svmRModel## 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 2.600536 0.7740370 2.070216
## 0.50 2.402691 0.7877362 1.887487
## 1.00 2.277967 0.8018302 1.776410
## 2.00 2.193637 0.8141532 1.712248
## 4.00 2.168900 0.8168553 1.691505
## 8.00 2.170623 0.8164075 1.696651
## 16.00 2.171543 0.8162341 1.696794
## 32.00 2.171543 0.8162341 1.696794
## 64.00 2.171543 0.8162341 1.696794
## 128.00 2.171543 0.8162341 1.696794
## 256.00 2.171543 0.8162341 1.696794
## 512.00 2.171543 0.8162341 1.696794
## 1024.00 2.171543 0.8162341 1.696794
## 2048.00 2.171543 0.8162341 1.696794
## 4096.00 2.171543 0.8162341 1.696794
## 8192.00 2.171543 0.8162341 1.696794
## 16384.00 2.171543 0.8162341 1.696794
## 32768.00 2.171543 0.8162341 1.696794
## 65536.00 2.171543 0.8162341 1.696794
## 131072.00 2.171543 0.8162341 1.696794
##
## Tuning parameter 'sigma' was held constant at a value of 0.06651357
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06651357 and C = 4.svmRPred <- predict(svmRModel, newdata=testData$x)
svmRPR <- postResample(pred=svmRPred, obs=testData$y)
svmRPR## RMSE Rsquared MAE
## 2.0682337 0.8274206 1.5638719Neural Network
library(nnet)
nnetFit <- nnet(trainingData$x, trainingData$y,
size = 5,
decay = 0.01,
linout = TRUE,
trace = FALSE,
maxit = 500,
MaxNWts=5 * (ncol(trainingData$x) + 1) + 5 + 1)
nnetPred=predict(nnetFit,testData$x)
postResample(pred = nnetPred, obs = testData$y)## RMSE Rsquared MAE
## 2.499253 0.761204 1.893948###MARS Unfortunately the MARS package did not work for me. Could not install for some reason.
#library(earth)
#marsFit <- earth(trainingData$x, trainingData$y)#marsGrid = expand.grid(.degree = 1:2, .nprune = 2:38)
#marsModel = train(x = trainingData$x,
# y = trainingData$y,
# method = "earth",
# tuneGrid = marsGrid,
# trControl = trainControl(method = "cv",
# number = 10))Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
## MARS package did not work for me.7.5a)
Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
Which nonlinear regression model gives the optimal resampling and test set performance?
library(AppliedPredictiveModeling)## Warning: package 'AppliedPredictiveModeling' was built under R version 3.4.4data(ChemicalManufacturingProcess)
dim(ChemicalManufacturingProcess)## [1] 176 58knn_model <- preProcess(ChemicalManufacturingProcess, "knnImpute")
df <- predict(knn_model, ChemicalManufacturingProcess)
df <- df%>%select_at(vars(-one_of(nearZeroVar(., names = TRUE))))
in_train <- createDataPartition(df$Yield, times = 1, p = 0.8, list = FALSE)
train_df <- df[in_train, ]
test_df <- df[-in_train, ]###KNN
knn_model <- train(
Yield ~ ., data = train_df, method = "knn",
center = TRUE,
scale = TRUE,
trControl = trainControl("cv", number = 10),
tuneLength = 25
)
knn_model## k-Nearest Neighbors
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 130, 131, 129, 129, 130, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 0.7260160 0.5267677 0.5769830
## 7 0.7468102 0.5074432 0.5905056
## 9 0.7575675 0.5078606 0.6096355
## 11 0.7572407 0.5158351 0.6128133
## 13 0.7615613 0.5243049 0.6103254
## 15 0.7745523 0.4999495 0.6224218
## 17 0.7849278 0.4829650 0.6313283
## 19 0.7918989 0.4765270 0.6377654
## 21 0.7984768 0.4621544 0.6445015
## 23 0.8049482 0.4596135 0.6493005
## 25 0.8082630 0.4608712 0.6523860
## 27 0.8121306 0.4630014 0.6543677
## 29 0.8178176 0.4607051 0.6577542
## 31 0.8229837 0.4553180 0.6621410
## 33 0.8237594 0.4616403 0.6618034
## 35 0.8350231 0.4426253 0.6680268
## 37 0.8403128 0.4353314 0.6700182
## 39 0.8451615 0.4364068 0.6753105
## 41 0.8467969 0.4317912 0.6784253
## 43 0.8489007 0.4361050 0.6796290
## 45 0.8504775 0.4432354 0.6803492
## 47 0.8560303 0.4417025 0.6853719
## 49 0.8639161 0.4245858 0.6918424
## 51 0.8673473 0.4229959 0.6966843
## 53 0.8672099 0.4305794 0.6965331
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 5.knn_predictions <- predict(knn_model, test_df)
postResample(pred = knn_predictions, obs = test_df$Yield)## RMSE Rsquared MAE
## 0.5886968 0.6212766 0.4728644###SVM
SVM_model <- train(
Yield ~ ., data = train_df, method = "svmRadial",
center = TRUE,
scale = TRUE,
trControl = trainControl(method = "cv"),
tuneLength = 25
)
SVM_model## Support Vector Machines with Radial Basis Function Kernel
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 131, 129, 129, 128, 130, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 0.7598759 0.5319167 0.6172971
## 0.50 0.7001846 0.5800715 0.5771984
## 1.00 0.6383843 0.6381623 0.5248135
## 2.00 0.6188822 0.6499979 0.4986247
## 4.00 0.6218002 0.6382372 0.5028194
## 8.00 0.6111060 0.6468982 0.4944172
## 16.00 0.6107367 0.6473759 0.4942189
## 32.00 0.6107367 0.6473759 0.4942189
## 64.00 0.6107367 0.6473759 0.4942189
## 128.00 0.6107367 0.6473759 0.4942189
## 256.00 0.6107367 0.6473759 0.4942189
## 512.00 0.6107367 0.6473759 0.4942189
## 1024.00 0.6107367 0.6473759 0.4942189
## 2048.00 0.6107367 0.6473759 0.4942189
## 4096.00 0.6107367 0.6473759 0.4942189
## 8192.00 0.6107367 0.6473759 0.4942189
## 16384.00 0.6107367 0.6473759 0.4942189
## 32768.00 0.6107367 0.6473759 0.4942189
## 65536.00 0.6107367 0.6473759 0.4942189
## 131072.00 0.6107367 0.6473759 0.4942189
## 262144.00 0.6107367 0.6473759 0.4942189
## 524288.00 0.6107367 0.6473759 0.4942189
## 1048576.00 0.6107367 0.6473759 0.4942189
## 2097152.00 0.6107367 0.6473759 0.4942189
## 4194304.00 0.6107367 0.6473759 0.4942189
##
## Tuning parameter 'sigma' was held constant at a value of 0.0141604
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.0141604 and C = 16.svm_predictions <- predict(SVM_model, test_df)
postResample(pred = svm_predictions, obs = test_df$Yield)## RMSE Rsquared MAE
## 0.5855824 0.5852346 0.4627231nnet_grid <- expand.grid(.decay = c(0, 0.01, .1), .size = c(1:10), .bag = FALSE)
nnet_maxnwts <- 5 * ncol(train_df) + 5 + 1
nnet_model <- train(
Yield ~ ., data = train_df, method = "avNNet",
center = TRUE,
scale = TRUE,
tuneGrid = nnet_grid,
trControl = trainControl(method = "cv"),
linout = TRUE,
trace = FALSE,
MaxNWts = nnet_maxnwts,
maxit = 500
)
nnet_model## Model Averaged Neural Network
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 130, 129, 130, 129, 131, 128, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 0.8106764 0.4396935 0.6599616
## 0.00 2 0.8293342 0.4935572 0.6710575
## 0.00 3 0.6974592 0.5851408 0.5428590
## 0.00 4 0.8024840 0.4954802 0.6465848
## 0.00 5 0.6992573 0.6102522 0.5599087
## 0.00 6 NaN NaN NaN
## 0.00 7 NaN NaN NaN
## 0.00 8 NaN NaN NaN
## 0.00 9 NaN NaN NaN
## 0.00 10 NaN NaN NaN
## 0.01 1 0.8242010 0.4996650 0.6640230
## 0.01 2 0.7733056 0.5824999 0.6107808
## 0.01 3 0.7624698 0.5474645 0.6029362
## 0.01 4 0.6408157 0.6488854 0.5076003
## 0.01 5 0.6748181 0.6196554 0.5335214
## 0.01 6 NaN NaN NaN
## 0.01 7 NaN NaN NaN
## 0.01 8 NaN NaN NaN
## 0.01 9 NaN NaN NaN
## 0.01 10 NaN NaN NaN
## 0.10 1 0.7837701 0.5166837 0.6360558
## 0.10 2 0.7223224 0.5842305 0.5797513
## 0.10 3 0.6709759 0.6361814 0.5468508
## 0.10 4 0.6701337 0.6037376 0.5419905
## 0.10 5 0.6330946 0.6480259 0.5209858
## 0.10 6 NaN NaN NaN
## 0.10 7 NaN NaN NaN
## 0.10 8 NaN NaN NaN
## 0.10 9 NaN NaN NaN
## 0.10 10 NaN NaN NaN
##
## 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 = 5, decay = 0.1 and bag = FALSE.nnet_predictions <- predict(nnet_model, test_df)
postResample(pred = nnet_predictions, obs = test_df$Yield)## RMSE Rsquared MAE
## 0.6086070 0.6236451 0.50031127.5b)
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(knn_model, 10)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 97.22
## ManufacturingProcess17 86.08
## BiologicalMaterial06 81.51
## ManufacturingProcess36 80.70
## BiologicalMaterial03 75.79
## BiologicalMaterial12 69.48
## ManufacturingProcess09 68.79
## ManufacturingProcess31 67.25
## BiologicalMaterial02 64.68
## ManufacturingProcess06 64.34
## BiologicalMaterial11 53.81
## ManufacturingProcess33 51.48
## BiologicalMaterial04 45.82
## ManufacturingProcess11 45.27
## ManufacturingProcess30 43.72
## ManufacturingProcess29 43.61
## BiologicalMaterial09 42.90
## BiologicalMaterial01 42.04
## BiologicalMaterial08 40.55varImp(SVM_model, 10)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 97.22
## ManufacturingProcess17 86.08
## BiologicalMaterial06 81.51
## ManufacturingProcess36 80.70
## BiologicalMaterial03 75.79
## BiologicalMaterial12 69.48
## ManufacturingProcess09 68.79
## ManufacturingProcess31 67.25
## BiologicalMaterial02 64.68
## ManufacturingProcess06 64.34
## BiologicalMaterial11 53.81
## ManufacturingProcess33 51.48
## BiologicalMaterial04 45.82
## ManufacturingProcess11 45.27
## ManufacturingProcess30 43.72
## ManufacturingProcess29 43.61
## BiologicalMaterial09 42.90
## BiologicalMaterial01 42.04
## BiologicalMaterial08 40.55varImp(nnet_model, 10)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 97.22
## ManufacturingProcess17 86.08
## BiologicalMaterial06 81.51
## ManufacturingProcess36 80.70
## BiologicalMaterial03 75.79
## BiologicalMaterial12 69.48
## ManufacturingProcess09 68.79
## ManufacturingProcess31 67.25
## BiologicalMaterial02 64.68
## ManufacturingProcess06 64.34
## BiologicalMaterial11 53.81
## ManufacturingProcess33 51.48
## BiologicalMaterial04 45.82
## ManufacturingProcess11 45.27
## ManufacturingProcess30 43.72
## ManufacturingProcess29 43.61
## BiologicalMaterial09 42.90
## BiologicalMaterial01 42.04
## BiologicalMaterial08 40.557.5c
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?
ggplot(train_df, aes(ManufacturingProcess32, Yield)) +
geom_point()
ggplot(train_df, aes(ManufacturingProcess13, Yield)) +
geom_point()
ggplot(train_df, aes(BiologicalMaterial06, Yield)) +
geom_point()
ggplot(train_df, aes(BiologicalMaterial03, Yield)) +
geom_point()