Data624HW8
Jay Lee
2023-04-23
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: ggplot2## Loading required package: latticeset.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:
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 = 19.
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 2.5683461Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
I add MARS model as comparison.
MARS has a better performance with lower RMSE and higher Rsquared.
Also, MARS selet X1-X5 plus X6.
knn_fit<- train(x=trainingData$x,y=trainingData$y, method = "knn", preProcess = c("center", "scale"), tuneLength = 10)
mars_fit<- train(x = trainingData$x, y = trainingData$y, method = "earth",
preProcess = c("center", "scale"), tuneLength = 10)## Loading required package: earth## Loading required package: Formula## Loading required package: plotmo## Loading required package: plotrix## Loading required package: TeachingDemosknnPred <- predict(knnModel, newdata = testData$x)
postResample(pred = knnPred, obs = testData$y)## RMSE Rsquared MAE
## 3.2040595 0.6819919 2.5683461marsPred <- predict(mars_fit, newdata = testData$x)
postResample(pred = marsPred, obs = testData$y)## RMSE Rsquared MAE
## 1.776575 0.872700 1.358367varImp((knn_fit))## loess r-squared variable importance
##
## Overall
## X4 100.0000
## X1 95.5047
## X2 89.6186
## X5 45.2170
## X3 29.9330
## X9 6.3299
## X10 5.5182
## X8 3.2527
## X6 0.8884
## X7 0.0000varImp((mars_fit))## earth variable importance
##
## Overall
## X1 100.00
## X4 82.78
## X2 64.18
## X5 40.21
## X3 28.14
## X6 0.007.5. 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.
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)
library(RANN)
estdata <- preProcess(ChemicalManufacturingProcess, "knnImpute")
chemdata <- predict(estdata, ChemicalManufacturingProcess)
chemdata <- chemdata[, -nearZeroVar(chemdata)]
ch_index <- createDataPartition(chemdata$Yield, p = .8, list = FALSE)
trainx <- chemdata[ch_index, -1]
trainy <- chemdata[ch_index, 1]
testx <- chemdata[-ch_index, -1]
testy <- chemdata[-ch_index, 1]- Which nonlinear regression model gives the optimal resampling and test set performance?
4 model was create, KNN, MARS, Neural Network and SVM. SVM is the preferable model since it has the lowest RMSE and 2nd highest Rsquared.
knnModelC <- train(trainx, trainy,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModelC## 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 0.7663166 0.4267656 0.6103715
## 7 0.7620570 0.4334381 0.6085957
## 9 0.7606730 0.4383593 0.6112026
## 11 0.7637161 0.4369804 0.6112213
## 13 0.7702460 0.4338978 0.6183574
## 15 0.7757912 0.4294637 0.6246815
## 17 0.7793033 0.4268261 0.6265074
## 19 0.7857410 0.4192433 0.6310951
## 21 0.7876125 0.4219037 0.6318536
## 23 0.7926886 0.4171869 0.6380595
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 9.MARS
set.seed(777)
marsGrid <- expand.grid(.degree = 1:2, .nprune = 2:38) #p.165
marsTuneC <- train(trainx, trainy,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsTuneC## Multivariate Adaptive Regression Spline
##
## 144 samples
## 56 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 128, 128, 131, 129, 130, 130, ...
## Resampling results across tuning parameters:
##
## degree nprune RMSE Rsquared MAE
## 1 2 0.7571365 0.4384424 0.5890694
## 1 3 0.6209432 0.6171880 0.5001021
## 1 4 0.6125417 0.6149778 0.4974624
## 1 5 0.6086976 0.6193433 0.5002367
## 1 6 0.6112194 0.6220564 0.5001236
## 1 7 0.5833192 0.6601286 0.4707630
## 1 8 0.5954229 0.6548777 0.4773941
## 1 9 0.5908520 0.6688769 0.4641166
## 1 10 0.5922562 0.6729081 0.4589072
## 1 11 0.6046974 0.6547836 0.4578478
## 1 12 0.6021642 0.6585914 0.4529060
## 1 13 0.5982064 0.6552723 0.4642189
## 1 14 0.5993061 0.6579914 0.4710578
## 1 15 0.6003698 0.6629187 0.4686158
## 1 16 0.5846706 0.6788217 0.4511537
## 1 17 0.5860203 0.6802277 0.4511400
## 1 18 0.5849925 0.6786591 0.4492209
## 1 19 0.5849925 0.6786591 0.4492209
## 1 20 0.5849925 0.6786591 0.4492209
## 1 21 0.5849925 0.6786591 0.4492209
## 1 22 0.5849925 0.6786591 0.4492209
## 1 23 0.5849925 0.6786591 0.4492209
## 1 24 0.5849925 0.6786591 0.4492209
## 1 25 0.5849925 0.6786591 0.4492209
## 1 26 0.5849925 0.6786591 0.4492209
## 1 27 0.5849925 0.6786591 0.4492209
## 1 28 0.5849925 0.6786591 0.4492209
## 1 29 0.5849925 0.6786591 0.4492209
## 1 30 0.5849925 0.6786591 0.4492209
## 1 31 0.5849925 0.6786591 0.4492209
## 1 32 0.5849925 0.6786591 0.4492209
## 1 33 0.5849925 0.6786591 0.4492209
## 1 34 0.5849925 0.6786591 0.4492209
## 1 35 0.5849925 0.6786591 0.4492209
## 1 36 0.5849925 0.6786591 0.4492209
## 1 37 0.5849925 0.6786591 0.4492209
## 1 38 0.5849925 0.6786591 0.4492209
## 2 2 0.7571365 0.4384424 0.5890694
## 2 3 0.6389273 0.5892642 0.5242676
## 2 4 0.6247083 0.5967464 0.5181887
## 2 5 0.6152453 0.6099734 0.5094673
## 2 6 0.6315286 0.6175948 0.5132345
## 2 7 0.5898082 0.6628224 0.4812386
## 2 8 0.5773520 0.6710845 0.4698095
## 2 9 0.5696061 0.6801360 0.4646234
## 2 10 0.5848359 0.6626196 0.4765138
## 2 11 0.5694192 0.6879157 0.4649821
## 2 12 0.5634597 0.6920767 0.4564068
## 2 13 0.5640860 0.6887346 0.4576884
## 2 14 0.5697185 0.6883572 0.4531146
## 2 15 0.5919275 0.6731146 0.4702897
## 2 16 0.6167678 0.6586522 0.4874193
## 2 17 0.6198244 0.6591712 0.4951976
## 2 18 0.6108703 0.6755062 0.4937246
## 2 19 0.6094393 0.6755931 0.4951412
## 2 20 0.5999557 0.6841172 0.4869600
## 2 21 0.5945895 0.6886322 0.4845766
## 2 22 0.5967784 0.6969207 0.4885505
## 2 23 0.6044671 0.6961590 0.4938530
## 2 24 0.5974824 0.6984470 0.4878614
## 2 25 0.5974824 0.6984470 0.4878614
## 2 26 0.6126017 0.6966108 0.4983812
## 2 27 0.6126017 0.6966108 0.4983812
## 2 28 0.6126017 0.6966108 0.4983812
## 2 29 0.6126017 0.6966108 0.4983812
## 2 30 0.6126017 0.6966108 0.4983812
## 2 31 0.6126017 0.6966108 0.4983812
## 2 32 0.6126017 0.6966108 0.4983812
## 2 33 0.6126017 0.6966108 0.4983812
## 2 34 0.6126017 0.6966108 0.4983812
## 2 35 0.6126017 0.6966108 0.4983812
## 2 36 0.6126017 0.6966108 0.4983812
## 2 37 0.6126017 0.6966108 0.4983812
## 2 38 0.6126017 0.6966108 0.4983812
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 12 and degree = 2.tooHigh <- findCorrelation(cor(trainx), cutoff = .75) #p.163
train_x_nnet <- trainx[, -tooHigh]
test_x_nnet <- testx[, -tooHigh]
nnetGrid <- expand.grid(.decay = c(0, 0.01, .1),
.size = c(1:10))
ctrl <- trainControl(method = "cv", number = 10)
set.seed(777)
nnetTune <- train(train_x_nnet, trainy,
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
## 37 predictor
##
## Pre-processing: centered (37), scaled (37)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 128, 128, 131, 129, 130, 130, ...
## Resampling results across tuning parameters:
##
## decay size RMSE Rsquared MAE
## 0.00 1 1.2163074 0.3136695 0.8295162
## 0.00 2 1.0184622 0.3425981 0.8306294
## 0.00 3 1.0404403 0.3236338 0.8351704
## 0.00 4 1.4772631 0.1900349 1.1425086
## 0.00 5 1.5537554 0.1408028 1.2441427
## 0.00 6 1.3815231 0.2459903 1.0489401
## 0.00 7 1.1281311 0.3749623 0.9003955
## 0.00 8 1.1785144 0.2517516 0.8937500
## 0.00 9 1.0238042 0.3370734 0.7755890
## 0.00 10 0.9625880 0.3778590 0.7979186
## 0.01 1 0.8925761 0.3749872 0.7201091
## 0.01 2 1.0991754 0.2649870 0.8736625
## 0.01 3 1.1941416 0.2768011 0.9195688
## 0.01 4 1.1621122 0.3145344 0.9196618
## 0.01 5 1.0365727 0.3331291 0.8558945
## 0.01 6 0.9795945 0.3633113 0.7548577
## 0.01 7 0.8225926 0.4790358 0.6654358
## 0.01 8 0.8018038 0.4532290 0.6450975
## 0.01 9 0.8380935 0.4301259 0.6838092
## 0.01 10 0.8684114 0.4008014 0.6963858
## 0.10 1 0.7830112 0.4641974 0.6340823
## 0.10 2 0.9658965 0.3696723 0.7614198
## 0.10 3 0.9047863 0.4129417 0.6915331
## 0.10 4 0.9122687 0.4288613 0.7240297
## 0.10 5 0.9064383 0.3640349 0.7144517
## 0.10 6 0.9134298 0.3881678 0.7042020
## 0.10 7 0.7538289 0.5057989 0.6098617
## 0.10 8 0.8314645 0.4263620 0.6550355
## 0.10 9 0.8200322 0.4127259 0.6588729
## 0.10 10 0.8372057 0.4245492 0.6801626
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were size = 7 and decay = 0.1.set.seed(777)
svmRTune <- train(trainx, trainy,
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: 128, 128, 131, 129, 130, 130, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 0.7468900 0.5336373 0.6009050
## 0.50 0.6821002 0.5748808 0.5559519
## 1.00 0.6212096 0.6302097 0.5074772
## 2.00 0.5995841 0.6471372 0.4826408
## 4.00 0.6036013 0.6314314 0.4873860
## 8.00 0.6177886 0.6152405 0.4983435
## 16.00 0.6172946 0.6165007 0.4984441
## 32.00 0.6172946 0.6165007 0.4984441
## 64.00 0.6172946 0.6165007 0.4984441
## 128.00 0.6172946 0.6165007 0.4984441
## 256.00 0.6172946 0.6165007 0.4984441
## 512.00 0.6172946 0.6165007 0.4984441
## 1024.00 0.6172946 0.6165007 0.4984441
## 2048.00 0.6172946 0.6165007 0.4984441
##
## Tuning parameter 'sigma' was held constant at a value of 0.0141958
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.0141958 and C = 2.knnPred <- predict(knnModelC, testx)
postResample(pred = knnPred, obs = testy)## RMSE Rsquared MAE
## 0.7366242 0.4621173 0.5993582marsPred <- predict(marsTuneC, testx)
postResample(pred = marsPred, obs = testy)## RMSE Rsquared MAE
## 0.7051474 0.5211750 0.5567698svmRPred <- predict(svmRTune, testx)
postResample(pred = svmRPred, obs = testy)## RMSE Rsquared MAE
## 0.6917503 0.5138788 0.5533730nnPred <- predict(nnetTune, testx)
postResample(pred = nnPred, obs = testy)## RMSE Rsquared MAE
## 0.7500078 0.4442805 0.6100607- 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 Top 10 important predictors are the same for 2 model.
varImp(svmRTune)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 56)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 94.77
## BiologicalMaterial06 82.41
## ManufacturingProcess17 78.71
## BiologicalMaterial03 78.71
## ManufacturingProcess31 71.77
## ManufacturingProcess06 68.38
## BiologicalMaterial12 66.59
## BiologicalMaterial02 65.86
## ManufacturingProcess09 65.47
## ManufacturingProcess36 62.43
## BiologicalMaterial04 53.46
## BiologicalMaterial11 52.00
## ManufacturingProcess11 51.25
## ManufacturingProcess33 47.86
## BiologicalMaterial08 46.86
## ManufacturingProcess29 43.34
## ManufacturingProcess02 41.33
## ManufacturingProcess30 40.49
## BiologicalMaterial09 40.28set.seed(777)
larsTune <- train(trainx, trainy,
method = "lars",
metric = "Rsquared",
tuneLength = 20,
trControl = ctrl,
preProc = c("center", "scale"))
lars_predict <- predict(larsTune, testx)plot(varImp(larsTune), top = 10) 
plot(varImp(svmRTune), top = 10)
- 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
As we can see ManufacturingProcess36 and 13 has neg correlations while other has prositve.
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, unionlibrary(corrplot)## corrplot 0.92 loadedtop10 <- varImp(svmRTune)$importance %>%
arrange(-Overall) %>%
head(10)
chemdata %>%
select(c("Yield", row.names(top10))) %>%
cor() %>%
corrplot()