Data624_Hw8
Vijaya Cherukuri
11/09/2020
7.2 Friedman (1991) introduced several benchmark data sets create by simulation. One of these simulations used the following nonlinear equation to create data:
\[ y = 10 sin(\pi x_1x_2) + 20(x_3 − 0.5)^2 + 10x_4 + 5x_5 + N(0, \sigma^2) \]
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:
library(mlbench)
library(kableExtra)
library(caret)## Loading required package: lattice## Loading required package: ggplot2set.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)Knn Model :
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.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 2.5683461knnAccuracy <- postResample(pred = knnPred, obs = testData$y)SVM Model :
SvmRadialModel <- train(x = trainingData$x,
y = trainingData$y,
method = "svmRadial",
tuneLength=10,
preProc = c("center", "scale"))
SvmRadialModel## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## Pre-processing: centered (10), scaled (10)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 2.545335 0.7804647 2.015121
## 0.50 2.319786 0.7965148 1.830009
## 1.00 2.188349 0.8119636 1.726027
## 2.00 2.103655 0.8241314 1.655842
## 4.00 2.066879 0.8294322 1.631051
## 8.00 2.052681 0.8313929 1.623550
## 16.00 2.049867 0.8318312 1.621820
## 32.00 2.049867 0.8318312 1.621820
## 64.00 2.049867 0.8318312 1.621820
## 128.00 2.049867 0.8318312 1.621820
##
## Tuning parameter 'sigma' was held constant at a value of 0.06802164
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06802164 and C = 16.svmRadialPred <- predict(SvmRadialModel, newdata = testData$x)
#Use postResample function to get the test set performance values
postResample(pred = svmRadialPred, obs = testData$y)## RMSE Rsquared MAE
## 2.0864652 0.8236735 1.5854649svmRadialAccuracy <- postResample(pred = svmRadialPred, obs = testData$y)Neural Networks Model :
nnetGrid <- expand.grid(.decay=c(0, 0.01, 0.1, 0.5, 0.9),
.size=c(1, 10, 15, 20),
.bag=FALSE)nnet <- train(x = trainingData$x,
y = trainingData$y,
method = "avNNet",
tuneGrid = nnetGrid,
preProc = c("center", "scale"),
trace=FALSE,
linout=TRUE,
maxit=500)
nnet## 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 2.589902 0.7324403 2.014450
## 0.00 10 3.082909 0.6581005 2.345346
## 0.00 15 2.666684 0.7202739 2.135432
## 0.00 20 2.634394 0.7273102 2.116201
## 0.01 1 2.567190 0.7363972 1.993021
## 0.01 10 2.714509 0.7118994 2.173845
## 0.01 15 2.435366 0.7634558 1.935705
## 0.01 20 2.346376 0.7801847 1.850974
## 0.10 1 2.580129 0.7336990 2.000459
## 0.10 10 2.528971 0.7492960 2.003431
## 0.10 15 2.309856 0.7879857 1.823430
## 0.10 20 2.289300 0.7922572 1.799799
## 0.50 1 2.620985 0.7251648 2.034073
## 0.50 10 2.389468 0.7734132 1.893293
## 0.50 15 2.248817 0.7979988 1.778851
## 0.50 20 2.257951 0.7973906 1.768133
## 0.90 1 2.649162 0.7195330 2.057453
## 0.90 10 2.339031 0.7803865 1.849270
## 0.90 15 2.247236 0.7980673 1.774157
## 0.90 20 2.248629 0.7989807 1.770371
##
## 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 = 0.9 and bag = FALSE.nnetPred <- predict(nnet, newdata = testData$x)
# Use postResample function to get the test set performance values
postResample(pred = nnetPred, obs = testData$y)## RMSE Rsquared MAE
## 1.894755 0.856176 1.441820nnetAccuracy <- postResample(pred = nnetPred, obs = testData$y)MARS Model :
marsGrid <- expand.grid(.degree=1:2,
.nprune=2:20)
marsModel <- train(x = trainingData$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: TeachingDemosmarsModel## Multivariate Adaptive Regression Spline
##
## 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:
##
## degree nprune RMSE Rsquared MAE
## 1 2 4.416233 0.2187486 3.630918
## 1 3 3.716956 0.4495842 2.981710
## 1 4 2.829383 0.6783492 2.285456
## 1 5 2.500824 0.7495068 2.002628
## 1 6 2.377484 0.7721684 1.897799
## 1 7 1.964112 0.8402449 1.540566
## 1 8 1.849850 0.8590078 1.446960
## 1 9 1.760287 0.8734284 1.379551
## 1 10 1.745434 0.8758361 1.358502
## 1 11 1.721573 0.8784527 1.333126
## 1 12 1.741003 0.8757510 1.341457
## 1 13 1.762562 0.8729306 1.355525
## 1 14 1.779852 0.8704986 1.376884
## 1 15 1.796118 0.8682289 1.386179
## 1 16 1.801970 0.8673854 1.392551
## 1 17 1.801970 0.8673854 1.392551
## 1 18 1.801970 0.8673854 1.392551
## 1 19 1.801970 0.8673854 1.392551
## 1 20 1.801970 0.8673854 1.392551
## 2 2 4.421087 0.2136318 3.622047
## 2 3 3.738888 0.4424544 3.004778
## 2 4 2.878704 0.6647839 2.315845
## 2 5 2.556082 0.7359640 2.039618
## 2 6 2.448842 0.7590378 1.941403
## 2 7 2.076809 0.8217323 1.631062
## 2 8 1.919449 0.8477097 1.506739
## 2 9 1.750995 0.8737507 1.383005
## 2 10 1.589003 0.8961730 1.262151
## 2 11 1.503925 0.9077370 1.182204
## 2 12 1.459184 0.9131975 1.139692
## 2 13 1.465062 0.9130346 1.140011
## 2 14 1.440347 0.9155789 1.123394
## 2 15 1.462849 0.9131866 1.141915
## 2 16 1.477622 0.9110707 1.148694
## 2 17 1.479198 0.9110804 1.152683
## 2 18 1.480900 0.9108744 1.151674
## 2 19 1.475717 0.9116473 1.146616
## 2 20 1.475717 0.9116473 1.146616
##
## 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.marsPred <- predict(marsModel, newdata = testData$x)
# Use postResample function to get the test set performance values
postResample(pred = marsPred, obs = testData$y)## RMSE Rsquared MAE
## 1.2779993 0.9338365 1.0147070marsAccuracy <- postResample(pred = marsPred, obs = testData$y)Compare the models to find the best one
accuracies <- rbind(marsAccuracy,svmRadialAccuracy,knnAccuracy,nnetAccuracy)
rownames(accuracies )<- c("MARS","SVM","KNN", "NeuralNet")
accuracies%>%
kable() %>%
kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| MARS | 1.277999 | 0.9338365 | 1.014707 |
| SVM | 2.086465 | 0.8236735 | 1.585465 |
| KNN | 3.204060 | 0.6819919 | 2.568346 |
| NeuralNet | 1.894755 | 0.8561760 | 1.441820 |
Based on the above results we can find that MARS has the best accuracy when compared to KNN, SVM and Neural Network. RMSE value of MARS model is much lower when compared to other models.
Does MARS select the informative predictors (those named X1–X5)?
We can use varImp function to find out the informative variables that MARS model selected.
varImp(marsModel)## earth variable importance
##
## Overall
## X1 100.00
## X4 75.40
## X2 49.00
## X5 15.72
## X3 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)
library(Amelia)## Loading required package: Rcpp## ##
## ## Amelia II: Multiple Imputation
## ## (Version 1.7.6, built: 2019-11-24)
## ## Copyright (C) 2005-2020 James Honaker, Gary King and Matthew Blackwell
## ## Refer to http://gking.harvard.edu/amelia/ for more information
## ##library(missForest)## Loading required package: randomForest## randomForest 4.6-14## Type rfNews() to see new features/changes/bug fixes.##
## Attaching package: 'randomForest'## The following object is masked from 'package:ggplot2':
##
## margin## Loading required package: foreach## Loading required package: itertools## Loading required package: iteratorslibrary(nnet)
library(corrgram)##
## Attaching package: 'corrgram'## The following object is masked from 'package:lattice':
##
## panel.filllibrary(ggplot2)Use missmap function to find the missing values
data(ChemicalManufacturingProcess)
missmap(ChemicalManufacturingProcess, col = c("red", "lightgreen"))
In previous instance i used bagImpute to replace missing values. This time i will use missForest function to impute missing values
Original_df <- ChemicalManufacturingProcess
Imputed_df <- missForest(Original_df)## missForest iteration 1 in progress...done!
## missForest iteration 2 in progress...done!
## missForest iteration 3 in progress...done!
## missForest iteration 4 in progress...done!
## missForest iteration 5 in progress...done!df <- Imputed_df$ximpSplit the data into test and training dataset
data <- df[, 2:58]
target <- df[,1]
train <- createDataPartition(target, p=0.75)
train_pred <- data[train$Resample1,]
train_target <- target[train$Resample]
test_pred <- data[-train$Resample1,]
test_target <- target[-train$Resample1]
control <- trainControl(method = "cv", number=10)(a) Which nonlinear regression model gives the optimal resampling and test set performance?
Knn Model :
set.seed(1)
knnModel <- train(x = train_pred,
y = train_target,
method = "knn",
tuneLength = 10)
knnModel## k-Nearest Neighbors
##
## 132 samples
## 57 predictor
##
## No pre-processing
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 132, 132, 132, 132, 132, 132, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.722106 0.2152919 1.400652
## 7 1.696309 0.2170071 1.382955
## 9 1.696599 0.2075752 1.378167
## 11 1.689217 0.2104431 1.371941
## 13 1.681288 0.2175028 1.371749
## 15 1.675921 0.2192990 1.365006
## 17 1.675337 0.2211266 1.358179
## 19 1.673974 0.2225404 1.359908
## 21 1.675164 0.2198840 1.358720
## 23 1.681794 0.2145043 1.356062
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 19.knn.Predict <- predict(knnModel, newdata = test_pred)
# Use PostResample function
postResample(pred = knn.Predict, obs = test_target)## RMSE Rsquared MAE
## 1.6591763 0.3388975 1.3013158knnAccuracy <- postResample(pred = knn.Predict, obs = test_target)Svm Model :
set.seed(1)
svmModel <- train(x = train_pred,
y = train_target,
method='svmRadial',
tuneLength=14,
trControl = trainControl(method = "cv"),
preProc = c("center", "scale"))
svmModel## Support Vector Machines with Radial Basis Function Kernel
##
## 132 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 119, 119, 119, 118, 119, 118, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.426859 0.4930403 1.1553090
## 0.50 1.299018 0.5499885 1.0420521
## 1.00 1.189160 0.6049125 0.9514879
## 2.00 1.141489 0.6329814 0.9128962
## 4.00 1.141144 0.6373573 0.9156004
## 8.00 1.115601 0.6531527 0.8996701
## 16.00 1.113618 0.6546783 0.8981607
## 32.00 1.113618 0.6546783 0.8981607
## 64.00 1.113618 0.6546783 0.8981607
## 128.00 1.113618 0.6546783 0.8981607
## 256.00 1.113618 0.6546783 0.8981607
## 512.00 1.113618 0.6546783 0.8981607
## 1024.00 1.113618 0.6546783 0.8981607
## 2048.00 1.113618 0.6546783 0.8981607
##
## Tuning parameter 'sigma' was held constant at a value of 0.01398119
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01398119 and C = 16.svm.Predict <- predict (svmModel, test_pred)
# Use PostResample function
postResample(pred = svm.Predict, obs = test_target)## RMSE Rsquared MAE
## 1.0326586 0.7074344 0.8384875svmRadialAccuracy <- postResample(pred = svm.Predict, obs = test_target)Neural Network Model :
set.seed(1)
nnetModel <- nnet(train_pred,
train_target,
size=5,
decay=0.01,
linout= T,
trace=F,
maxit = 500 ,
MaxNWts = 5 * (ncol(train_pred) + 1) + 5 + 1)
nnetPredict <- predict(nnetModel, test_pred)
# Use PostResample function
postResample(pred = nnetPredict, obs = test_target)## RMSE Rsquared MAE
## 2.9292442 0.1673242 2.3167136nnetAccuracy <- postResample(pred = nnetPredict, obs = test_target)MARS Model :
set.seed(1)
marsModel <- train(x = train_pred,
y = train_target,
method='earth',
tuneLength=10,
trControl = trainControl(method = "cv"))
marsModel## Multivariate Adaptive Regression Spline
##
## 132 samples
## 57 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 119, 119, 119, 118, 119, 118, ...
## Resampling results across tuning parameters:
##
## nprune RMSE Rsquared MAE
## 2 1.414798 0.4364493 1.1195825
## 3 1.190013 0.5992537 0.9679315
## 5 1.119823 0.6159515 0.9068656
## 7 1.147665 0.6119270 0.9529042
## 8 1.167424 0.6026739 0.9707559
## 10 1.159271 0.6099304 0.9829971
## 12 1.226075 0.5719064 1.0406675
## 13 1.259002 0.5538101 1.0751039
## 15 1.305346 0.5356789 1.0979933
## 17 1.305006 0.5382960 1.1034235
##
## Tuning parameter 'degree' was held constant at a value of 1
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 5 and degree = 1.mars.Predict <- predict (marsModel, test_pred)
# Use PostResample function
postResample(pred = mars.Predict, obs = test_target)## RMSE Rsquared MAE
## 1.2250658 0.6062770 0.9706621marsAccuracy <- postResample(pred = mars.Predict, obs = test_target)Compare Models
accuracies1 <- rbind(marsAccuracy,svmRadialAccuracy,knnAccuracy,nnetAccuracy)
rownames(accuracies1)<- c("MARS","SVM","KNN", "NeuralNet")
accuracies1%>%
kable() %>%
kable_styling()| RMSE | Rsquared | MAE | |
|---|---|---|---|
| MARS | 1.225066 | 0.6062770 | 0.9706621 |
| SVM | 1.032659 | 0.7074344 | 0.8384875 |
| KNN | 1.659176 | 0.3388975 | 1.3013158 |
| NeuralNet | 2.929244 | 0.1673242 | 2.3167136 |
SVM model was slightly performing better than MARS model. Here we can observe the least RMSE for SVM.
(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?
varImp(svmModel)## loess r-squared variable importance
##
## only 20 most important variables shown (out of 57)
##
## Overall
## ManufacturingProcess32 100.00
## ManufacturingProcess13 94.26
## ManufacturingProcess17 80.60
## BiologicalMaterial06 67.90
## ManufacturingProcess09 64.43
## BiologicalMaterial03 62.21
## BiologicalMaterial12 58.04
## ManufacturingProcess36 55.82
## ManufacturingProcess06 51.42
## BiologicalMaterial02 50.70
## ManufacturingProcess31 46.60
## ManufacturingProcess11 43.88
## ManufacturingProcess30 39.49
## ManufacturingProcess33 38.76
## ManufacturingProcess12 37.05
## ManufacturingProcess29 36.88
## BiologicalMaterial08 36.68
## BiologicalMaterial11 36.49
## BiologicalMaterial09 33.27
## BiologicalMaterial04 33.09plot(varImp(svmModel), top=10)
Manufacturing process 32 and Maunfacturing process 13 are in top 3 list for the selected SVM model. In top 10 we got 6 Manufacturing processes and 4 Biological processes. By this we can say Manufacturing processes dominates Biological processes.
(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?
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:randomForest':
##
## combine## The following object is masked from 'package:kableExtra':
##
## group_rows## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unioncor_model <- train_pred %>% select(ManufacturingProcess32, ManufacturingProcess13, ManufacturingProcess17, BiologicalMaterial12, BiologicalMaterial03, ManufacturingProcess09, BiologicalMaterial06, ManufacturingProcess36, ManufacturingProcess06, BiologicalMaterial02)
corrgram(cor_model, order=TRUE, upper.panel=panel.cor)