Data HW 8
Mikhail Broomes
2024-11-05
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(AppliedPredictiveModeling)## Warning: package 'AppliedPredictiveModeling' was built under R version 4.3.3library(mlbench)## Warning: package 'mlbench' was built under R version 4.3.3library(caret)## Warning: package 'caret' was built under R version 4.3.3## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 4.3.3## Loading required package: lattice## Warning: package 'lattice' was built under R version 4.3.3library(tidyverse)## Warning: package 'tidyverse' was built under R version 4.3.3## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.1
## ✔ lubridate 1.9.3 ✔ tibble 3.2.1
## ✔ purrr 1.0.2 ✔ tidyr 1.3.1## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ purrr::lift() masks caret::lift()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsset.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:
#KNN
library(caret)
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 2.5683461#MARS
library(earth)## Warning: package 'earth' was built under R version 4.3.3## Loading required package: Formula## Loading required package: plotmo## Warning: package 'plotmo' was built under R version 4.3.3## Loading required package: plotrixmarsModel <- train(x = trainingData$x, y = trainingData$y, method = "earth", tuneLength = 10)
marsModel## Multivariate Adaptive Regression Spline
##
## 200 samples
## 10 predictor
##
## No pre-processing
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 200, 200, 200, 200, 200, 200, ...
## Resampling results across tuning parameters:
##
## nprune RMSE Rsquared MAE
## 2 4.383438 0.2405683 3.597961
## 3 3.645469 0.4745962 2.930453
## 4 2.727602 0.7035031 2.184240
## 6 2.331605 0.7835496 1.833420
## 7 1.976830 0.8421599 1.562591
## 9 1.804342 0.8683110 1.410395
## 10 1.787676 0.8711960 1.386944
## 12 1.821005 0.8670619 1.419893
## 13 1.858688 0.8617344 1.445459
## 15 1.871033 0.8607099 1.457618
##
## 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 = 10 and degree = 1.marsPred <- predict(marsModel, newdata = testData$x)Neural Networks
library(neuralnet)## Warning: package 'neuralnet' was built under R version 4.3.3##
## Attaching package: 'neuralnet'## The following object is masked from 'package:dplyr':
##
## computemodel = neuralnet(
y ~ .,
data = trainingData,
hidden = c(5, 5),
linear.output = FALSE
)
nnPred <- predict(model, newdata = testData$x)SVM
library(kernlab)## Warning: package 'kernlab' was built under R version 4.3.3##
## Attaching package: 'kernlab'## The following object is masked from 'package:purrr':
##
## cross## The following object is masked from 'package:ggplot2':
##
## alphasvmModel <- train(x = trainingData$x, y = trainingData$y, method = "svmRadial", tuneLength = 10)
svmModel## Support Vector Machines with Radial Basis Function Kernel
##
## 200 samples
## 10 predictor
##
## No pre-processing
## 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 2.605094 0.7645344 2.068745
## 0.50 2.383375 0.7824840 1.866226
## 1.00 2.261899 0.7977780 1.760744
## 2.00 2.184218 0.8098223 1.697706
## 4.00 2.156007 0.8134055 1.673019
## 8.00 2.143625 0.8152672 1.669552
## 16.00 2.141395 0.8156733 1.668470
## 32.00 2.141315 0.8156906 1.668308
## 64.00 2.141315 0.8156906 1.668308
## 128.00 2.141315 0.8156906 1.668308
##
## Tuning parameter 'sigma' was held constant at a value of 0.06269697
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.06269697 and C = 32.svmPred <- predict(svmModel, newdata = testData$x)svmPred <- predict(svmModel, newdata = testData$x)
postResample(pred = svmPred, obs = testData$y)## RMSE Rsquared MAE
## 2.0729792 0.8257729 1.5746258Which models appear to give the best performance? Does MARS select the informative predictors (those named X1–X5)?
as.data.frame(rbind(
knn = postResample(pred = knnPred, obs = testData$y),
mars = postResample(pred = predict(marsModel, newdata = testData$x), obs = testData$y),
nn = postResample(pred = predict(model, newdata = testData$x), obs = testData$y),
svm = postResample(pred = svmPred, obs = testData$y)
) )## RMSE Rsquared MAE
## knn 3.204059 0.68199191 2.568346
## mars 1.776575 0.87269996 1.358367
## nn 14.276930 0.01587806 13.386911
## svm 2.072979 0.82577292 1.574626The MARS model has the best performance across all three metrics. It has the lowest RMSE and MAE, and the highest R-squared.
varImp(marsModel)## earth variable importance
##
## Overall
## X1 100.00
## X4 82.78
## X2 64.18
## X5 40.21
## X3 28.14
## X6 0.00The mars model does select the informative predictors. The top 5 predictors are X1, X2, X3, X4, and X5.
7.5. Exercise 6.3 describes data for a chemical manufacturing process. Use the same data imputation, data splitting, and pre-processing steps as before and train several nonlinear regression models.
Data
library(AppliedPredictiveModeling)
data(ChemicalManufacturingProcess)Models
library(caret)
set.seed(200)
data("ChemicalManufacturingProcess")
pre_process_model <- preProcess(ChemicalManufacturingProcess,
method = c("BoxCox", "knnImpute", "center", "scale"))
processed_data <- predict(pre_process_model, ChemicalManufacturingProcess)
processed_data$Yield <- ChemicalManufacturingProcess$Yield
train_indices <- sample(seq_len(nrow(processed_data)), size = floor(0.85 * nrow(processed_data)))
train_data <- processed_data[train_indices, ]
test_data <- processed_data[-train_indices, ]KNN
knn_model <- train(x = train_data[, -1], y = train_data$Yield, method = "knn", preProc = c("center", "scale"), tuneLength = 10)## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07
## Warning in preProcess.default(thresh = 0.95, k = 5, freqCut = 19, uniqueCut =
## 10, : These variables have zero variances: BiologicalMaterial07knn_model## k-Nearest Neighbors
##
## 149 samples
## 57 predictor
##
## Pre-processing: centered (57), scaled (57)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 149, 149, 149, 149, 149, 149, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 1.355320 0.4107029 1.066563
## 7 1.329797 0.4302767 1.045199
## 9 1.324941 0.4390097 1.046048
## 11 1.344199 0.4237490 1.069445
## 13 1.343844 0.4310569 1.073263
## 15 1.357754 0.4248850 1.085066
## 17 1.384597 0.4029840 1.104304
## 19 1.391522 0.4013591 1.107286
## 21 1.397203 0.4016020 1.107565
## 23 1.406312 0.3971716 1.115442
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 9.knn_pred <- predict(knn_model, newdata = test_data[, -1])MARS
mars_Model <- train(x = train_data[, -1], y = train_data$Yield, method = "earth", tuneLength = 10)
mars_Model## Multivariate Adaptive Regression Spline
##
## 149 samples
## 57 predictor
##
## No pre-processing
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 149, 149, 149, 149, 149, 149, ...
## Resampling results across tuning parameters:
##
## nprune RMSE Rsquared MAE
## 2 1.485273 0.3243963 1.164782
## 3 1.329175 0.4628585 1.063063
## 5 1.611336 0.4683647 1.092185
## 7 2.285255 0.4293546 1.211014
## 8 2.425555 0.4193465 1.235860
## 10 2.835064 0.4000325 1.319180
## 12 3.162953 0.3716991 1.402142
## 13 3.180676 0.3571767 1.406669
## 15 3.630383 0.3334311 1.479677
## 17 4.050221 0.3151682 1.545117
##
## Tuning parameter 'degree' was held constant at a value of 1
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 3 and degree = 1.mars_pred <- predict(mars_Model, newdata = test_data[, -1])Neural Networks
library(neuralnet)
model = neuralnet(
Yield ~ .,
data = train_data,
hidden = c(5, 5),
linear.output = FALSE
)nn_pred <- predict(model, newdata = test_data[, -1])SVM
svm_model <- train(x = train_data[, -1], y = train_data$Yield, method = "svmRadial", tuneLength = 10)## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.
## Warning in .local(x, ...): Variable(s) `' constant. Cannot scale data.svm_model## Support Vector Machines with Radial Basis Function Kernel
##
## 149 samples
## 57 predictor
##
## No pre-processing
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 149, 149, 149, 149, 149, 149, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 1.417383 0.4738932 1.1124693
## 0.50 1.308322 0.5290979 1.0197167
## 1.00 1.233713 0.5649867 0.9579879
## 2.00 1.179980 0.5917616 0.9125943
## 4.00 1.162134 0.5986932 0.9020097
## 8.00 1.160849 0.5991925 0.9001530
## 16.00 1.160561 0.5994268 0.8998274
## 32.00 1.160561 0.5994268 0.8998274
## 64.00 1.160561 0.5994268 0.8998274
## 128.00 1.160561 0.5994268 0.8998274
##
## Tuning parameter 'sigma' was held constant at a value of 0.01567433
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.01567433 and C = 16.svm_pred <- predict(svm_model, newdata = test_data[, -1])- Which nonlinear regression model gives the optimal resampling and test set performance?
as.data.frame(rbind(
knn = postResample(pred = knn_pred, obs = test_data$Yield),
mars = postResample(pred = mars_pred, obs = test_data$Yield),
nn = postResample(pred = nn_pred, obs = test_data$Yield),
svm = postResample(pred = svm_pred, obs = test_data$Yield)
) )## RMSE Rsquared MAE
## knn 1.578356 0.4705665 1.4018519
## mars 1.249131 0.6706054 0.9052919
## nn 39.813943 0.1436116 39.7559269
## svm 1.419354 0.6100339 1.2341218The MARS model has the best performance across all three metrics. It has the lowest RMSE and MAE, and the highest R-squared.
- 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(mars_Model)## earth variable importance
##
## Overall
## ManufacturingProcess32 100
## ManufacturingProcess13 0- 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?
preProcess <- preProcess(ChemicalManufacturingProcess,
method = c("BoxCox", "knnImpute", "center", "scale"))
predPreProcess <- predict(preProcess, ChemicalManufacturingProcess)varImp_mars <- varImp(mars_Model)
top_predictors <- varImp_mars$importance %>%
as.data.frame() %>%
rownames_to_column("Predictor") %>%
arrange(desc(Overall)) %>%
head(10) %>%
select(Predictor)
variables <- top_predictors$Predictor
featurePlot(
x = predPreProcess[, variables],
y = predPreProcess$Yield,
plot = "scatter"
)
There seems to be a clear trend in ManufacturingProcess32 but there isn’t a clear trend in the other predictors. However we can see that they are all clusters.