Lab 9 Homework

3. Consider the Gini index, classification error, and entropy in a simple classification setting with two classes. Create a single plot that displays each of these quantities as a function of p_m1 . The x-axis should display p_m1, ranging from 0 to 1, and the y-axis should display the value of the Gini index, classification error, and entropy. Hint: In a setting with two classes, p_m1 = 1 − p_m2. You could make this plot by hand, but it will be much easier to make in R.

p_m1 <- seq(from = 0, to = 1, length.out = 100)
gini_index <- p_m1 * (1-p_m1)
plot(p_m1, gini_index, type = 'l', main = "Gini Index vs p_m1")

p_m1 <- seq(from = 0, to = 1, length.out = 100)
classification_error <-  1 - pmax(p_m1, 1-p_m1)
plot(p_m1, classification_error, type = 'l', main = "Classification Error vs p_m1")

p_m1 <- seq(from = 0, to = 1, length.out = 100)
cross_entropy <- - p_m1 * log(p_m1) - (1-p_m1) * log((1-p_m1))
plot(p_m1, cross_entropy, type = 'l', main = "Cross Entropy vs p_m1")

8. In the lab, a classification tree was applied to the Carseats data set after converting Sales into a qualitative response variable. Now we will seek to predict Sales using regression trees and related approaches, treating the response as a quantitative variable.

Regression Tree

library(tree)
library(ISLR2)
attach(Carseats)

Carseats <- Carseats %>% 
  dplyr::select(Sales, CompPrice, Income, Advertising,Population, Price, 
                ShelveLoc, Age, Education, Urban, US)

set.seed(1)
train <- sample(1:nrow(Carseats), nrow(Carseats)/2)

tree.carseats <- tree(Sales ~ ., Carseats, subset = train)
summary(tree.carseats)

Regression tree:
tree(formula = Sales ~ ., data = Carseats, subset = train)
Variables actually used in tree construction:
[1] "ShelveLoc"   "Price"       "Age"         "Advertising" "CompPrice"   "US"         
Number of terminal nodes:  18 
Residual mean deviance:  2.167 = 394.3 / 182 
Distribution of residuals:
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-3.88200 -0.88200 -0.08712  0.00000  0.89590  4.09900 

plot(tree.carseats)
text(tree.carseats, pretty = 0)

Comments: Variables used in tree construction include “ShelveLoc”, “Price”, “Age”, “Advertising”, “CompPrice”, “US”, with 18 terminal nodes.

yhat <- predict(tree.carseats, newdata = Carseats[-train, ])
carseats.test <- Carseats[-train, "Sales"]
plot(yhat, carseats.test)
abline(0, 1)

mean((yhat-carseats.test)^2)

[1] 4.922039

sqrt(mean((yhat-carseats.test)^2))

[1] 2.218567

Comments: The test MSE is 4.92, and the square root of the MSE is around 2.21, meaning test predictions are (on average) within around $2.21 of the actual sales value.

Regression Tree with Cross Validation

cv.carseats <- cv.tree(tree.carseats)
plot(cv.carseats$size, cv.carseats$dev, type = "b")

prune.carseats <- prune.tree(tree.carseats, best = 11)
plot(prune.carseats)
text(prune.carseats, pretty = 0)

yhat_pru <- predict(prune.carseats, newdata = Carseats[-train, ])
carseats.test <- Carseats[-train, "Sales"]
plot(yhat_pru, carseats.test)
abline(0, 1)

mean((yhat_pru - carseats.test)^2)

[1] 4.757881

Comments: Yes, after pruning the tree, the test MSE improved.

Bagging

library(randomForest)

set.seed(1)
bag.carseats <- randomForest(Sales ~ ., data = Carseats, 
                             subset = train, mtry = 10, importance = TRUE)

bag.carseats

Call:
 randomForest(formula = Sales ~ ., data = Carseats, mtry = 10,      importance = TRUE, subset = train) 
               Type of random forest: regression
                     Number of trees: 500
No. of variables tried at each split: 10

          Mean of squared residuals: 2.889221
                    % Var explained: 63.26

yhat.bag <- predict(bag.carseats, newdata = Carseats[-train, ])
plot(yhat.bag, carseats.test)
abline(0, 1)

mean((yhat.bag - carseats.test)^2)

[1] 2.605253

Comments: The test set MSE of the bagged regression tree is 2.605, about 55% of that obtained using an optimally-pruned single tree.

importance(bag.carseats)

               %IncMSE IncNodePurity
CompPrice   24.8888481    170.182937
Income       4.7121131     91.264880
Advertising 12.7692401     97.164338
Population  -1.8074075     58.244596
Price       56.3326252    502.903407
ShelveLoc   48.8886689    380.032715
Age         17.7275460    157.846774
Education    0.5962186     44.598731
Urban        0.1728373      9.822082
US           4.2172102     18.073863

varImpPlot(bag.carseats)

Comments: The most crucial variable is Price, which the company charges for car seats at each site.

Random Forest

set.seed(1)
rf.carseats <- randomForest(Sales ~ ., data = Carseats,
    subset = train, mtry = 4, importance = TRUE) # p/3 for regression, sqrt(p) for classification
rf.carseats

Call:
 randomForest(formula = Sales ~ ., data = Carseats, mtry = 4,      importance = TRUE, subset = train) 
               Type of random forest: regression
                     Number of trees: 500
No. of variables tried at each split: 4

          Mean of squared residuals: 3.140932
                    % Var explained: 60.06

yhat.rf <- predict(rf.carseats, newdata = Carseats[-train, ])
plot(yhat.rf, carseats.test)
abline(0, 1)

mean((yhat.rf - carseats.test)^2)

[1] 2.787584

Comments: When m = 4, the MSE of random forest regression is a bit larger than bagging.

importance(rf.carseats)

               %IncMSE IncNodePurity
CompPrice   15.7891655     160.57944
Income       4.1275509     121.12953
Advertising  9.6425758     111.54581
Population  -1.3596645      85.92575
Price       43.4055391     423.06225
ShelveLoc   37.8850232     311.97119
Age         13.8924424     174.18229
Education    0.1960888      62.77782
Urban        0.1393816      12.92952
US           6.3532441      30.42255

varImpPlot(rf.carseats)

Comments: The most important variable is still Price.

set.seed(1)
for (m in 1:10) {
  rf.carseats <- randomForest(Sales ~ ., data = Carseats, subset = train, mtry = m)
  yhat.rf <- predict(rf.carseats, newdata = Carseats[-train, ])
  mse[m] <- mean((yhat.rf - carseats.test)^2)
}

# Plot
plot(1:10, mse, type = "b", xlab = "mtry", ylab = "Test MSE")
points(which.min(mse), mse[which.min(mse)], col = "red", pch = 19)

mse[8]

[1] 2.580009

Comments: As the mtry increases, random forest regression performs better. The smallest test MSE value appears when 8 variables considered at each split equal.

BART

library(BART)

x <- Carseats[, 1:10]
y <- Carseats[, "Sales"]
xtrain <- x[train, ]
ytrain <- y[train]
xtest <- x[-train, ]
ytest <- y[-train]

set.seed(1)
bartfit <- gbart(xtrain, ytrain, x.test = xtest)

*****Calling gbart: type=1
*****Data:
data:n,p,np: 200, 13, 200
y1,yn: 2.781850, 1.091850
x1,x[n*p]: 10.360000, 1.000000
xp1,xp[np*p]: 11.220000, 1.000000
*****Number of Trees: 200
*****Number of Cut Points: 100 ... 1
*****burn,nd,thin: 100,1000,1
*****Prior:beta,alpha,tau,nu,lambda,offset: 2,0.95,0.273474,3,6.77063e-30,7.57815
*****sigma: 0.000000
*****w (weights): 1.000000 ... 1.000000
*****Dirichlet:sparse,theta,omega,a,b,rho,augment: 0,0,1,0.5,1,13,0
*****printevery: 100

MCMC
done 0 (out of 1100)
done 100 (out of 1100)
done 200 (out of 1100)
done 300 (out of 1100)
done 400 (out of 1100)
done 500 (out of 1100)
done 600 (out of 1100)
done 700 (out of 1100)
done 800 (out of 1100)
done 900 (out of 1100)
done 1000 (out of 1100)
time: 3s
trcnt,tecnt: 1000,1000

yhat.bart <- bartfit$yhat.test.mean
mean((ytest - yhat.bart)^2)

[1] 0.184202

Comments: On this data set, the test MSE of BART is lower than the test MSE of random forests and bagging.