Assignment 7
Nicholas Mata jnm705
2025-05-05
Chapter 8 numbers 3, 8, and 9
library(ISLR2)
library(caret)## Loading required package: ggplot2## Loading required package: latticelibrary(rpart)
library(rattle)## Loading required package: tibble## Loading required package: bitops## Rattle: A free graphical interface for data science with R.
## Version 5.5.1 Copyright (c) 2006-2021 Togaware Pty Ltd.
## Type 'rattle()' to shake, rattle, and roll your data.library(BART)## Loading required package: nlme## Loading required package: survival##
## Attaching package: 'survival'## The following object is masked from 'package:caret':
##
## clusterlibrary(tree)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 \(\hat{p}_{m1}\). The x-axis should display \(\hat{p}_{m1}\), 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, \(\hat{p}_{m1} = 1 - \hat{p}_{m2}\). You could make this plot by hand, but it will be much easier to make in R.
# Define a sequence of values for p_hat
p <- seq(0, 1, by = 0.01)
# Calculate Gini index
gini <- 2 * p * (1 - p)
# Calculate classification error
class_error <- 1 - pmax(p, 1 - p)
# Calculate entropy (handle log(0) as 0)
entropy <- -p * log2(p) - (1 - p) * log2(1 - p)
entropy[is.nan(entropy)] <- 0 # replace NaNs with 0 (when p is 0 or 1)
# Plot all three measures
plot(p, gini, type = "l", col = "blue", lwd = 2,
ylab = "Impurity Measure",
xlab = expression(hat(p)[m1]),
main = "Gini Index, Classification Error, and Entropy")
lines(p, class_error, col = "red", lwd = 2, lty = 2)
lines(p, entropy, col = "darkgreen", lwd = 2, lty = 3)
legend("top", legend = c("Gini Index", "Classification Error", "Entropy"),
col = c("blue", "red", "darkgreen"), lty = 1:3, lwd = 2)
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.
data("Carseats")(a) Split the data set into a training set and a test set.
set.seed(1)
train_index = createDataPartition(Carseats$Sales, p = 0.5, list = FALSE)
train = Carseats[train_index, ]
test = Carseats[-train_index, ](b) Fit a regression tree to the training set. Plot the tree, and interpret the results. What test MSE do you obtain?
tree_model = rpart(Sales ~ ., data = train, method = "anova", control = rpart.control(minsplit = 15, cp = 0))
fancyRpartPlot(tree_model, sub = "", cex = 0.6)
tree_preds = predict(tree_model, newdata = test)
mean((test$Sales - tree_preds)^2)## [1] 4.319363MSE is 4.319363
(c) Use cross-validation in order to determine the optimal level of tree complexity. Does pruning the tree improve the test MSE?
printcp(tree_model)##
## Regression tree:
## rpart(formula = Sales ~ ., data = train, method = "anova", control = rpart.control(minsplit = 15,
## cp = 0))
##
## Variables actually used in tree construction:
## [1] Advertising Age CompPrice Income Population Price
## [7] ShelveLoc Urban
##
## Root node error: 1476.1/201 = 7.3439
##
## n= 201
##
## CP nsplit rel error xerror xstd
## 1 0.2084483 0 1.00000 1.00879 0.087596
## 2 0.1138324 1 0.79155 0.81341 0.077576
## 3 0.0636946 2 0.67772 0.82897 0.079899
## 4 0.0528339 3 0.61402 0.84903 0.081825
## 5 0.0490121 4 0.56119 0.84508 0.081696
## 6 0.0325569 5 0.51218 0.84964 0.082104
## 7 0.0315159 6 0.47962 0.81428 0.078891
## 8 0.0311924 7 0.44811 0.81657 0.078878
## 9 0.0282239 8 0.41691 0.79175 0.078147
## 10 0.0206688 9 0.38869 0.72616 0.070345
## 11 0.0192129 10 0.36802 0.71050 0.067181
## 12 0.0172196 11 0.34881 0.69697 0.067441
## 13 0.0171463 12 0.33159 0.69277 0.068072
## 14 0.0113740 13 0.31444 0.66972 0.066233
## 15 0.0110334 15 0.29169 0.67344 0.064341
## 16 0.0100634 16 0.28066 0.67963 0.064373
## 17 0.0092325 17 0.27060 0.66379 0.062633
## 18 0.0084827 18 0.26136 0.66739 0.063314
## 19 0.0071732 19 0.25288 0.66970 0.063334
## 20 0.0054995 21 0.23854 0.65064 0.061569
## 21 0.0054015 22 0.23304 0.64926 0.061590
## 22 0.0000000 23 0.22763 0.64257 0.060180plotcp(tree_model)
prune_model = prune(tree_model, cp = tree_model$cptable[which.min(tree_model$cptable[,"xerror"]), "CP"])
fancyRpartPlot(prune_model, cex = 0.6)
prune_preds = predict(prune_model, newdata = test)
mean((test$Sales - prune_preds)^2)## [1] 4.319363(d) Use the bagging approach in order to analyze this data. What test MSE do you obtain? Use the importance() function to determine which variables are most important.
set.seed(1)
train_control = trainControl("cv", number = 10)
p = ncol(train) - 1
bag_model = train(Sales ~ ., data = train, method = "rf", trControl = train_control, tuneGrid = data.frame(mtry = p), importance = T)
bag_preds = predict(bag_model, newdata = test)
mean((test$Sales - bag_preds)^2)## [1] 2.648432varImp(bag_model)## rf variable importance
##
## Overall
## Price 100.00000
## ShelveLocGood 86.57993
## ShelveLocMedium 44.76368
## CompPrice 44.56485
## Advertising 36.63108
## Age 34.25622
## Income 20.61102
## USYes 12.16029
## Education 4.97453
## UrbanYes 0.01901
## Population 0.00000Test MSE is 2.648432
(e) Use random forests to analyze this data. What test MSE do you obtain? Use the importance() function to determine which variables are most important. Describe the effect of m, the number of variables considered at each split, on the error rate obtained.
set.seed(1)
train_control = trainControl("cv", number = 10)
rf_model = train(Sales ~ ., data = train, method = "rf", trControl = train_control, importance = T)
rf_preds = predict(rf_model, newdata = test)
mean((test$Sales - rf_preds)^2)## [1] 2.646104varImp(rf_model)## rf variable importance
##
## Overall
## Price 100.000
## ShelveLocGood 84.578
## CompPrice 46.822
## ShelveLocMedium 38.684
## Age 34.358
## Advertising 32.218
## Income 23.379
## USYes 8.078
## Education 2.773
## UrbanYes 1.873
## Population 0.000MSE is 2.646104
(f) Now analyze the data using BART, and report your results.
set.seed(1)
bart_model = gbart(x.train = train[, 2:11], y.train = train[, "Sales"], x.test = test[, 2:11]) ## *****Calling gbart: type=1
## *****Data:
## data:n,p,np: 201, 14, 199
## y1,yn: 2.007711, -1.552289
## x1,x[n*p]: 138.000000, 1.000000
## xp1,xp[np*p]: 113.000000, 1.000000
## *****Number of Trees: 200
## *****Number of Cut Points: 64 ... 1
## *****burn,nd,thin: 100,1000,1
## *****Prior:beta,alpha,tau,nu,lambda,offset: 2,0.95,0.2512,3,0.22362,7.49229
## *****sigma: 1.071447
## *****w (weights): 1.000000 ... 1.000000
## *****Dirichlet:sparse,theta,omega,a,b,rho,augment: 0,0,1,0.5,1,14,0
## *****printevery: 100
##
## MCMC
## done 0 (out of 1100)
## done 100 (out of 1100)
## done 200 (out of 1100)
## done 300 (out of 1100)
## done 400 (out of 1100)
## done 500 (out of 1100)
## done 600 (out of 1100)
## done 700 (out of 1100)
## done 800 (out of 1100)
## done 900 (out of 1100)
## done 1000 (out of 1100)
## time: 4s
## trcnt,tecnt: 1000,1000yhat_bart = bart_model$yhat.test.mean
mean((test[, "Sales"] - yhat_bart)^2)## [1] 1.540608ord = order(bart_model$varcount.mean, decreasing = T)
bart_model$varcount.mean[ord]## Price CompPrice Age ShelveLoc1 ShelveLoc2 ShelveLoc3
## 24.042 18.028 17.555 17.025 17.025 16.589
## US1 Urban2 Income US2 Education Population
## 16.516 16.361 16.299 16.210 15.774 15.619
## Urban1 Advertising
## 15.010 14.139The MSE is 1.540608
The output above shows which variables were used the most when splitting the trees. Price was used the most out of the variables.
9
This problem involves the OJ data set which is part of the ISLR2 package.
data(OJ)(a) Create a training set containing a random sample of 800 observations, and a test set containing the remaining observations.
set.seed(1)
oj_index = createDataPartition(OJ$Purchase, p = 0.7, list = FALSE)
oj_train = OJ[oj_index, ]
oj_test = OJ[-oj_index, ](b) Fit a tree to the training data, with Purchase as the response and the other variables as predictors. Use the summary() function to produce summary statistics about the tree, and describe the results obtained. What is the training error rate? How many terminal nodes does the tree have?
tree_oj = rpart(Purchase ~ ., data = oj_train, method = "class", control = rpart.control(minsplit = 15, cp = 0))
summary(tree_oj)## Call:
## rpart(formula = Purchase ~ ., data = oj_train, method = "class",
## control = rpart.control(minsplit = 15, cp = 0))
## n= 750
##
## CP nsplit rel error xerror xstd
## 1 0.4828767123 0 1.0000000 1.0000000 0.04573100
## 2 0.0256849315 1 0.5171233 0.5479452 0.03842134
## 3 0.0239726027 3 0.4657534 0.5650685 0.03885138
## 4 0.0205479452 4 0.4417808 0.5513699 0.03850854
## 5 0.0119863014 5 0.4212329 0.5308219 0.03797614
## 6 0.0102739726 7 0.3972603 0.5136986 0.03751524
## 7 0.0085616438 8 0.3869863 0.5136986 0.03751524
## 8 0.0068493151 10 0.3698630 0.5136986 0.03751524
## 9 0.0057077626 11 0.3630137 0.4931507 0.03694061
## 10 0.0034246575 18 0.3184932 0.4828767 0.03664416
## 11 0.0008561644 22 0.3047945 0.4863014 0.03674367
## 12 0.0000000000 26 0.3013699 0.4863014 0.03674367
##
## Variable importance
## LoyalCH PriceDiff SalePriceMM StoreID PriceMM
## 39 10 8 7 6
## ListPriceDiff DiscMM PctDiscMM PriceCH WeekofPurchase
## 5 5 5 4 4
## SalePriceCH STORE SpecialCH Store7
## 3 3 1 1
##
## Node number 1: 750 observations, complexity param=0.4828767
## predicted class=CH expected loss=0.3893333 P(node) =1
## class counts: 458 292
## probabilities: 0.611 0.389
## left son=2 (425 obs) right son=3 (325 obs)
## Primary splits:
## LoyalCH < 0.5036 to the right, improve=123.09670, (0 missing)
## StoreID < 3.5 to the right, improve= 32.56409, (0 missing)
## PriceDiff < 0.31 to the right, improve= 19.87539, (0 missing)
## SalePriceMM < 2.04 to the right, improve= 17.34553, (0 missing)
## DiscCH < 0.255 to the right, improve= 15.60231, (0 missing)
## Surrogate splits:
## StoreID < 3.5 to the right, agree=0.648, adj=0.188, (0 split)
## PriceMM < 1.89 to the right, agree=0.596, adj=0.068, (0 split)
## WeekofPurchase < 246.5 to the right, agree=0.595, adj=0.065, (0 split)
## ListPriceDiff < 0.035 to the right, agree=0.591, adj=0.055, (0 split)
## PriceCH < 1.72 to the right, agree=0.588, adj=0.049, (0 split)
##
## Node number 2: 425 observations, complexity param=0.0239726
## predicted class=CH expected loss=0.1388235 P(node) =0.5666667
## class counts: 366 59
## probabilities: 0.861 0.139
## left son=4 (404 obs) right son=5 (21 obs)
## Primary splits:
## PriceDiff < -0.39 to the right, improve=12.310240, (0 missing)
## LoyalCH < 0.7645725 to the right, improve= 9.114571, (0 missing)
## DiscMM < 0.57 to the left, improve= 8.216284, (0 missing)
## PctDiscMM < 0.264375 to the left, improve= 8.216284, (0 missing)
## SalePriceMM < 1.585 to the right, improve= 7.859655, (0 missing)
## Surrogate splits:
## DiscMM < 0.72 to the left, agree=0.976, adj=0.524, (0 split)
## SalePriceMM < 1.435 to the right, agree=0.976, adj=0.524, (0 split)
## PctDiscMM < 0.3342595 to the left, agree=0.976, adj=0.524, (0 split)
## SalePriceCH < 2.075 to the left, agree=0.958, adj=0.143, (0 split)
##
## Node number 3: 325 observations, complexity param=0.02568493
## predicted class=MM expected loss=0.2830769 P(node) =0.4333333
## class counts: 92 233
## probabilities: 0.283 0.717
## left son=6 (181 obs) right son=7 (144 obs)
## Primary splits:
## LoyalCH < 0.282272 to the right, improve=14.082430, (0 missing)
## PriceDiff < 0.31 to the right, improve= 8.237351, (0 missing)
## SpecialCH < 0.5 to the right, improve= 5.851637, (0 missing)
## DiscCH < 0.255 to the right, improve= 5.607657, (0 missing)
## PctDiscCH < 0.132882 to the right, improve= 5.607657, (0 missing)
## Surrogate splits:
## STORE < 2.5 to the left, agree=0.609, adj=0.118, (0 split)
## PriceCH < 1.875 to the left, agree=0.594, adj=0.083, (0 split)
## SalePriceCH < 1.875 to the left, agree=0.594, adj=0.083, (0 split)
## StoreID < 3.5 to the right, agree=0.569, adj=0.028, (0 split)
## PriceMM < 2.205 to the left, agree=0.566, adj=0.021, (0 split)
##
## Node number 4: 404 observations, complexity param=0.005707763
## predicted class=CH expected loss=0.1113861 P(node) =0.5386667
## class counts: 359 45
## probabilities: 0.889 0.111
## left son=8 (281 obs) right son=9 (123 obs)
## Primary splits:
## LoyalCH < 0.7053255 to the right, improve=6.210817, (0 missing)
## SalePriceMM < 2.125 to the right, improve=3.358289, (0 missing)
## PriceDiff < 0.31 to the right, improve=3.035729, (0 missing)
## PriceMM < 2.11 to the right, improve=2.730502, (0 missing)
## DiscCH < 0.115 to the right, improve=2.360569, (0 missing)
## Surrogate splits:
## WeekofPurchase < 236.5 to the right, agree=0.705, adj=0.033, (0 split)
## PriceCH < 1.72 to the right, agree=0.703, adj=0.024, (0 split)
## SalePriceMM < 1.585 to the right, agree=0.698, adj=0.008, (0 split)
##
## Node number 5: 21 observations, complexity param=0.01027397
## predicted class=MM expected loss=0.3333333 P(node) =0.028
## class counts: 7 14
## probabilities: 0.333 0.667
## left son=10 (9 obs) right son=11 (12 obs)
## Primary splits:
## LoyalCH < 0.742157 to the right, improve=3.5000000, (0 missing)
## STORE < 1.5 to the right, improve=3.0476190, (0 missing)
## SpecialCH < 0.5 to the left, improve=1.8666670, (0 missing)
## PriceMM < 2.11 to the left, improve=0.9333333, (0 missing)
## ListPriceDiff < 0.135 to the left, improve=0.9333333, (0 missing)
## Surrogate splits:
## PriceCH < 2.04 to the right, agree=0.714, adj=0.333, (0 split)
## DiscMM < 0.47 to the left, agree=0.714, adj=0.333, (0 split)
## SalePriceMM < 1.64 to the right, agree=0.714, adj=0.333, (0 split)
## SalePriceCH < 2.04 to the right, agree=0.714, adj=0.333, (0 split)
## PctDiscMM < 0.2224545 to the left, agree=0.714, adj=0.333, (0 split)
##
## Node number 6: 181 observations, complexity param=0.02568493
## predicted class=MM expected loss=0.4143646 P(node) =0.2413333
## class counts: 75 106
## probabilities: 0.414 0.586
## left son=12 (107 obs) right son=13 (74 obs)
## Primary splits:
## PriceDiff < 0.05 to the right, improve=12.694000, (0 missing)
## SalePriceMM < 1.94 to the right, improve=10.544100, (0 missing)
## DiscMM < 0.22 to the left, improve= 6.766872, (0 missing)
## PctDiscMM < 0.0729725 to the left, improve= 5.775128, (0 missing)
## ListPriceDiff < 0.235 to the right, improve= 4.614858, (0 missing)
## Surrogate splits:
## SalePriceMM < 1.94 to the right, agree=0.950, adj=0.878, (0 split)
## DiscMM < 0.08 to the left, agree=0.834, adj=0.595, (0 split)
## PctDiscMM < 0.038887 to the left, agree=0.834, adj=0.595, (0 split)
## ListPriceDiff < 0.135 to the right, agree=0.779, adj=0.459, (0 split)
## PriceMM < 2.04 to the right, agree=0.773, adj=0.446, (0 split)
##
## Node number 7: 144 observations, complexity param=0.0008561644
## predicted class=MM expected loss=0.1180556 P(node) =0.192
## class counts: 17 127
## probabilities: 0.118 0.882
## left son=14 (94 obs) right son=15 (50 obs)
## Primary splits:
## LoyalCH < 0.0356415 to the right, improve=1.4729200, (0 missing)
## PriceCH < 1.72 to the left, improve=0.7395995, (0 missing)
## PriceMM < 1.74 to the left, improve=0.7114842, (0 missing)
## WeekofPurchase < 273.5 to the left, improve=0.5734127, (0 missing)
## PctDiscMM < 0.1961965 to the left, improve=0.4667313, (0 missing)
## Surrogate splits:
## STORE < 2.5 to the left, agree=0.771, adj=0.34, (0 split)
## PriceCH < 1.975 to the left, agree=0.729, adj=0.22, (0 split)
## SalePriceMM < 2.205 to the left, agree=0.722, adj=0.20, (0 split)
## PriceMM < 2.205 to the left, agree=0.715, adj=0.18, (0 split)
## SalePriceCH < 1.875 to the left, agree=0.694, adj=0.12, (0 split)
##
## Node number 8: 281 observations
## predicted class=CH expected loss=0.05338078 P(node) =0.3746667
## class counts: 266 15
## probabilities: 0.947 0.053
##
## Node number 9: 123 observations, complexity param=0.005707763
## predicted class=CH expected loss=0.2439024 P(node) =0.164
## class counts: 93 30
## probabilities: 0.756 0.244
## left son=18 (60 obs) right son=19 (63 obs)
## Primary splits:
## PriceDiff < 0.265 to the right, improve=7.359504, (0 missing)
## LoyalCH < 0.6919315 to the left, improve=5.959074, (0 missing)
## ListPriceDiff < 0.135 to the right, improve=5.841506, (0 missing)
## SalePriceMM < 2.125 to the right, improve=5.646196, (0 missing)
## PriceMM < 2.11 to the right, improve=3.845598, (0 missing)
## Surrogate splits:
## ListPriceDiff < 0.255 to the right, agree=0.837, adj=0.667, (0 split)
## SalePriceMM < 1.94 to the right, agree=0.813, adj=0.617, (0 split)
## DiscMM < 0.03 to the left, agree=0.780, adj=0.550, (0 split)
## PctDiscMM < 0.0137615 to the left, agree=0.780, adj=0.550, (0 split)
## PriceMM < 2.04 to the right, agree=0.691, adj=0.367, (0 split)
##
## Node number 10: 9 observations
## predicted class=CH expected loss=0.3333333 P(node) =0.012
## class counts: 6 3
## probabilities: 0.667 0.333
##
## Node number 11: 12 observations
## predicted class=MM expected loss=0.08333333 P(node) =0.016
## class counts: 1 11
## probabilities: 0.083 0.917
##
## Node number 12: 107 observations, complexity param=0.02054795
## predicted class=CH expected loss=0.4299065 P(node) =0.1426667
## class counts: 61 46
## probabilities: 0.570 0.430
## left son=24 (53 obs) right son=25 (54 obs)
## Primary splits:
## STORE < 1.5 to the left, improve=3.442309, (0 missing)
## LoyalCH < 0.3084325 to the left, improve=2.349588, (0 missing)
## PriceDiff < 0.49 to the right, improve=1.997351, (0 missing)
## PriceMM < 2.205 to the left, improve=1.879220, (0 missing)
## SalePriceMM < 2.205 to the left, improve=1.879220, (0 missing)
## Surrogate splits:
## StoreID < 5.5 to the right, agree=0.794, adj=0.585, (0 split)
## Store7 splits as RL, agree=0.794, adj=0.585, (0 split)
## SalePriceCH < 1.775 to the left, agree=0.748, adj=0.491, (0 split)
## PriceMM < 2.155 to the left, agree=0.720, adj=0.434, (0 split)
## PriceCH < 1.875 to the left, agree=0.710, adj=0.415, (0 split)
##
## Node number 13: 74 observations, complexity param=0.003424658
## predicted class=MM expected loss=0.1891892 P(node) =0.09866667
## class counts: 14 60
## probabilities: 0.189 0.811
## left son=26 (11 obs) right son=27 (63 obs)
## Primary splits:
## SpecialCH < 0.5 to the right, improve=3.279903, (0 missing)
## LoyalCH < 0.37888 to the right, improve=1.829975, (0 missing)
## PriceDiff < -0.24 to the right, improve=1.579896, (0 missing)
## DiscMM < 0.47 to the left, improve=1.128932, (0 missing)
## PctDiscMM < 0.227263 to the left, improve=1.128932, (0 missing)
##
## Node number 14: 94 observations, complexity param=0.0008561644
## predicted class=MM expected loss=0.1702128 P(node) =0.1253333
## class counts: 16 78
## probabilities: 0.170 0.830
## left son=28 (6 obs) right son=29 (88 obs)
## Primary splits:
## SalePriceMM < 2.205 to the right, improve=1.3941010, (0 missing)
## PriceMM < 2.205 to the right, improve=1.0096780, (0 missing)
## WeekofPurchase < 273.5 to the left, improve=0.9531915, (0 missing)
## ListPriceDiff < 0.135 to the left, improve=0.9508105, (0 missing)
## PctDiscMM < 0.1961965 to the left, improve=0.8741791, (0 missing)
## Surrogate splits:
## PriceMM < 2.205 to the right, agree=0.989, adj=0.833, (0 split)
## ListPriceDiff < 0.43 to the right, agree=0.957, adj=0.333, (0 split)
## PriceDiff < 0.43 to the right, agree=0.947, adj=0.167, (0 split)
##
## Node number 15: 50 observations
## predicted class=MM expected loss=0.02 P(node) =0.06666667
## class counts: 1 49
## probabilities: 0.020 0.980
##
## Node number 18: 60 observations
## predicted class=CH expected loss=0.06666667 P(node) =0.08
## class counts: 56 4
## probabilities: 0.933 0.067
##
## Node number 19: 63 observations, complexity param=0.005707763
## predicted class=CH expected loss=0.4126984 P(node) =0.084
## class counts: 37 26
## probabilities: 0.587 0.413
## left son=38 (58 obs) right son=39 (5 obs)
## Primary splits:
## LoyalCH < 0.6919315 to the left, improve=3.746579, (0 missing)
## StoreID < 5.5 to the right, improve=3.050794, (0 missing)
## Store7 splits as RL, improve=3.050794, (0 missing)
## STORE < 0.5 to the left, improve=3.050794, (0 missing)
## SalePriceMM < 2.155 to the right, improve=1.850027, (0 missing)
##
## Node number 24: 53 observations, complexity param=0.008561644
## predicted class=CH expected loss=0.3018868 P(node) =0.07066667
## class counts: 37 16
## probabilities: 0.698 0.302
## left son=48 (32 obs) right son=49 (21 obs)
## Primary splits:
## PriceDiff < 0.31 to the right, improve=3.4259320, (0 missing)
## ListPriceDiff < 0.235 to the right, improve=2.1396230, (0 missing)
## WeekofPurchase < 249 to the right, improve=1.2176710, (0 missing)
## PriceMM < 2.155 to the right, improve=0.9623333, (0 missing)
## SalePriceMM < 2.155 to the right, improve=0.9623333, (0 missing)
## Surrogate splits:
## SalePriceMM < 2.04 to the right, agree=0.868, adj=0.667, (0 split)
## WeekofPurchase < 249 to the right, agree=0.830, adj=0.571, (0 split)
## PriceMM < 2.04 to the right, agree=0.792, adj=0.476, (0 split)
## PriceCH < 1.755 to the right, agree=0.755, adj=0.381, (0 split)
## DiscMM < 0.05 to the left, agree=0.736, adj=0.333, (0 split)
##
## Node number 25: 54 observations, complexity param=0.0119863
## predicted class=MM expected loss=0.4444444 P(node) =0.072
## class counts: 24 30
## probabilities: 0.444 0.556
## left son=50 (48 obs) right son=51 (6 obs)
## Primary splits:
## PriceDiff < 0.37 to the left, improve=2.666667, (0 missing)
## DiscCH < 0.115 to the left, improve=2.176871, (0 missing)
## PctDiscCH < 0.059517 to the left, improve=2.176871, (0 missing)
## LoyalCH < 0.49 to the right, improve=1.500000, (0 missing)
## SalePriceCH < 1.775 to the right, improve=1.463019, (0 missing)
## Surrogate splits:
## ListPriceDiff < 0.38 to the left, agree=0.944, adj=0.500, (0 split)
## DiscCH < 0.115 to the left, agree=0.907, adj=0.167, (0 split)
## PctDiscCH < 0.059517 to the left, agree=0.907, adj=0.167, (0 split)
##
## Node number 26: 11 observations
## predicted class=CH expected loss=0.4545455 P(node) =0.01466667
## class counts: 6 5
## probabilities: 0.545 0.455
##
## Node number 27: 63 observations
## predicted class=MM expected loss=0.1269841 P(node) =0.084
## class counts: 8 55
## probabilities: 0.127 0.873
##
## Node number 28: 6 observations
## predicted class=CH expected loss=0.5 P(node) =0.008
## class counts: 3 3
## probabilities: 0.500 0.500
##
## Node number 29: 88 observations, complexity param=0.0008561644
## predicted class=MM expected loss=0.1477273 P(node) =0.1173333
## class counts: 13 75
## probabilities: 0.148 0.852
## left son=58 (24 obs) right son=59 (64 obs)
## Primary splits:
## ListPriceDiff < 0.135 to the left, improve=1.3674240, (0 missing)
## LoyalCH < 0.203377 to the left, improve=0.7305195, (0 missing)
## WeekofPurchase < 273.5 to the left, improve=0.7266585, (0 missing)
## PctDiscMM < 0.1961965 to the left, improve=0.6657576, (0 missing)
## PriceDiff < 0.31 to the right, improve=0.6339944, (0 missing)
## Surrogate splits:
## PriceMM < 1.89 to the left, agree=0.886, adj=0.583, (0 split)
## WeekofPurchase < 229.5 to the left, agree=0.852, adj=0.458, (0 split)
## PriceCH < 1.72 to the left, agree=0.784, adj=0.208, (0 split)
## PriceDiff < 0.015 to the left, agree=0.773, adj=0.167, (0 split)
## SalePriceMM < 1.84 to the left, agree=0.761, adj=0.125, (0 split)
##
## Node number 38: 58 observations, complexity param=0.005707763
## predicted class=CH expected loss=0.362069 P(node) =0.07733333
## class counts: 37 21
## probabilities: 0.638 0.362
## left son=76 (17 obs) right son=77 (41 obs)
## Primary splits:
## StoreID < 5.5 to the right, improve=2.873448, (0 missing)
## Store7 splits as RL, improve=2.873448, (0 missing)
## STORE < 0.5 to the left, improve=2.873448, (0 missing)
## ListPriceDiff < 0.235 to the right, improve=2.364532, (0 missing)
## SalePriceCH < 1.755 to the left, improve=1.595588, (0 missing)
## Surrogate splits:
## STORE < 0.5 to the left, agree=1.000, adj=1.000, (0 split)
## SalePriceMM < 1.64 to the left, agree=0.845, adj=0.471, (0 split)
## PctDiscMM < 0.1961965 to the right, agree=0.845, adj=0.471, (0 split)
## SpecialCH < 0.5 to the right, agree=0.793, adj=0.294, (0 split)
## DiscCH < 0.335 to the right, agree=0.759, adj=0.176, (0 split)
##
## Node number 39: 5 observations
## predicted class=MM expected loss=0 P(node) =0.006666667
## class counts: 0 5
## probabilities: 0.000 1.000
##
## Node number 48: 32 observations
## predicted class=CH expected loss=0.15625 P(node) =0.04266667
## class counts: 27 5
## probabilities: 0.844 0.156
##
## Node number 49: 21 observations, complexity param=0.008561644
## predicted class=MM expected loss=0.4761905 P(node) =0.028
## class counts: 10 11
## probabilities: 0.476 0.524
## left son=98 (14 obs) right son=99 (7 obs)
## Primary splits:
## SalePriceMM < 2.04 to the left, improve=2.333333, (0 missing)
## SalePriceCH < 1.81 to the left, improve=2.333333, (0 missing)
## WeekofPurchase < 239 to the left, improve=1.937729, (0 missing)
## PriceCH < 1.755 to the left, improve=1.937729, (0 missing)
## PriceMM < 2.04 to the left, improve=1.912554, (0 missing)
## Surrogate splits:
## SalePriceCH < 1.81 to the left, agree=1.000, adj=1.000, (0 split)
## PriceCH < 1.81 to the left, agree=0.810, adj=0.429, (0 split)
## PriceMM < 2.04 to the left, agree=0.810, adj=0.429, (0 split)
## WeekofPurchase < 275 to the left, agree=0.762, adj=0.286, (0 split)
## ListPriceDiff < 0.235 to the right, agree=0.762, adj=0.286, (0 split)
##
## Node number 50: 48 observations, complexity param=0.0119863
## predicted class=CH expected loss=0.5 P(node) =0.064
## class counts: 24 24
## probabilities: 0.500 0.500
## left son=100 (15 obs) right son=101 (33 obs)
## Primary splits:
## LoyalCH < 0.49 to the right, improve=2.3757580, (0 missing)
## SpecialMM < 0.5 to the left, improve=0.7526132, (0 missing)
## WeekofPurchase < 269.5 to the right, improve=0.6153846, (0 missing)
## SalePriceMM < 2.15 to the left, improve=0.3920145, (0 missing)
## StoreID < 3.5 to the right, improve=0.3809524, (0 missing)
## Surrogate splits:
## StoreID < 3.5 to the right, agree=0.729, adj=0.133, (0 split)
## STORE < 3.5 to the right, agree=0.729, adj=0.133, (0 split)
##
## Node number 51: 6 observations
## predicted class=MM expected loss=0 P(node) =0.008
## class counts: 0 6
## probabilities: 0.000 1.000
##
## Node number 58: 24 observations, complexity param=0.0008561644
## predicted class=MM expected loss=0.2916667 P(node) =0.032
## class counts: 7 17
## probabilities: 0.292 0.708
## left son=116 (9 obs) right son=117 (15 obs)
## Primary splits:
## STORE < 1.5 to the left, improve=2.005556, (0 missing)
## PriceDiff < -0.12 to the left, improve=1.546919, (0 missing)
## PctDiscMM < 0.109423 to the right, improve=1.546919, (0 missing)
## StoreID < 5 to the right, improve=1.200877, (0 missing)
## DiscMM < 0.25 to the right, improve=1.200877, (0 missing)
## Surrogate splits:
## StoreID < 5 to the right, agree=0.833, adj=0.556, (0 split)
## SpecialMM < 0.5 to the right, agree=0.833, adj=0.556, (0 split)
## Store7 splits as RL, agree=0.833, adj=0.556, (0 split)
## DiscMM < 0.1 to the right, agree=0.792, adj=0.444, (0 split)
## PriceDiff < -0.035 to the left, agree=0.792, adj=0.444, (0 split)
##
## Node number 59: 64 observations
## predicted class=MM expected loss=0.09375 P(node) =0.08533333
## class counts: 6 58
## probabilities: 0.094 0.906
##
## Node number 76: 17 observations
## predicted class=CH expected loss=0.1176471 P(node) =0.02266667
## class counts: 15 2
## probabilities: 0.882 0.118
##
## Node number 77: 41 observations, complexity param=0.005707763
## predicted class=CH expected loss=0.4634146 P(node) =0.05466667
## class counts: 22 19
## probabilities: 0.537 0.463
## left son=154 (5 obs) right son=155 (36 obs)
## Primary splits:
## SalePriceMM < 2.15 to the right, improve=2.445799, (0 missing)
## SalePriceCH < 1.94 to the right, improve=2.276608, (0 missing)
## WeekofPurchase < 236.5 to the right, improve=1.480566, (0 missing)
## PriceDiff < -0.065 to the right, improve=1.401738, (0 missing)
## PriceCH < 1.875 to the right, improve=1.342625, (0 missing)
## Surrogate splits:
## PriceMM < 2.205 to the right, agree=0.951, adj=0.6, (0 split)
## SalePriceCH < 1.94 to the right, agree=0.927, adj=0.4, (0 split)
## WeekofPurchase < 275 to the right, agree=0.902, adj=0.2, (0 split)
##
## Node number 98: 14 observations
## predicted class=CH expected loss=0.3571429 P(node) =0.01866667
## class counts: 9 5
## probabilities: 0.643 0.357
##
## Node number 99: 7 observations
## predicted class=MM expected loss=0.1428571 P(node) =0.009333333
## class counts: 1 6
## probabilities: 0.143 0.857
##
## Node number 100: 15 observations
## predicted class=CH expected loss=0.2666667 P(node) =0.02
## class counts: 11 4
## probabilities: 0.733 0.267
##
## Node number 101: 33 observations, complexity param=0.006849315
## predicted class=MM expected loss=0.3939394 P(node) =0.044
## class counts: 13 20
## probabilities: 0.394 0.606
## left son=202 (8 obs) right son=203 (25 obs)
## Primary splits:
## WeekofPurchase < 269.5 to the right, improve=1.1275760, (0 missing)
## SpecialMM < 0.5 to the left, improve=0.7575758, (0 missing)
## PriceCH < 1.825 to the right, improve=0.4432900, (0 missing)
## SalePriceCH < 1.825 to the right, improve=0.4432900, (0 missing)
## LoyalCH < 0.3618285 to the right, improve=0.3191142, (0 missing)
## Surrogate splits:
## DiscMM < 0.03 to the right, agree=0.818, adj=0.250, (0 split)
## PctDiscMM < 0.0137615 to the right, agree=0.818, adj=0.250, (0 split)
## LoyalCH < 0.3136 to the left, agree=0.788, adj=0.125, (0 split)
##
## Node number 116: 9 observations
## predicted class=CH expected loss=0.4444444 P(node) =0.012
## class counts: 5 4
## probabilities: 0.556 0.444
##
## Node number 117: 15 observations
## predicted class=MM expected loss=0.1333333 P(node) =0.02
## class counts: 2 13
## probabilities: 0.133 0.867
##
## Node number 154: 5 observations
## predicted class=CH expected loss=0 P(node) =0.006666667
## class counts: 5 0
## probabilities: 1.000 0.000
##
## Node number 155: 36 observations, complexity param=0.005707763
## predicted class=MM expected loss=0.4722222 P(node) =0.048
## class counts: 17 19
## probabilities: 0.472 0.528
## left son=310 (5 obs) right son=311 (31 obs)
## Primary splits:
## LoyalCH < 0.54608 to the left, improve=1.2476700, (0 missing)
## WeekofPurchase < 236.5 to the right, improve=0.8213675, (0 missing)
## PriceDiff < -0.065 to the right, improve=0.6944444, (0 missing)
## SpecialMM < 0.5 to the left, improve=0.4629630, (0 missing)
## PriceMM < 2.04 to the right, improve=0.1920635, (0 missing)
##
## Node number 202: 8 observations
## predicted class=CH expected loss=0.375 P(node) =0.01066667
## class counts: 5 3
## probabilities: 0.625 0.375
##
## Node number 203: 25 observations, complexity param=0.003424658
## predicted class=MM expected loss=0.32 P(node) =0.03333333
## class counts: 8 17
## probabilities: 0.320 0.680
## left son=406 (17 obs) right son=407 (8 obs)
## Primary splits:
## LoyalCH < 0.3618285 to the right, improve=0.8947059, (0 missing)
## WeekofPurchase < 243 to the left, improve=0.2292063, (0 missing)
## StoreID < 2.5 to the right, improve=0.2261538, (0 missing)
## PriceCH < 1.875 to the right, improve=0.2261538, (0 missing)
## SalePriceCH < 1.875 to the right, improve=0.2261538, (0 missing)
## Surrogate splits:
## PriceMM < 2.26 to the left, agree=0.72, adj=0.125, (0 split)
## SalePriceMM < 2.26 to the left, agree=0.72, adj=0.125, (0 split)
##
## Node number 310: 5 observations
## predicted class=CH expected loss=0.2 P(node) =0.006666667
## class counts: 4 1
## probabilities: 0.800 0.200
##
## Node number 311: 31 observations, complexity param=0.005707763
## predicted class=MM expected loss=0.4193548 P(node) =0.04133333
## class counts: 13 18
## probabilities: 0.419 0.581
## left son=622 (13 obs) right son=623 (18 obs)
## Primary splits:
## LoyalCH < 0.62608 to the right, improve=1.7207060, (0 missing)
## WeekofPurchase < 261 to the left, improve=1.1230900, (0 missing)
## PriceDiff < 0.235 to the left, improve=0.5736973, (0 missing)
## SpecialMM < 0.5 to the left, improve=0.1876833, (0 missing)
## PriceCH < 1.875 to the right, improve=0.1402525, (0 missing)
## Surrogate splits:
## WeekofPurchase < 265 to the right, agree=0.710, adj=0.308, (0 split)
## PriceCH < 1.94 to the right, agree=0.677, adj=0.231, (0 split)
## PriceMM < 2.04 to the right, agree=0.677, adj=0.231, (0 split)
## SpecialCH < 0.5 to the right, agree=0.677, adj=0.231, (0 split)
## StoreID < 2.5 to the right, agree=0.645, adj=0.154, (0 split)
##
## Node number 406: 17 observations, complexity param=0.003424658
## predicted class=MM expected loss=0.4117647 P(node) =0.02266667
## class counts: 7 10
## probabilities: 0.412 0.588
## left son=812 (10 obs) right son=813 (7 obs)
## Primary splits:
## LoyalCH < 0.420437 to the left, improve=1.7210080, (0 missing)
## PriceCH < 1.875 to the right, improve=0.7908497, (0 missing)
## SalePriceCH < 1.875 to the right, improve=0.7908497, (0 missing)
## WeekofPurchase < 254.5 to the right, improve=0.6067227, (0 missing)
## StoreID < 2.5 to the right, improve=0.2352941, (0 missing)
## Surrogate splits:
## WeekofPurchase < 238.5 to the right, agree=0.706, adj=0.286, (0 split)
## PriceCH < 1.825 to the right, agree=0.647, adj=0.143, (0 split)
## SalePriceCH < 1.825 to the right, agree=0.647, adj=0.143, (0 split)
##
## Node number 407: 8 observations
## predicted class=MM expected loss=0.125 P(node) =0.01066667
## class counts: 1 7
## probabilities: 0.125 0.875
##
## Node number 622: 13 observations
## predicted class=CH expected loss=0.3846154 P(node) =0.01733333
## class counts: 8 5
## probabilities: 0.615 0.385
##
## Node number 623: 18 observations, complexity param=0.003424658
## predicted class=MM expected loss=0.2777778 P(node) =0.024
## class counts: 5 13
## probabilities: 0.278 0.722
## left son=1246 (5 obs) right son=1247 (13 obs)
## Primary splits:
## PriceCH < 1.755 to the left, improve=1.4376070, (0 missing)
## SalePriceCH < 1.755 to the left, improve=1.4376070, (0 missing)
## WeekofPurchase < 261 to the left, improve=1.3888890, (0 missing)
## PriceDiff < -0.065 to the right, improve=1.0683760, (0 missing)
## ListPriceDiff < 0.165 to the right, improve=0.4170274, (0 missing)
## Surrogate splits:
## SalePriceCH < 1.755 to the left, agree=1.000, adj=1.0, (0 split)
## WeekofPurchase < 240.5 to the left, agree=0.889, adj=0.6, (0 split)
## PriceDiff < 0.22 to the right, agree=0.833, adj=0.4, (0 split)
## ListPriceDiff < 0.235 to the right, agree=0.833, adj=0.4, (0 split)
##
## Node number 812: 10 observations
## predicted class=CH expected loss=0.4 P(node) =0.01333333
## class counts: 6 4
## probabilities: 0.600 0.400
##
## Node number 813: 7 observations
## predicted class=MM expected loss=0.1428571 P(node) =0.009333333
## class counts: 1 6
## probabilities: 0.143 0.857
##
## Node number 1246: 5 observations
## predicted class=CH expected loss=0.4 P(node) =0.006666667
## class counts: 3 2
## probabilities: 0.600 0.400
##
## Node number 1247: 13 observations
## predicted class=MM expected loss=0.1538462 P(node) =0.01733333
## class counts: 2 11
## probabilities: 0.154 0.846sum(tree_oj$frame$var == "<leaf>")## [1] 27There are 27 terminal nodes.
oj_preds = predict(tree_oj, newdata = oj_train, type = "class")
mean(oj_preds != oj_train$Purchase)## [1] 0.1173333There is an 11.73% training classification error.
# tree made using tree() from tree library
tree_oj_2 = tree::tree(Purchase ~ ., data = oj_train)
oj_preds_2 = predict(tree_oj_2, newdata = oj_train, type = "class")
mean(oj_preds_2 != oj_train$Purchase)## [1] 0.1586667summary(tree_oj_2)##
## Classification tree:
## tree::tree(formula = Purchase ~ ., data = oj_train)
## Variables actually used in tree construction:
## [1] "LoyalCH" "PriceDiff" "ListPriceDiff" "DiscMM"
## [5] "SalePriceMM"
## Number of terminal nodes: 11
## Residual mean deviance: 0.7306 = 539.9 / 739
## Misclassification error rate: 0.1587 = 119 / 750Using tree::tree(), the number of terminal nodes is 11
and the misclassification error rate is 15.87%.
(c) Type in the name of the tree object in order to get a detailed text output. Pick one of the terminal nodes, and interpret the information displayed.
tree_oj## n= 750
##
## node), split, n, loss, yval, (yprob)
## * denotes terminal node
##
## 1) root 750 292 CH (0.61066667 0.38933333)
## 2) LoyalCH>=0.5036 425 59 CH (0.86117647 0.13882353)
## 4) PriceDiff>=-0.39 404 45 CH (0.88861386 0.11138614)
## 8) LoyalCH>=0.7053255 281 15 CH (0.94661922 0.05338078) *
## 9) LoyalCH< 0.7053255 123 30 CH (0.75609756 0.24390244)
## 18) PriceDiff>=0.265 60 4 CH (0.93333333 0.06666667) *
## 19) PriceDiff< 0.265 63 26 CH (0.58730159 0.41269841)
## 38) LoyalCH< 0.6919315 58 21 CH (0.63793103 0.36206897)
## 76) StoreID>=5.5 17 2 CH (0.88235294 0.11764706) *
## 77) StoreID< 5.5 41 19 CH (0.53658537 0.46341463)
## 154) SalePriceMM>=2.15 5 0 CH (1.00000000 0.00000000) *
## 155) SalePriceMM< 2.15 36 17 MM (0.47222222 0.52777778)
## 310) LoyalCH< 0.54608 5 1 CH (0.80000000 0.20000000) *
## 311) LoyalCH>=0.54608 31 13 MM (0.41935484 0.58064516)
## 622) LoyalCH>=0.62608 13 5 CH (0.61538462 0.38461538) *
## 623) LoyalCH< 0.62608 18 5 MM (0.27777778 0.72222222)
## 1246) PriceCH< 1.755 5 2 CH (0.60000000 0.40000000) *
## 1247) PriceCH>=1.755 13 2 MM (0.15384615 0.84615385) *
## 39) LoyalCH>=0.6919315 5 0 MM (0.00000000 1.00000000) *
## 5) PriceDiff< -0.39 21 7 MM (0.33333333 0.66666667)
## 10) LoyalCH>=0.742157 9 3 CH (0.66666667 0.33333333) *
## 11) LoyalCH< 0.742157 12 1 MM (0.08333333 0.91666667) *
## 3) LoyalCH< 0.5036 325 92 MM (0.28307692 0.71692308)
## 6) LoyalCH>=0.282272 181 75 MM (0.41436464 0.58563536)
## 12) PriceDiff>=0.05 107 46 CH (0.57009346 0.42990654)
## 24) STORE< 1.5 53 16 CH (0.69811321 0.30188679)
## 48) PriceDiff>=0.31 32 5 CH (0.84375000 0.15625000) *
## 49) PriceDiff< 0.31 21 10 MM (0.47619048 0.52380952)
## 98) SalePriceMM< 2.04 14 5 CH (0.64285714 0.35714286) *
## 99) SalePriceMM>=2.04 7 1 MM (0.14285714 0.85714286) *
## 25) STORE>=1.5 54 24 MM (0.44444444 0.55555556)
## 50) PriceDiff< 0.37 48 24 CH (0.50000000 0.50000000)
## 100) LoyalCH>=0.49 15 4 CH (0.73333333 0.26666667) *
## 101) LoyalCH< 0.49 33 13 MM (0.39393939 0.60606061)
## 202) WeekofPurchase>=269.5 8 3 CH (0.62500000 0.37500000) *
## 203) WeekofPurchase< 269.5 25 8 MM (0.32000000 0.68000000)
## 406) LoyalCH>=0.3618285 17 7 MM (0.41176471 0.58823529)
## 812) LoyalCH< 0.420437 10 4 CH (0.60000000 0.40000000) *
## 813) LoyalCH>=0.420437 7 1 MM (0.14285714 0.85714286) *
## 407) LoyalCH< 0.3618285 8 1 MM (0.12500000 0.87500000) *
## 51) PriceDiff>=0.37 6 0 MM (0.00000000 1.00000000) *
## 13) PriceDiff< 0.05 74 14 MM (0.18918919 0.81081081)
## 26) SpecialCH>=0.5 11 5 CH (0.54545455 0.45454545) *
## 27) SpecialCH< 0.5 63 8 MM (0.12698413 0.87301587) *
## 7) LoyalCH< 0.282272 144 17 MM (0.11805556 0.88194444)
## 14) LoyalCH>=0.0356415 94 16 MM (0.17021277 0.82978723)
## 28) SalePriceMM>=2.205 6 3 CH (0.50000000 0.50000000) *
## 29) SalePriceMM< 2.205 88 13 MM (0.14772727 0.85227273)
## 58) ListPriceDiff< 0.135 24 7 MM (0.29166667 0.70833333)
## 116) STORE< 1.5 9 4 CH (0.55555556 0.44444444) *
## 117) STORE>=1.5 15 2 MM (0.13333333 0.86666667) *
## 59) ListPriceDiff>=0.135 64 6 MM (0.09375000 0.90625000) *
## 15) LoyalCH< 0.0356415 50 1 MM (0.02000000 0.98000000) *Node 8: PriceDiff>=-0.39 and LoyalCH >= 0.7053255 281 15 CH (0.94661922 0.05338078) *
281 observations in this node with 15 misclassified. 94.66% chance CH and 5.34% chance MM
tree_oj_2## node), split, n, deviance, yval, (yprob)
## * denotes terminal node
##
## 1) root 750 1003.000 CH ( 0.61067 0.38933 )
## 2) LoyalCH < 0.5036 325 387.300 MM ( 0.28308 0.71692 )
## 4) LoyalCH < 0.282272 144 104.600 MM ( 0.11806 0.88194 ) *
## 5) LoyalCH > 0.282272 181 245.600 MM ( 0.41436 0.58564 )
## 10) PriceDiff < 0.05 74 71.790 MM ( 0.18919 0.81081 ) *
## 11) PriceDiff > 0.05 107 146.200 CH ( 0.57009 0.42991 ) *
## 3) LoyalCH > 0.5036 425 342.400 CH ( 0.86118 0.13882 )
## 6) LoyalCH < 0.764572 169 195.900 CH ( 0.73373 0.26627 )
## 12) PriceDiff < 0.265 96 130.400 CH ( 0.58333 0.41667 )
## 24) PriceDiff < -0.165 23 26.400 MM ( 0.26087 0.73913 )
## 48) ListPriceDiff < 0.115 7 8.376 CH ( 0.71429 0.28571 ) *
## 49) ListPriceDiff > 0.115 16 7.481 MM ( 0.06250 0.93750 ) *
## 25) PriceDiff > -0.165 73 90.970 CH ( 0.68493 0.31507 )
## 50) DiscMM < 0.22 49 66.920 CH ( 0.57143 0.42857 )
## 100) SalePriceMM < 2.155 39 53.830 MM ( 0.46154 0.53846 ) *
## 101) SalePriceMM > 2.155 10 0.000 CH ( 1.00000 0.00000 ) *
## 51) DiscMM > 0.22 24 13.770 CH ( 0.91667 0.08333 ) *
## 13) PriceDiff > 0.265 73 36.460 CH ( 0.93151 0.06849 ) *
## 7) LoyalCH > 0.764572 256 108.600 CH ( 0.94531 0.05469 )
## 14) PriceDiff < 0.31 174 97.400 CH ( 0.91954 0.08046 ) *
## 15) PriceDiff > 0.31 82 0.000 CH ( 1.00000 0.00000 ) *(d) Create a plot of the tree, and interpret the results.
fancyRpartPlot(tree_oj, cex = 0.6)
plot(tree_oj_2)
text(tree_oj_2, pretty = 0)
(e) Predict the response on the test data, and produce a confusion matrix comparing the test labels to the predicted test labels. What is the test error rate?
oj_preds = predict(tree_oj, newdata = oj_test, type = "class")
confusionMatrix(oj_preds, oj_test$Purchase)## Confusion Matrix and Statistics
##
## Reference
## Prediction CH MM
## CH 160 33
## MM 35 92
##
## Accuracy : 0.7875
## 95% CI : (0.7385, 0.831)
## No Information Rate : 0.6094
## P-Value [Acc > NIR] : 7.64e-12
##
## Kappa : 0.5549
##
## Mcnemar's Test P-Value : 0.9035
##
## Sensitivity : 0.8205
## Specificity : 0.7360
## Pos Pred Value : 0.8290
## Neg Pred Value : 0.7244
## Prevalence : 0.6094
## Detection Rate : 0.5000
## Detection Prevalence : 0.6031
## Balanced Accuracy : 0.7783
##
## 'Positive' Class : CH
## mean(oj_preds != oj_test$Purchase)## [1] 0.2125The test error rate was 21.25%
# same but with tree model from tree()
oj_preds_2 = predict(tree_oj_2, newdata = oj_test, type = "class")
confusionMatrix(oj_preds_2, oj_test$Purchase)## Confusion Matrix and Statistics
##
## Reference
## Prediction CH MM
## CH 158 24
## MM 37 101
##
## Accuracy : 0.8094
## 95% CI : (0.762, 0.8509)
## No Information Rate : 0.6094
## P-Value [Acc > NIR] : 1.07e-14
##
## Kappa : 0.6069
##
## Mcnemar's Test P-Value : 0.1244
##
## Sensitivity : 0.8103
## Specificity : 0.8080
## Pos Pred Value : 0.8681
## Neg Pred Value : 0.7319
## Prevalence : 0.6094
## Detection Rate : 0.4938
## Detection Prevalence : 0.5687
## Balanced Accuracy : 0.8091
##
## 'Positive' Class : CH
## mean(oj_preds_2 != oj_test$Purchase)## [1] 0.190625(f) Apply the cv.tree() function to the training set in order to determine the optimal tree size.
set.seed(1)
cv_oj = tree::cv.tree(tree_oj_2, FUN = prune.misclass)
cv_oj## $size
## [1] 11 10 8 7 4 2 1
##
## $dev
## [1] 155 155 155 159 163 169 292
##
## $k
## [1] -Inf 0.000000 1.500000 3.000000 3.666667 7.500000 141.000000
##
## $method
## [1] "misclass"
##
## attr(,"class")
## [1] "prune" "tree.sequence"best_size = cv_oj$size[which.min(cv_oj$dev)]
best_size## [1] 11(g) Produce a plot with tree size on the x-axis and cross-validated classification error rate on the y-axis.
plot(cv_oj$size, cv_oj$dev, type = "b",
xlab = "Tree Size (Number of Terminal Nodes)",
ylab = "Cross-Validated Misclassification Error",
main = "CV Error vs Tree Size")
(h) Which tree size corresponds to the lowest cross-validated classification error rate? 11
(i) Produce a pruned tree corresponding to the optimal tree size obtained using cross-validation. If cross-validation does not lead to selection of a pruned tree, then create a pruned tree with five terminal nodes.
pruned_tree = prune.misclass(tree_oj_2, best = 5)
plot(pruned_tree)
text(pruned_tree, pretty = 0)
(j) Compare the training error rates between the pruned and unpruned trees. Which is higher?
prune_preds = predict(pruned_tree, newdata = oj_train, type = "class")
mean(prune_preds != oj_train$Purchase)## [1] 0.1666667Pruned training error: 16.67%
Unpruned training error: 15.87% (from the tree() function)
(k) Compare the test error rates between the pruned and unpruned trees. Which is higher?
prune_preds = predict(pruned_tree, newdata = oj_test, type = "class")
mean(prune_preds != oj_test$Purchase)## [1] 0.159375Pruned test error: 15.84%
Unpruned test error: 19.10% (from the tree() function)