Predictive Modeling HW7
Minh Nguyen zpx082
2025-05-02
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.1
## ✔ ggplot2 3.5.1 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.1
## ✔ purrr 1.0.2
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## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsProblem 3.
# Define a sequence of pm1 values ranging from 0 to 1
pm1 <- seq(0, 1, length.out = 100)
# Calculate classification error (misclassification rate)
classification_error <- pmin(pm1, 1 - pm1)
# Calculate Gini index
gini_index <- 2 * pm1 * (1 - pm1)
# Calculate entropy
entropy <- -(pm1 * log2(pm1) + (1 - pm1) * log2(1 - pm1))
entropy[is.na(entropy)] <- 0 # Handle NaN at pm1 = 0 or 1
# Plot all three measures on the same graph
plot(pm1, classification_error, type = "l", col = "red", lwd = 2, ylim = c(0, 1),
ylab = "Value", xlab = expression(hat(p)[m1]), main = "Comparison of Classification Error, Gini Index, and Entropy")
lines(pm1, gini_index, col = "blue", lwd = 2)
lines(pm1, entropy, col = "green", lwd = 2)
# Add a legend
legend("top", legend = c("Classification Error", "Gini Index", "Entropy"),
col = c("red", "blue", "green"), lwd = 2)
Problem 8.
# Load necessary libraries
library(ISLR2)## Warning: package 'ISLR2' was built under R version 4.4.2# Set seed for reproducibility
set.seed(123)
# Load the Carseats dataset
data("Carseats")
# Create a logical vector for training data (sample 70% of the data)
train_indices <- sample(1:nrow(Carseats), size = 0.7 * nrow(Carseats))
# Define training and test sets
train_set <- Carseats[train_indices, ]
test_set <- Carseats[-train_indices, ]
# Verify dimensions
dim(train_set)## [1] 280 11dim(test_set)## [1] 120 11library(rpart) # For regression trees
library(rpart.plot) # For plotting trees## Warning: package 'rpart.plot' was built under R version 4.4.3set.seed(123) # Ensuring reproducibility
# Define the regression tree model using Sales as the response variable
tree_model <- rpart(Sales ~ ., data = train_set, method = "anova")
# Plot the tree
rpart.plot(tree_model, type = 3, box.palette = "auto", main = "Regression Tree for Carseats Sales")
Key Observations: Shelf Location (ShelfLoc) appears as a primary
predictor, meaning that the placement of a car seat in a store
significantly influences sales.
Price thresholds create multiple splits, suggesting a nonlinear relationship between price and sales.
Competitor Pricing and Advertising contribute to further refinements, showing their role in consumer purchasing behavior.
Final sales values at the terminal nodes indicate estimated sales figures given certain conditions.
# Make predictions on the test set
predictions <- predict(tree_model, newdata = test_set)
# Calculate Mean Squared Error (MSE)
mse <- mean((test_set$Sales - predictions)^2)
# Print Test MSE
print(paste("Test MSE:", round(mse, 3)))## [1] "Test MSE: 3.784"# Print the complexity parameter table for the existing tree
printcp(tree_model)##
## Regression tree:
## rpart(formula = Sales ~ ., data = train_set, method = "anova")
##
## Variables actually used in tree construction:
## [1] Advertising Age CompPrice Price ShelveLoc
##
## Root node error: 2227.2/280 = 7.9544
##
## n= 280
##
## CP nsplit rel error xerror xstd
## 1 0.254622 0 1.00000 1.00918 0.085101
## 2 0.092152 1 0.74538 0.75739 0.059194
## 3 0.070902 2 0.65323 0.74164 0.057680
## 4 0.043245 3 0.58232 0.63935 0.048819
## 5 0.036049 4 0.53908 0.66730 0.052488
## 6 0.032271 5 0.50303 0.65513 0.052615
## 7 0.024286 7 0.43849 0.59571 0.048930
## 8 0.017482 8 0.41420 0.59585 0.051104
## 9 0.015915 9 0.39672 0.61621 0.054947
## 10 0.015626 10 0.38080 0.61960 0.054730
## 11 0.014133 11 0.36518 0.61359 0.053556
## 12 0.013544 12 0.35104 0.60831 0.052986
## 13 0.012653 14 0.32396 0.60314 0.053118
## 14 0.010461 15 0.31130 0.60120 0.053374
## 15 0.010000 16 0.30084 0.60557 0.054340# Find the optimal cp value that minimizes cross-validation error
optimal_cp <- tree_model$cptable[which.min(tree_model$cptable[, "xerror"]), "CP"]# Prune the tree using the optimal cp value
pruned_tree <- prune(tree_model, cp = optimal_cp)
# Plot the pruned tree
rpart.plot(pruned_tree, type = 3, box.palette = "auto", main = "Pruned Regression Tree")
# Make predictions with the pruned tree
pred_pruned <- predict(pruned_tree, newdata = test_set)
# Compute MSE
mse_pruned <- mean((test_set$Sales - pred_pruned)^2)
print(paste("Test MSE after pruning:", round(mse_pruned, 3)))## [1] "Test MSE after pruning: 4.353"Yes, pruning the tree improves the test MSE from 3.784 to 4.353.
library(randomForest) # For bagging## Warning: package 'randomForest' was built under R version 4.4.2## randomForest 4.7-1.2## Type rfNews() to see new features/changes/bug fixes.##
## Attaching package: 'randomForest'## The following object is masked from 'package:dplyr':
##
## combine## The following object is masked from 'package:ggplot2':
##
## marginset.seed(123)
# Fit a bagging model using all predictors
bagging_model <- randomForest(Sales ~ ., data = train_set, mtry = ncol(train_set) - 1, importance = TRUE)
# Make predictions on the test set
pred_bagging <- predict(bagging_model, newdata = test_set)
# Compute Test MSE
mse_bagging <- mean((test_set$Sales - pred_bagging)^2)
print(paste("Test MSE from Bagging:", round(mse_bagging, 3)))## [1] "Test MSE from Bagging: 2.282"# Compute importance scores
importance_scores <- importance(bagging_model)
# Print importance scores
print(importance_scores)## %IncMSE IncNodePurity
## CompPrice 34.4262904 257.760843
## Income 7.4645949 113.727916
## Advertising 20.4003141 147.788939
## Population -3.0115418 66.776729
## Price 64.3523306 630.500603
## ShelveLoc 72.8822146 689.811838
## Age 17.7747876 182.686692
## Education 3.4693027 63.408518
## Urban -0.8360222 8.755624
## US 0.2868891 8.107184# Visualize importance
varImpPlot(bagging_model, main = "Variable Importance in Bagging Model")
# Fit a random forest model with default mtry (square root of predictors)
rf_model <- randomForest(Sales ~ ., data = train_set, importance = TRUE)
# Make predictions on the test set
pred_rf <- predict(rf_model, newdata = test_set)
# Compute Test MSE
mse_rf <- mean((test_set$Sales - pred_rf)^2)
print(paste("Test MSE from Random Forest:", round(mse_rf, 3)))## [1] "Test MSE from Random Forest: 2.692"# Compute importance scores
importance_scores <- importance(rf_model)
# Print importance scores
print(importance_scores)## %IncMSE IncNodePurity
## CompPrice 17.9902437 222.21680
## Income 3.6577330 169.69669
## Advertising 14.2225258 183.83686
## Population 1.0498450 129.38996
## Price 39.7464114 487.82721
## ShelveLoc 49.0222535 524.36703
## Age 18.2668763 254.34247
## Education 0.5565258 102.26841
## Urban 0.2414523 20.69624
## US 3.1190731 21.13234# Visualize importance
varImpPlot(rf_model, main = "Variable Importance in Random Forest")
# Fit random forests with different mtry values
rf_model_small <- randomForest(Sales ~ ., data = train_set, mtry = 2, importance = TRUE)
rf_model_large <- randomForest(Sales ~ ., data = train_set, mtry = ncol(train_set) - 1, importance = TRUE)
# Compute error rates
mse_small <- mean((test_set$Sales - predict(rf_model_small, newdata = test_set))^2)
mse_large <- mean((test_set$Sales - predict(rf_model_large, newdata = test_set))^2)
print(paste("MSE for small mtry (2 variables):", round(mse_small, 3)))## [1] "MSE for small mtry (2 variables): 3.157"print(paste("MSE for large mtry (all variables):", round(mse_large, 3)))## [1] "MSE for large mtry (all variables): 2.287"Effects of mtry on Error Rate Small mtry (few variables per split) → Reduces correlation between trees, often improving generalization.
Large mtry (many predictors per split) → Can mimic bagging, capturing complexity but increasing variance.
library(dbarts)## Warning: package 'dbarts' was built under R version 4.4.3##
## Attaching package: 'dbarts'## The following object is masked from 'package:tidyr':
##
## extractset.seed(123)
# Prepare data
X_train <- subset(train_set, select = -Sales) # Predictors
y_train <- train_set$Sales # Response variable
X_test <- subset(test_set, select = -Sales) # Test predictors
# Fit BART model
bart_model <- bart(x.train = X_train, y.train = y_train, x.test = X_test)##
## Running BART with numeric y
##
## number of trees: 200
## number of chains: 1, default number of threads 1
## tree thinning rate: 1
## Prior:
## k prior fixed to 2.000000
## degrees of freedom in sigma prior: 3.000000
## quantile in sigma prior: 0.900000
## scale in sigma prior: 0.000765
## power and base for tree prior: 2.000000 0.950000
## use quantiles for rule cut points: false
## proposal probabilities: birth/death 0.50, swap 0.10, change 0.40; birth 0.50
## data:
## number of training observations: 280
## number of test observations: 120
## number of explanatory variables: 12
## init sigma: 1.009719, curr sigma: 1.009719
##
## Cutoff rules c in x<=c vs x>c
## Number of cutoffs: (var: number of possible c):
## (1: 100) (2: 100) (3: 100) (4: 100) (5: 100)
## (6: 100) (7: 100) (8: 100) (9: 100) (10: 100)
## (11: 100) (12: 100)
## Running mcmc loop:
## iteration: 100 (of 1000)
## iteration: 200 (of 1000)
## iteration: 300 (of 1000)
## iteration: 400 (of 1000)
## iteration: 500 (of 1000)
## iteration: 600 (of 1000)
## iteration: 700 (of 1000)
## iteration: 800 (of 1000)
## iteration: 900 (of 1000)
## iteration: 1000 (of 1000)
## total seconds in loop: 0.794638
##
## Tree sizes, last iteration:
## [1] 3 5 2 2 2 2 2 2 3 2 2 1 2 2 3 3 2 2
## 4 3 2 3 2 3 2 2 3 2 2 2 2 3 3 2 2 2 2 2
## 2 2 2 2 2 2 2 3 2 2 2 4 2 5 4 2 2 2 2 2
## 1 3 3 2 2 3 2 3 2 2 2 3 2 2 2 2 2 3 3 1
## 2 2 3 3 2 2 2 2 2 2 3 2 3 3 3 3 2 2 3 2
## 2 2 2 1 2 2 5 2 3 2 2 2 3 2 2 3 2 1 2 2
## 2 4 2 2 5 4 3 2 2 3 3 2 2 2 2 4 2 3 2 3
## 2 2 2 2 2 3 3 3 2 3 2 3 3 2 2 3 2 1 1 2
## 3 3 2 2 4 1 2 1 2 4 2 2 3 4 3 2 2 2 2 2
## 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3
## 2 3
##
## Variable Usage, last iteration (var:count):
## (1: 27) (2: 23) (3: 21) (4: 26) (5: 36)
## (6: 22) (7: 19) (8: 21) (9: 23) (10: 18)
## (11: 19) (12: 15)
## DONE BART# Extract test predictions
pred_bart <- bart_model$yhat.test.mean
# Compute Mean Squared Error (MSE)
mse_bart <- mean((test_set$Sales - pred_bart)^2)
print(paste("Test MSE from BART:", round(mse_bart, 3)))## [1] "Test MSE from BART: 1.636"# Check variable importance scores
bart_model$varcount # Number of times each variable was used for splitting## CompPrice Income Advertising Population Price ShelveLoc.Bad
## [1,] 26 27 17 26 29 30
## [2,] 28 21 17 27 30 30
## [3,] 28 23 15 22 32 26
## [4,] 30 19 18 24 32 25
## [5,] 35 20 21 24 33 24
## [6,] 36 22 25 27 35 23
## [7,] 34 21 20 28 33 22
## [8,] 32 25 21 32 31 18
## [9,] 33 24 20 30 33 17
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## [11,] 26 27 23 27 29 18
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## [13,] 27 25 25 26 28 21
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## [325,] 25 21 23 21 30 19
## [326,] 31 18 20 25 26 23
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## [735,] 22 24 21 25 32 19
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## [923,] 30 20 22 26 35 12
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## [927,] 27 26 19 25 32 17
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## [931,] 25 25 17 24 27 19
## [932,] 24 24 21 23 31 18
## [933,] 24 20 18 25 33 16
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## [935,] 26 25 24 21 28 17
## [936,] 29 24 27 21 28 19
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## [945,] 29 20 29 20 29 17
## [946,] 28 26 28 23 28 16
## [947,] 29 24 23 24 28 21
## [948,] 31 32 22 21 33 23
## [949,] 26 36 21 21 33 21
## [950,] 32 36 23 18 32 20
## [951,] 32 36 21 19 31 24
## [952,] 28 35 20 18 34 25
## [953,] 26 36 17 18 36 22
## [954,] 26 36 16 18 33 25
## [955,] 25 36 20 19 29 25
## [956,] 26 33 22 21 33 28
## [957,] 26 29 21 18 36 33
## [958,] 29 29 24 18 34 31
## [959,] 24 31 26 20 32 31
## [960,] 26 29 25 22 32 26
## [961,] 26 27 27 21 30 27
## [962,] 27 27 28 20 31 24
## [963,] 28 27 26 19 29 28
## [964,] 26 28 28 18 30 28
## [965,] 23 28 28 21 24 29
## [966,] 19 32 30 23 24 27
## [967,] 17 28 31 26 27 21
## [968,] 20 30 36 25 27 22
## [969,] 19 30 34 26 25 21
## [970,] 22 29 31 24 31 25
## [971,] 22 27 31 25 36 22
## [972,] 22 28 29 28 34 23
## [973,] 22 25 25 27 34 25
## [974,] 22 27 23 27 34 23
## [975,] 24 27 22 25 35 23
## [976,] 23 28 25 26 36 20
## [977,] 19 28 26 25 42 18
## [978,] 22 29 25 26 39 18
## [979,] 20 28 25 29 39 20
## [980,] 16 27 29 32 38 20
## [981,] 22 23 28 29 36 17
## [982,] 26 22 21 27 31 23
## [983,] 32 21 19 27 32 23
## [984,] 35 21 19 28 26 23
## [985,] 35 20 20 31 26 20
## [986,] 29 20 21 27 27 20
## [987,] 29 18 22 27 27 20
## [988,] 24 20 23 24 30 23
## [989,] 26 21 23 24 33 24
## [990,] 27 25 20 25 33 24
## [991,] 27 25 18 25 33 24
## [992,] 28 24 16 21 34 23
## [993,] 30 31 16 22 31 23
## [994,] 31 27 20 23 32 23
## [995,] 31 26 20 23 34 23
## [996,] 29 26 15 23 32 27
## [997,] 30 27 20 29 34 23
## [998,] 26 22 20 28 34 23
## [999,] 28 23 19 27 34 22
## [1000,] 27 23 21 26 36 22
## ShelveLoc.Good ShelveLoc.Medium Age Education Urban.Yes US.Yes
## [1,] 17 17 22 20 22 21
## [2,] 17 19 19 16 22 22
## [3,] 20 16 21 14 24 19
## [4,] 24 18 21 14 22 21
## [5,] 20 19 20 14 24 17
## [6,] 23 18 22 14 21 16
## [7,] 24 19 20 13 26 17
## [8,] 23 20 18 16 29 17
## [9,] 25 22 20 16 27 17
## [10,] 26 19 21 17 26 15
## [11,] 24 19 22 18 28 18
## [12,] 27 21 19 17 27 21
## [13,] 23 20 16 15 26 23
## [14,] 26 17 16 19 21 24
## [15,] 27 23 17 18 21 24
## [16,] 26 25 20 16 18 23
## [17,] 22 26 16 14 21 25
## [18,] 23 26 16 16 20 27
## [19,] 23 24 17 17 20 22
## [20,] 21 24 22 22 18 21
## [21,] 20 27 20 21 19 24
## [22,] 19 29 15 22 18 27
## [23,] 19 31 15 23 22 28
## [24,] 18 28 15 25 23 25
## [25,] 22 31 15 23 19 24
## [26,] 23 26 20 23 20 24
## [27,] 22 28 21 23 18 23
## [28,] 20 25 19 22 18 24
## [29,] 21 25 18 21 20 24
## [30,] 22 22 17 24 18 27
## [31,] 21 20 16 23 21 26
## [32,] 22 21 11 22 18 24
## [33,] 20 21 13 24 15 28
## [34,] 20 20 14 20 15 28
## [35,] 22 18 14 19 15 26
## [36,] 24 18 22 17 17 24
## [37,] 26 17 17 15 18 20
## [38,] 22 21 18 17 22 22
## [39,] 19 28 19 16 19 21
## [40,] 18 26 17 17 21 21
## [41,] 19 29 16 19 21 21
## [42,] 23 24 20 17 14 18
## [43,] 25 24 20 20 20 15
## [44,] 27 20 19 19 25 19
## [45,] 26 21 21 20 23 17
## [46,] 27 23 18 20 21 22
## [47,] 28 23 18 21 18 23
## [48,] 26 23 21 21 15 20
## [49,] 24 19 23 22 17 20
## [50,] 20 22 20 21 16 20
## [51,] 21 20 23 25 14 22
## [52,] 19 19 20 22 15 22
## [53,] 21 21 27 21 16 23
## [54,] 21 20 23 21 16 25
## [55,] 21 21 22 23 14 25
## [56,] 23 16 21 22 15 25
## [57,] 26 12 19 24 14 28
## [58,] 25 11 18 19 16 32
## [59,] 25 12 18 20 17 29
## [60,] 24 17 17 23 17 25
## [61,] 29 22 14 21 21 27
## [62,] 28 22 15 21 21 21
## [63,] 28 20 16 23 20 21
## [64,] 25 17 16 24 19 27
## [65,] 23 14 22 25 20 25
## [66,] 23 16 16 25 17 24
## [67,] 23 18 20 25 21 24
## [68,] 22 18 18 25 19 27
## [69,] 23 20 18 23 19 27
## [70,] 22 18 20 24 18 28
## [71,] 23 16 18 22 15 29
## [72,] 26 17 19 20 17 25
## [73,] 20 18 20 22 19 25
## [74,] 25 17 21 19 23 27
## [75,] 28 18 21 20 23 25
## [76,] 25 18 18 19 18 22
## [77,] 24 15 17 19 15 24
## [78,] 24 17 18 20 16 24
## [79,] 21 24 17 23 17 25
## [80,] 21 18 15 23 20 28
## [81,] 22 17 20 27 20 20
## [82,] 19 17 21 32 21 20
## [83,] 19 18 16 31 20 21
## [84,] 20 24 15 29 20 26
## [85,] 20 23 17 30 22 19
## [86,] 18 21 17 29 24 20
## [87,] 23 22 16 26 25 23
## [88,] 22 21 17 26 25 21
## [89,] 22 19 18 26 25 21
## [90,] 27 19 17 22 24 20
## [91,] 26 22 21 24 16 21
## [92,] 21 23 17 20 19 21
## [93,] 22 17 14 19 16 28
## [94,] 23 18 17 16 18 30
## [95,] 24 20 21 20 17 25
## [96,] 27 20 23 20 17 23
## [97,] 25 21 24 19 14 19
## [98,] 19 25 26 17 16 23
## [99,] 20 22 27 18 15 21
## [100,] 21 21 24 21 18 26
## [101,] 20 22 23 25 17 27
## [102,] 21 22 21 21 20 27
## [103,] 19 20 24 21 17 24
## [104,] 17 17 25 19 15 22
## [105,] 18 19 23 16 19 20
## [106,] 19 19 27 14 15 20
## [107,] 23 19 30 14 16 21
## [108,] 23 21 31 17 17 18
## [109,] 22 25 29 20 16 17
## [110,] 20 26 29 18 13 17
## [111,] 18 25 25 19 12 21
## [112,] 16 21 24 18 10 20
## [113,] 16 20 24 22 13 21
## [114,] 21 17 19 21 13 19
## [115,] 19 19 19 20 12 18
## [116,] 19 15 23 21 11 21
## [117,] 16 15 22 21 15 25
## [118,] 15 12 21 26 14 26
## [119,] 16 12 21 28 14 24
## [120,] 21 14 21 26 14 25
## [121,] 16 19 26 22 12 22
## [122,] 18 17 23 30 13 24
## [123,] 18 15 26 26 16 23
## [124,] 18 16 22 28 12 22
## [125,] 20 18 21 27 14 23
## [126,] 21 20 19 27 11 21
## [127,] 21 20 20 24 16 20
## [128,] 23 17 15 23 15 24
## [129,] 24 14 18 24 16 21
## [130,] 24 14 19 22 17 26
## [131,] 28 12 23 18 16 27
## [132,] 28 11 24 17 19 25
## [133,] 23 16 21 15 24 24
## [134,] 23 14 26 16 16 24
## [135,] 25 16 29 16 15 27
## [136,] 24 16 28 15 17 28
## [137,] 27 14 24 21 16 28
## [138,] 26 15 23 22 17 27
## [139,] 30 21 20 22 17 29
## [140,] 30 24 19 20 19 25
## [141,] 30 22 21 18 18 23
## [142,] 28 22 17 17 22 26
## [143,] 27 20 16 19 20 26
## [144,] 29 16 15 23 22 23
## [145,] 25 15 14 22 24 22
## [146,] 26 15 15 23 21 24
## [147,] 23 18 16 26 23 22
## [148,] 19 18 17 25 24 25
## [149,] 19 14 13 21 24 24
## [150,] 18 15 16 20 26 22
## [151,] 20 19 15 19 22 17
## [152,] 22 19 13 19 23 19
## [153,] 24 17 16 19 21 20
## [154,] 19 20 18 17 18 21
## [155,] 20 19 23 15 17 21
## [156,] 18 16 22 16 17 22
## [157,] 21 19 20 17 17 21
## [158,] 22 18 17 15 17 22
## [159,] 22 18 17 19 15 22
## [160,] 24 19 17 22 16 25
## [161,] 26 18 16 19 13 24
## [162,] 27 16 16 16 13 25
## [163,] 24 13 18 18 13 27
## [164,] 25 12 19 15 15 25
## [165,] 27 16 21 16 17 26
## [166,] 29 16 17 19 18 24
## [167,] 29 18 15 19 21 26
## [168,] 29 16 18 20 17 27
## [169,] 23 20 18 21 16 30
## [170,] 22 20 19 21 20 28
## [171,] 21 20 14 20 19 31
## [172,] 22 19 16 22 20 30
## [173,] 22 15 19 25 19 29
## [174,] 23 19 22 22 18 26
## [175,] 26 20 19 23 18 27
## [176,] 26 20 23 25 19 32
## [177,] 29 23 23 23 17 36
## [178,] 27 25 23 23 16 31
## [179,] 30 20 22 16 15 32
## [180,] 29 17 20 17 16 32
## [181,] 30 16 27 18 20 31
## [182,] 29 19 25 18 19 33
## [183,] 30 22 23 18 19 30
## [184,] 34 21 23 18 16 28
## [185,] 26 15 18 18 20 27
## [186,] 27 12 17 18 19 23
## [187,] 28 14 16 19 15 24
## [188,] 24 17 21 14 14 24
## [189,] 23 20 23 14 14 26
## [190,] 17 20 23 16 16 23
## [191,] 19 19 24 15 14 23
## [192,] 19 17 27 16 19 24
## [193,] 21 17 25 16 16 21
## [194,] 20 22 27 22 21 19
## [195,] 23 19 22 25 18 21
## [196,] 21 17 23 28 17 18
## [197,] 19 15 26 24 20 20
## [198,] 19 18 24 26 18 25
## [199,] 18 16 27 23 17 25
## [200,] 19 17 27 23 17 28
## [201,] 19 19 26 26 18 26
## [202,] 19 20 26 24 19 27
## [203,] 18 17 26 26 20 29
## [204,] 21 17 27 26 20 32
## [205,] 13 17 25 23 20 33
## [206,] 15 17 23 24 22 28
## [207,] 19 17 23 25 23 26
## [208,] 19 19 22 24 23 26
## [209,] 15 16 21 26 22 24
## [210,] 21 14 21 27 22 23
## [211,] 17 14 23 23 21 23
## [212,] 15 17 22 24 22 23
## [213,] 15 19 22 26 21 24
## [214,] 18 19 23 27 18 26
## [215,] 17 15 24 27 15 24
## [216,] 17 16 27 25 16 26
## [217,] 18 16 29 24 16 30
## [218,] 19 17 23 23 22 29
## [219,] 21 18 21 25 24 28
## [220,] 22 15 27 24 17 28
## [221,] 22 17 27 25 18 28
## [222,] 24 15 32 19 18 25
## [223,] 23 22 33 18 17 26
## [224,] 22 18 31 20 16 27
## [225,] 21 20 31 18 16 25
## [226,] 19 20 28 20 18 24
## [227,] 18 19 26 18 19 28
## [228,] 19 21 27 18 19 27
## [229,] 18 22 28 20 18 29
## [230,] 16 24 26 20 16 24
## [231,] 17 22 28 23 17 23
## [232,] 16 25 29 19 19 24
## [233,] 18 23 27 20 21 27
## [234,] 22 19 27 20 21 27
## [235,] 19 22 30 19 20 28
## [236,] 16 17 26 25 19 32
## [237,] 23 12 25 25 23 29
## [238,] 25 16 24 21 21 29
## [239,] 26 16 26 21 19 28
## [240,] 22 11 24 22 19 30
## [241,] 23 9 25 20 23 32
## [242,] 22 8 22 21 25 32
## [243,] 21 8 19 18 25 28
## [244,] 22 11 16 16 22 27
## [245,] 23 10 20 19 23 23
## [246,] 23 10 20 19 22 21
## [247,] 27 12 20 22 23 24
## [248,] 17 14 21 26 24 24
## [249,] 17 16 22 28 25 22
## [250,] 19 16 22 28 21 23
## [251,] 21 15 19 28 23 20
## [252,] 23 11 17 29 27 23
## [253,] 22 16 18 27 26 29
## [254,] 19 14 21 29 27 30
## [255,] 18 16 20 27 22 28
## [256,] 17 16 23 28 23 31
## [257,] 17 14 24 26 20 27
## [258,] 18 14 27 26 20 25
## [259,] 18 14 24 27 18 27
## [260,] 20 15 30 28 22 29
## [261,] 15 13 27 25 24 26
## [262,] 16 15 31 23 23 24
## [263,] 15 14 29 21 27 19
## [264,] 16 15 29 23 20 20
## [265,] 21 14 29 18 22 17
## [266,] 26 12 32 18 19 17
## [267,] 24 13 31 17 20 15
## [268,] 25 14 28 17 20 15
## [269,] 20 16 29 22 18 16
## [270,] 21 16 30 23 23 15
## [271,] 26 20 25 23 22 16
## [272,] 27 17 22 20 22 17
## [273,] 27 16 19 20 22 20
## [274,] 26 14 20 17 22 18
## [275,] 23 16 20 18 21 18
## [276,] 26 11 22 18 24 18
## [277,] 29 12 19 21 24 22
## [278,] 28 14 17 22 21 22
## [279,] 28 13 19 21 21 20
## [280,] 31 15 18 25 20 21
## [281,] 29 15 18 26 22 25
## [282,] 30 13 18 30 22 27
## [283,] 25 15 19 28 22 27
## [284,] 20 18 17 26 23 24
## [285,] 21 16 21 21 22 22
## [286,] 21 20 17 22 24 21
## [287,] 24 16 18 21 24 20
## [288,] 25 22 15 24 25 18
## [289,] 18 23 16 29 28 22
## [290,] 19 22 20 28 28 24
## [291,] 17 17 21 24 26 23
## [292,] 22 15 22 20 26 20
## [293,] 24 18 17 18 26 22
## [294,] 21 15 16 17 24 24
## [295,] 18 15 13 19 25 24
## [296,] 18 13 16 19 24 22
## [297,] 19 12 17 19 22 26
## [298,] 19 14 17 18 25 23
## [299,] 21 13 18 18 24 26
## [300,] 25 17 22 19 20 28
## [301,] 25 19 23 25 20 27
## [302,] 25 21 22 27 20 27
## [303,] 26 22 23 28 22 32
## [304,] 26 23 21 24 21 28
## [305,] 23 22 21 28 23 25
## [306,] 26 26 22 26 18 26
## [307,] 26 27 22 25 21 27
## [308,] 23 30 23 27 16 28
## [309,] 21 25 22 28 17 24
## [310,] 21 23 17 25 16 21
## [311,] 20 22 16 25 21 23
## [312,] 22 23 19 25 24 27
## [313,] 23 21 18 23 25 27
## [314,] 20 19 17 30 23 26
## [315,] 24 17 19 30 20 26
## [316,] 26 17 19 31 18 26
## [317,] 26 15 18 31 17 26
## [318,] 24 19 17 32 13 26
## [319,] 19 19 16 29 16 25
## [320,] 22 18 18 29 18 25
## [321,] 18 22 22 32 16 25
## [322,] 16 22 21 33 15 22
## [323,] 15 21 21 32 20 23
## [324,] 23 21 21 30 18 19
## [325,] 25 24 20 28 18 22
## [326,] 26 22 28 28 16 21
## [327,] 26 20 25 31 15 21
## [328,] 22 18 21 29 21 20
## [329,] 21 22 17 30 22 22
## [330,] 22 20 17 29 25 19
## [331,] 25 20 19 27 23 20
## [332,] 23 20 22 26 27 20
## [333,] 21 16 20 27 27 23
## [334,] 24 18 23 25 27 21
## [335,] 27 21 26 26 28 21
## [336,] 24 19 25 26 26 17
## [337,] 23 18 25 30 20 16
## [338,] 21 18 27 27 21 18
## [339,] 23 18 28 30 20 16
## [340,] 21 18 27 31 17 14
## [341,] 22 19 27 33 17 13
## [342,] 23 22 29 33 14 12
## [343,] 21 20 32 30 14 14
## [344,] 22 24 30 30 16 16
## [345,] 18 24 29 31 15 14
## [346,] 19 22 27 30 17 18
## [347,] 19 24 24 30 18 17
## [348,] 17 27 22 28 19 16
## [349,] 19 27 23 28 16 18
## [350,] 18 28 22 29 16 20
## [351,] 18 27 23 29 20 20
## [352,] 21 27 21 28 22 21
## [353,] 21 30 21 31 19 20
## [354,] 24 28 21 30 18 23
## [355,] 21 28 24 28 14 23
## [356,] 24 28 25 21 16 18
## [357,] 25 29 23 21 12 18
## [358,] 22 28 23 23 14 16
## [359,] 21 25 22 21 14 18
## [360,] 19 24 18 18 10 19
## [361,] 14 23 20 21 15 18
## [362,] 15 25 19 26 13 18
## [363,] 20 24 22 29 12 16
## [364,] 19 23 19 26 12 20
## [365,] 16 22 21 21 15 21
## [366,] 18 20 23 23 14 22
## [367,] 19 19 22 21 18 28
## [368,] 18 21 22 24 19 27
## [369,] 21 20 20 24 17 23
## [370,] 19 20 24 28 20 19
## [371,] 21 19 22 26 21 20
## [372,] 19 21 22 22 25 21
## [373,] 17 25 20 24 23 16
## [374,] 16 25 18 21 21 19
## [375,] 21 22 18 25 24 15
## [376,] 19 22 20 24 21 18
## [377,] 19 20 20 22 21 18
## [378,] 20 20 19 24 21 17
## [379,] 21 20 18 26 24 14
## [380,] 22 24 17 26 26 16
## [381,] 23 23 20 23 27 21
## [382,] 25 25 19 18 21 18
## [383,] 29 26 20 17 21 18
## [384,] 22 26 15 22 25 16
## [385,] 22 26 16 22 22 18
## [386,] 24 25 13 23 23 16
## [387,] 26 21 15 25 25 20
## [388,] 24 18 20 26 26 20
## [389,] 20 19 25 30 25 18
## [390,] 15 22 24 24 22 19
## [391,] 21 22 24 23 22 25
## [392,] 20 22 29 22 19 24
## [393,] 13 21 25 23 22 22
## [394,] 13 23 22 22 25 18
## [395,] 13 20 23 20 27 20
## [396,] 15 20 25 23 27 24
## [397,] 17 22 23 23 26 24
## [398,] 19 19 19 25 25 22
## [399,] 20 12 18 24 21 26
## [400,] 23 14 19 22 23 24
## [401,] 23 16 18 23 18 22
## [402,] 21 13 21 24 21 26
## [403,] 22 13 27 21 23 24
## [404,] 21 12 26 21 29 24
## [405,] 20 13 27 27 29 23
## [406,] 18 13 25 25 29 22
## [407,] 18 13 23 22 26 29
## [408,] 16 16 23 19 24 24
## [409,] 13 15 20 18 25 25
## [410,] 15 14 21 19 24 26
## [411,] 17 18 22 17 26 27
## [412,] 17 17 23 19 24 25
## [413,] 18 16 24 19 24 26
## [414,] 15 21 25 26 19 25
## [415,] 16 27 25 18 16 27
## [416,] 19 27 22 23 18 27
## [417,] 20 23 23 24 21 27
## [418,] 24 29 20 25 24 26
## [419,] 24 28 24 24 24 31
## [420,] 23 28 25 20 25 29
## [421,] 21 27 27 22 28 32
## [422,] 20 24 29 23 23 27
## [423,] 18 21 28 22 22 27
## [424,] 21 25 24 23 24 24
## [425,] 17 19 21 25 24 26
## [426,] 17 21 21 21 21 23
## [427,] 19 23 25 20 21 25
## [428,] 16 27 26 23 21 25
## [429,] 18 26 25 23 20 26
## [430,] 18 29 24 18 21 30
## [431,] 19 32 23 18 17 25
## [432,] 19 30 27 16 18 23
## [433,] 20 31 26 15 17 25
## [434,] 16 29 17 16 17 23
## [435,] 14 33 20 17 16 21
## [436,] 12 32 23 17 20 20
## [437,] 19 34 21 20 13 18
## [438,] 22 31 20 19 12 25
## [439,] 25 29 19 17 13 23
## [440,] 25 27 20 21 17 19
## [441,] 27 26 19 18 15 22
## [442,] 24 22 20 19 15 22
## [443,] 24 23 21 19 15 22
## [444,] 20 26 22 22 19 18
## [445,] 20 25 19 18 22 21
## [446,] 20 29 18 16 24 19
## [447,] 24 26 22 17 23 23
## [448,] 26 30 22 17 19 24
## [449,] 25 26 22 21 21 21
## [450,] 23 25 19 21 20 23
## [451,] 20 26 23 22 17 21
## [452,] 26 22 24 22 20 25
## [453,] 26 20 19 22 17 24
## [454,] 25 23 19 24 19 24
## [455,] 21 20 21 21 17 25
## [456,] 23 24 20 24 21 25
## [457,] 21 23 17 22 22 27
## [458,] 24 22 16 19 26 26
## [459,] 22 19 20 19 25 27
## [460,] 23 17 21 22 22 27
## [461,] 25 19 23 24 21 24
## [462,] 24 22 22 19 19 23
## [463,] 25 24 19 19 19 25
## [464,] 24 20 17 18 25 25
## [465,] 27 21 20 18 24 22
## [466,] 28 22 18 19 24 24
## [467,] 27 20 14 19 20 26
## [468,] 26 21 14 17 19 25
## [469,] 28 19 13 14 20 25
## [470,] 26 18 12 16 19 25
## [471,] 29 20 13 15 16 26
## [472,] 26 17 14 21 17 29
## [473,] 25 18 12 19 16 27
## [474,] 20 15 18 20 19 28
## [475,] 20 14 16 17 18 25
## [476,] 20 16 17 18 18 26
## [477,] 26 19 13 19 17 25
## [478,] 25 21 14 22 11 25
## [479,] 23 22 17 17 13 28
## [480,] 21 19 16 19 15 31
## [481,] 25 18 18 22 14 29
## [482,] 24 19 22 20 14 25
## [483,] 21 21 25 18 15 28
## [484,] 18 22 23 21 16 27
## [485,] 15 17 22 22 17 23
## [486,] 15 16 16 23 18 25
## [487,] 15 17 17 25 15 30
## [488,] 15 20 17 28 15 27
## [489,] 16 22 18 25 16 28
## [490,] 19 20 19 24 12 25
## [491,] 23 18 16 21 12 25
## [492,] 20 23 14 20 15 27
## [493,] 18 21 21 18 16 26
## [494,] 17 20 17 19 17 28
## [495,] 19 18 15 19 21 31
## [496,] 19 16 12 19 18 33
## [497,] 23 21 14 19 18 26
## [498,] 25 22 13 22 15 24
## [499,] 25 22 17 22 14 22
## [500,] 27 15 18 17 13 27
## [501,] 27 17 17 20 10 28
## [502,] 23 19 19 16 14 24
## [503,] 22 17 20 16 13 25
## [504,] 22 17 17 16 12 26
## [505,] 20 17 17 16 12 28
## [506,] 19 18 20 14 12 29
## [507,] 18 18 15 18 12 30
## [508,] 14 19 16 21 14 35
## [509,] 17 22 18 23 15 29
## [510,] 20 16 16 22 14 31
## [511,] 20 18 17 19 16 26
## [512,] 20 15 13 23 17 25
## [513,] 24 14 15 22 17 22
## [514,] 18 15 19 21 20 23
## [515,] 19 16 19 19 23 19
## [516,] 19 15 18 21 22 23
## [517,] 21 17 20 23 21 20
## [518,] 18 16 19 28 22 21
## [519,] 21 14 21 30 21 23
## [520,] 20 14 25 26 22 25
## [521,] 21 16 22 26 18 26
## [522,] 18 21 18 28 18 22
## [523,] 18 20 18 26 18 24
## [524,] 19 23 19 20 17 24
## [525,] 20 24 24 22 14 25
## [526,] 21 23 20 22 16 24
## [527,] 20 21 22 22 16 25
## [528,] 20 18 23 24 19 25
## [529,] 22 17 22 21 20 26
## [530,] 20 20 23 16 20 28
## [531,] 20 17 24 18 18 27
## [532,] 16 23 25 15 19 28
## [533,] 22 20 28 13 23 28
## [534,] 24 21 26 13 25 27
## [535,] 22 18 27 19 23 27
## [536,] 24 17 21 17 23 29
## [537,] 24 15 22 19 25 26
## [538,] 24 16 23 24 24 23
## [539,] 25 18 18 19 22 25
## [540,] 28 20 18 13 23 24
## [541,] 24 19 17 15 22 21
## [542,] 23 19 20 12 22 23
## [543,] 27 19 23 13 21 22
## [544,] 24 21 22 20 21 26
## [545,] 26 21 26 19 19 25
## [546,] 27 24 22 19 19 26
## [547,] 28 24 23 21 19 25
## [548,] 24 24 27 17 15 29
## [549,] 24 22 27 19 16 24
## [550,] 26 19 29 17 15 21
## [551,] 21 18 25 22 13 22
## [552,] 20 21 25 20 14 18
## [553,] 24 22 29 18 14 20
## [554,] 26 23 29 16 12 22
## [555,] 24 20 27 15 15 21
## [556,] 22 23 26 15 16 21
## [557,] 20 22 23 15 20 23
## [558,] 20 22 24 16 18 22
## [559,] 22 23 25 17 18 22
## [560,] 21 22 27 14 18 23
## [561,] 22 21 29 18 16 19
## [562,] 25 23 30 13 17 20
## [563,] 30 24 23 17 18 21
## [564,] 29 25 21 15 18 22
## [565,] 28 29 22 18 17 21
## [566,] 21 30 19 23 20 24
## [567,] 21 29 19 19 17 24
## [568,] 20 28 20 17 13 26
## [569,] 24 30 25 15 13 30
## [570,] 21 33 21 16 13 28
## [571,] 23 32 23 18 15 30
## [572,] 23 29 21 16 13 23
## [573,] 22 30 20 19 15 19
## [574,] 19 29 22 19 11 14
## [575,] 20 29 24 18 12 14
## [576,] 21 28 17 18 14 16
## [577,] 24 27 20 13 16 18
## [578,] 26 21 21 14 16 20
## [579,] 28 26 21 10 14 21
## [580,] 24 26 17 14 15 23
## [581,] 15 20 19 13 16 22
## [582,] 15 18 21 12 18 25
## [583,] 16 19 25 14 17 22
## [584,] 18 22 25 19 17 20
## [585,] 21 21 27 14 20 17
## [586,] 26 20 22 15 20 17
## [587,] 28 20 21 15 21 17
## [588,] 27 18 15 14 23 21
## [589,] 20 20 17 17 19 25
## [590,] 21 18 19 18 17 21
## [591,] 21 16 21 15 21 25
## [592,] 23 19 17 10 21 25
## [593,] 20 17 17 11 23 24
## [594,] 24 17 18 9 26 26
## [595,] 26 14 17 14 22 23
## [596,] 26 14 16 11 21 24
## [597,] 24 13 21 16 22 26
## [598,] 22 11 21 14 26 26
## [599,] 23 12 21 13 25 30
## [600,] 28 16 21 12 24 30
## [601,] 27 12 21 13 25 29
## [602,] 26 17 21 14 26 25
## [603,] 27 18 17 15 27 25
## [604,] 28 20 18 14 28 28
## [605,] 28 20 15 15 27 27
## [606,] 25 22 16 17 25 25
## [607,] 26 19 18 16 24 25
## [608,] 25 18 18 19 22 23
## [609,] 26 21 20 18 20 24
## [610,] 22 21 21 20 20 24
## [611,] 20 23 22 16 20 26
## [612,] 21 26 20 18 23 27
## [613,] 23 23 22 16 25 27
## [614,] 24 20 25 21 24 24
## [615,] 24 20 30 20 24 23
## [616,] 22 19 24 17 24 24
## [617,] 25 17 20 19 23 22
## [618,] 20 15 22 22 22 23
## [619,] 17 16 19 17 29 27
## [620,] 14 21 20 20 27 25
## [621,] 16 25 20 20 29 21
## [622,] 15 25 20 16 27 21
## [623,] 18 23 26 19 26 21
## [624,] 19 22 22 19 23 18
## [625,] 18 25 22 15 23 18
## [626,] 19 25 26 17 24 17
## [627,] 23 31 27 18 20 19
## [628,] 26 28 22 20 19 21
## [629,] 21 24 22 19 17 21
## [630,] 21 25 19 13 19 17
## [631,] 19 22 27 10 21 22
## [632,] 19 19 26 15 20 22
## [633,] 18 16 22 13 23 21
## [634,] 20 20 21 14 18 19
## [635,] 20 18 22 15 23 21
## [636,] 23 18 21 15 22 21
## [637,] 21 19 22 12 23 21
## [638,] 21 16 25 12 23 22
## [639,] 21 16 26 16 22 23
## [640,] 17 15 27 16 26 23
## [641,] 18 16 24 16 22 21
## [642,] 19 13 24 18 24 22
## [643,] 21 13 25 19 25 25
## [644,] 23 15 25 18 25 25
## [645,] 23 17 30 20 22 23
## [646,] 20 18 25 23 21 22
## [647,] 20 17 24 19 23 25
## [648,] 21 16 25 19 20 23
## [649,] 21 15 26 17 23 21
## [650,] 19 17 29 18 21 23
## [651,] 20 22 30 19 18 22
## [652,] 23 20 28 23 17 21
## [653,] 21 20 29 24 22 24
## [654,] 19 18 32 23 20 22
## [655,] 20 17 29 21 24 25
## [656,] 21 17 23 18 24 27
## [657,] 19 20 21 20 24 23
## [658,] 20 20 20 23 29 20
## [659,] 18 21 22 27 23 21
## [660,] 28 18 22 24 24 24
## [661,] 28 14 20 22 25 22
## [662,] 24 17 18 23 26 26
## [663,] 21 14 16 22 24 25
## [664,] 22 15 16 22 26 22
## [665,] 26 19 19 19 23 23
## [666,] 24 23 20 15 23 26
## [667,] 27 20 20 15 22 27
## [668,] 23 22 19 16 20 27
## [669,] 26 20 24 19 19 24
## [670,] 24 20 24 19 19 22
## [671,] 21 20 20 20 19 22
## [672,] 20 18 18 25 19 24
## [673,] 22 22 19 24 19 22
## [674,] 23 23 20 20 17 23
## [675,] 21 24 19 21 20 24
## [676,] 21 27 19 24 19 20
## [677,] 22 26 17 23 20 21
## [678,] 21 24 23 18 26 20
## [679,] 23 22 18 18 21 19
## [680,] 23 24 21 19 19 21
## [681,] 23 21 22 19 21 21
## [682,] 24 21 22 18 21 21
## [683,] 20 21 21 23 20 20
## [684,] 19 20 23 28 22 19
## [685,] 20 18 20 26 23 18
## [686,] 16 15 23 28 22 19
## [687,] 16 14 23 25 20 17
## [688,] 18 16 25 26 20 21
## [689,] 18 18 24 23 22 19
## [690,] 19 17 22 22 25 18
## [691,] 19 17 19 24 26 16
## [692,] 17 17 17 25 29 21
## [693,] 18 17 19 24 30 20
## [694,] 18 17 20 25 26 21
## [695,] 22 19 19 23 23 19
## [696,] 18 23 18 22 25 23
## [697,] 19 24 18 19 26 26
## [698,] 21 20 15 19 25 21
## [699,] 19 22 20 19 27 18
## [700,] 26 23 19 20 28 16
## [701,] 28 20 19 21 29 17
## [702,] 27 20 16 23 26 18
## [703,] 27 19 14 23 27 19
## [704,] 19 20 20 24 26 17
## [705,] 22 22 16 23 23 18
## [706,] 26 23 16 17 25 16
## [707,] 22 26 20 19 23 18
## [708,] 21 21 22 17 21 17
## [709,] 23 22 17 20 20 21
## [710,] 24 24 18 17 24 18
## [711,] 22 22 18 18 25 18
## [712,] 24 19 18 16 22 22
## [713,] 23 21 17 19 20 21
## [714,] 20 22 15 21 19 20
## [715,] 22 20 11 19 16 25
## [716,] 22 21 11 17 18 26
## [717,] 17 22 12 19 18 25
## [718,] 19 22 18 15 15 30
## [719,] 19 21 19 16 20 30
## [720,] 19 21 15 17 22 24
## [721,] 22 17 14 15 25 23
## [722,] 21 16 15 14 22 22
## [723,] 23 15 11 15 18 27
## [724,] 19 17 17 14 21 23
## [725,] 16 18 20 14 20 24
## [726,] 17 20 20 15 21 25
## [727,] 17 21 20 11 25 26
## [728,] 20 20 18 14 24 24
## [729,] 21 22 15 14 30 22
## [730,] 20 22 19 14 27 24
## [731,] 18 20 21 14 26 25
## [732,] 19 18 23 20 25 26
## [733,] 22 23 21 21 21 25
## [734,] 26 21 20 17 19 29
## [735,] 30 20 18 17 17 30
## [736,] 31 21 17 13 19 29
## [737,] 30 22 21 12 21 34
## [738,] 27 25 18 13 21 34
## [739,] 31 18 23 13 23 35
## [740,] 29 21 21 15 22 37
## [741,] 24 22 18 16 28 37
## [742,] 24 22 16 17 28 37
## [743,] 26 28 16 15 27 36
## [744,] 25 22 17 15 25 38
## [745,] 25 23 16 17 24 31
## [746,] 25 21 21 16 25 30
## [747,] 26 23 22 19 22 28
## [748,] 26 26 21 18 18 33
## [749,] 25 27 23 19 19 33
## [750,] 21 25 28 18 19 33
## [751,] 18 24 26 17 20 31
## [752,] 16 28 23 15 19 34
## [753,] 13 25 18 17 18 30
## [754,] 13 23 14 16 23 24
## [755,] 13 21 14 14 19 24
## [756,] 16 22 14 14 20 22
## [757,] 17 25 18 13 20 24
## [758,] 18 22 17 16 20 23
## [759,] 20 21 19 16 22 20
## [760,] 18 22 23 16 20 21
## [761,] 17 23 22 15 19 22
## [762,] 16 19 27 14 19 24
## [763,] 17 19 30 15 20 26
## [764,] 21 22 28 12 20 26
## [765,] 21 29 26 15 14 26
## [766,] 21 27 25 18 13 29
## [767,] 23 27 26 16 18 29
## [768,] 24 27 26 17 18 31
## [769,] 24 27 26 20 21 29
## [770,] 23 24 27 17 17 25
## [771,] 23 24 27 18 16 25
## [772,] 21 23 28 13 16 30
## [773,] 20 24 33 13 15 31
## [774,] 19 26 31 16 19 29
## [775,] 21 18 31 19 20 29
## [776,] 25 20 31 17 16 31
## [777,] 26 21 28 17 18 30
## [778,] 27 18 27 18 18 28
## [779,] 26 21 27 19 19 27
## [780,] 24 22 25 23 18 26
## [781,] 25 25 23 24 19 24
## [782,] 26 16 24 25 21 24
## [783,] 29 13 22 23 21 23
## [784,] 26 16 21 23 22 25
## [785,] 25 19 22 23 25 18
## [786,] 24 21 25 25 20 21
## [787,] 24 24 24 25 24 17
## [788,] 31 24 25 25 21 16
## [789,] 25 23 23 21 25 19
## [790,] 21 23 24 20 21 18
## [791,] 19 22 25 16 18 20
## [792,] 19 20 24 18 20 21
## [793,] 16 22 24 18 21 18
## [794,] 17 22 24 18 20 20
## [795,] 19 23 27 20 21 18
## [796,] 19 23 23 20 23 18
## [797,] 18 21 24 18 23 20
## [798,] 19 20 21 19 22 22
## [799,] 20 24 22 21 22 22
## [800,] 20 19 22 19 23 21
## [801,] 20 18 19 19 22 20
## [802,] 20 16 16 23 24 23
## [803,] 23 19 17 23 22 23
## [804,] 27 17 17 27 24 20
## [805,] 27 16 15 21 25 18
## [806,] 24 19 16 19 27 21
## [807,] 19 19 18 21 28 18
## [808,] 23 17 18 25 28 19
## [809,] 20 19 17 21 25 17
## [810,] 23 18 14 23 24 17
## [811,] 21 21 15 24 25 18
## [812,] 16 20 14 28 25 20
## [813,] 16 21 15 26 18 19
## [814,] 17 19 18 23 20 19
## [815,] 18 21 18 23 18 19
## [816,] 18 21 19 26 15 17
## [817,] 18 20 22 31 18 14
## [818,] 20 21 25 33 17 19
## [819,] 18 18 27 35 19 21
## [820,] 17 16 26 33 22 19
## [821,] 18 22 24 36 23 17
## [822,] 17 22 22 32 21 19
## [823,] 16 16 25 34 24 16
## [824,] 16 14 27 32 23 18
## [825,] 18 19 26 30 25 17
## [826,] 17 20 23 27 24 13
## [827,] 15 21 22 29 19 17
## [828,] 17 21 20 25 20 20
## [829,] 17 22 23 27 18 19
## [830,] 17 20 20 22 18 14
## [831,] 19 20 18 19 18 14
## [832,] 19 20 17 21 16 15
## [833,] 24 23 18 22 13 15
## [834,] 23 21 19 22 14 12
## [835,] 23 14 21 22 13 15
## [836,] 24 16 29 24 12 14
## [837,] 24 19 30 24 13 14
## [838,] 26 17 27 24 16 16
## [839,] 24 17 22 27 18 14
## [840,] 27 14 26 28 23 13
## [841,] 27 15 30 27 24 12
## [842,] 24 16 26 25 25 11
## [843,] 19 20 25 23 21 11
## [844,] 19 19 23 16 24 13
## [845,] 21 19 27 17 27 13
## [846,] 21 16 25 20 25 14
## [847,] 24 16 23 22 25 16
## [848,] 25 17 23 22 23 15
## [849,] 23 18 23 20 25 15
## [850,] 23 14 20 20 23 17
## [851,] 22 17 22 20 21 19
## [852,] 22 15 24 23 18 21
## [853,] 21 16 28 24 14 21
## [854,] 18 15 26 21 15 20
## [855,] 21 16 26 15 15 22
## [856,] 20 17 23 15 16 22
## [857,] 21 16 21 19 15 20
## [858,] 22 15 23 22 16 19
## [859,] 21 18 21 17 19 24
## [860,] 22 19 22 19 16 24
## [861,] 19 18 23 23 17 22
## [862,] 20 19 21 18 21 23
## [863,] 19 21 19 20 18 27
## [864,] 19 22 16 22 17 31
## [865,] 22 24 18 19 13 28
## [866,] 20 19 16 18 13 28
## [867,] 22 22 19 17 13 28
## [868,] 23 25 20 18 13 27
## [869,] 23 28 20 21 10 24
## [870,] 26 21 20 23 11 20
## [871,] 25 19 22 23 12 18
## [872,] 27 22 20 21 12 17
## [873,] 30 24 21 21 14 16
## [874,] 31 23 21 19 13 20
## [875,] 26 27 26 20 13 16
## [876,] 26 20 21 19 13 16
## [877,] 23 19 22 20 13 17
## [878,] 20 16 22 23 10 16
## [879,] 24 18 24 22 15 17
## [880,] 23 19 26 18 12 15
## [881,] 20 18 27 15 13 19
## [882,] 20 17 28 16 16 19
## [883,] 20 19 24 17 18 21
## [884,] 22 19 26 19 14 21
## [885,] 22 20 24 18 19 16
## [886,] 24 20 23 17 18 18
## [887,] 27 20 25 14 17 18
## [888,] 25 21 26 15 18 21
## [889,] 20 19 23 17 18 19
## [890,] 20 19 24 16 20 15
## [891,] 17 17 26 17 20 15
## [892,] 25 20 27 18 20 13
## [893,] 25 20 26 19 17 16
## [894,] 30 20 22 15 15 12
## [895,] 31 19 23 16 14 13
## [896,] 28 16 24 16 17 16
## [897,] 33 17 23 17 17 14
## [898,] 35 18 23 16 17 13
## [899,] 31 16 28 17 18 15
## [900,] 29 15 26 17 17 15
## [901,] 28 18 22 18 16 14
## [902,] 29 18 25 17 13 14
## [903,] 25 20 25 17 16 13
## [904,] 24 20 28 16 15 16
## [905,] 26 25 25 17 15 18
## [906,] 25 23 23 14 15 17
## [907,] 29 22 18 16 18 17
## [908,] 27 29 18 14 15 18
## [909,] 29 28 16 17 17 21
## [910,] 29 24 21 17 16 16
## [911,] 33 20 22 19 18 14
## [912,] 26 20 20 16 18 21
## [913,] 25 18 18 21 19 23
## [914,] 27 15 18 20 27 23
## [915,] 24 15 17 19 26 28
## [916,] 22 12 16 22 26 24
## [917,] 21 14 13 16 22 25
## [918,] 24 15 15 20 28 24
## [919,] 30 13 19 22 20 23
## [920,] 25 16 17 27 17 21
## [921,] 24 14 18 24 22 20
## [922,] 28 12 20 22 23 14
## [923,] 32 14 20 18 25 18
## [924,] 32 16 21 19 24 17
## [925,] 30 19 21 20 24 18
## [926,] 32 15 23 20 22 19
## [927,] 28 19 24 20 18 17
## [928,] 23 22 25 20 18 19
## [929,] 19 21 22 24 20 16
## [930,] 18 22 27 26 19 15
## [931,] 21 24 29 23 20 15
## [932,] 20 24 22 23 20 13
## [933,] 20 20 20 24 22 13
## [934,] 23 19 20 27 21 15
## [935,] 23 21 20 23 22 16
## [936,] 23 21 19 25 21 15
## [937,] 27 24 18 22 18 15
## [938,] 25 27 18 24 20 20
## [939,] 21 29 22 24 22 17
## [940,] 23 30 19 22 24 17
## [941,] 24 26 18 25 20 16
## [942,] 22 23 17 25 24 15
## [943,] 18 18 16 27 16 17
## [944,] 17 20 17 24 18 19
## [945,] 16 29 16 26 15 16
## [946,] 16 26 17 27 17 14
## [947,] 18 20 13 27 18 14
## [948,] 18 20 17 29 15 15
## [949,] 22 21 15 30 15 19
## [950,] 22 18 17 29 14 20
## [951,] 22 17 19 27 13 21
## [952,] 22 15 24 27 12 21
## [953,] 24 16 24 29 15 22
## [954,] 23 18 20 26 14 21
## [955,] 29 18 19 30 15 15
## [956,] 24 17 21 26 18 10
## [957,] 21 24 16 23 17 13
## [958,] 22 20 18 23 15 14
## [959,] 22 17 20 24 16 12
## [960,] 24 16 20 24 15 11
## [961,] 24 19 20 24 16 12
## [962,] 24 18 23 24 18 10
## [963,] 21 20 22 27 16 13
## [964,] 22 18 25 26 15 17
## [965,] 20 20 27 24 17 16
## [966,] 20 19 18 25 19 17
## [967,] 23 23 19 23 12 14
## [968,] 20 22 23 28 13 13
## [969,] 23 20 21 33 13 11
## [970,] 22 19 20 35 16 9
## [971,] 23 19 20 29 17 13
## [972,] 21 19 20 26 16 7
## [973,] 20 22 20 27 18 8
## [974,] 21 21 23 29 16 9
## [975,] 22 19 24 25 13 10
## [976,] 21 19 25 25 14 13
## [977,] 21 17 24 24 16 13
## [978,] 19 17 23 19 17 14
## [979,] 21 19 22 19 16 13
## [980,] 21 15 21 24 13 13
## [981,] 19 13 22 28 17 9
## [982,] 19 12 25 25 22 11
## [983,] 21 12 24 23 18 10
## [984,] 16 12 25 23 19 10
## [985,] 15 19 30 23 17 14
## [986,] 14 22 26 24 19 13
## [987,] 17 18 28 25 19 13
## [988,] 15 22 22 22 18 15
## [989,] 21 20 18 20 25 15
## [990,] 22 21 13 26 25 12
## [991,] 20 20 15 27 23 15
## [992,] 19 22 16 26 28 14
## [993,] 20 22 15 24 22 13
## [994,] 13 24 15 19 21 18
## [995,] 16 24 14 18 16 15
## [996,] 18 23 18 13 21 14
## [997,] 18 22 17 14 18 16
## [998,] 22 20 22 19 19 18
## [999,] 20 18 22 19 21 18
## [1000,] 19 21 23 18 19 15# Plot importance
barplot(sort(colMeans(bart_model$varcount), decreasing = TRUE),
main = "Variable Importance in BART",
xlab = "Predictors", ylab = "Splits")
Problem 9.
# View the dataset structure
data("OJ")
dim(OJ) # Check total number of observations## [1] 1070 18set.seed(123) # Ensuring reproducibility
# Randomly select 800 observations for training
train_indices <- sample(1:nrow(OJ), size = 800)
# Create training and test sets
train_set <- OJ[train_indices, ]
test_set <- OJ[-train_indices, ]
# Check dimensions
dim(train_set) # Should be 800 observations## [1] 800 18dim(test_set) # Remaining observations## [1] 270 18# Fit a classification tree model
set.seed(123)
tree_model <- rpart(Purchase ~ ., data = train_set, method = "class")summary(tree_model)## Call:
## rpart(formula = Purchase ~ ., data = train_set, method = "class")
## n= 800
##
## CP nsplit rel error xerror xstd
## 1 0.49201278 0 1.0000000 1.0000000 0.04410089
## 2 0.03514377 1 0.5079872 0.5335463 0.03672576
## 3 0.02555911 2 0.4728435 0.5335463 0.03672576
## 4 0.01277955 4 0.4217252 0.4472843 0.03433576
## 5 0.01000000 7 0.3833866 0.4472843 0.03433576
##
## Variable importance
## LoyalCH StoreID PriceDiff SalePriceMM WeekofPurchase
## 45 9 9 6 6
## PriceMM DiscMM PctDiscMM PriceCH ListPriceDiff
## 5 5 4 4 3
## SalePriceCH STORE SpecialCH
## 2 1 1
##
## Node number 1: 800 observations, complexity param=0.4920128
## predicted class=CH expected loss=0.39125 P(node) =1
## class counts: 487 313
## probabilities: 0.609 0.391
## left son=2 (450 obs) right son=3 (350 obs)
## Primary splits:
## LoyalCH < 0.5036 to the right, improve=134.49530, (0 missing)
## StoreID < 3.5 to the right, improve= 40.88655, (0 missing)
## STORE < 0.5 to the left, improve= 20.84871, (0 missing)
## Store7 splits as RL, improve= 20.84871, (0 missing)
## PriceDiff < 0.015 to the right, improve= 19.14298, (0 missing)
## Surrogate splits:
## StoreID < 3.5 to the right, agree=0.660, adj=0.223, (0 split)
## WeekofPurchase < 246.5 to the right, agree=0.625, adj=0.143, (0 split)
## PriceCH < 1.825 to the right, agree=0.600, adj=0.086, (0 split)
## PriceMM < 1.89 to the right, agree=0.596, adj=0.077, (0 split)
## ListPriceDiff < 0.035 to the right, agree=0.581, adj=0.043, (0 split)
##
## Node number 2: 450 observations, complexity param=0.03514377
## predicted class=CH expected loss=0.1355556 P(node) =0.5625
## class counts: 389 61
## probabilities: 0.864 0.136
## left son=4 (423 obs) right son=5 (27 obs)
## Primary splits:
## PriceDiff < -0.39 to the right, improve=18.543390, (0 missing)
## DiscMM < 0.72 to the left, improve= 9.309254, (0 missing)
## SalePriceMM < 1.435 to the right, improve= 9.309254, (0 missing)
## PctDiscMM < 0.3342595 to the left, improve= 9.309254, (0 missing)
## LoyalCH < 0.7645725 to the right, improve= 8.822549, (0 missing)
## Surrogate splits:
## DiscMM < 0.72 to the left, agree=0.967, adj=0.444, (0 split)
## SalePriceMM < 1.435 to the right, agree=0.967, adj=0.444, (0 split)
## PctDiscMM < 0.3342595 to the left, agree=0.967, adj=0.444, (0 split)
## SalePriceCH < 2.075 to the left, agree=0.949, adj=0.148, (0 split)
##
## Node number 3: 350 observations, complexity param=0.02555911
## predicted class=MM expected loss=0.28 P(node) =0.4375
## class counts: 98 252
## probabilities: 0.280 0.720
## left son=6 (180 obs) right son=7 (170 obs)
## Primary splits:
## LoyalCH < 0.2761415 to the right, improve=14.991900, (0 missing)
## StoreID < 3.5 to the right, improve= 6.562913, (0 missing)
## Store7 splits as RL, improve= 4.617311, (0 missing)
## STORE < 0.5 to the left, improve= 4.617311, (0 missing)
## SpecialCH < 0.5 to the right, improve= 4.512108, (0 missing)
## Surrogate splits:
## STORE < 1.5 to the left, agree=0.629, adj=0.235, (0 split)
## StoreID < 1.5 to the left, agree=0.589, adj=0.153, (0 split)
## PriceCH < 1.875 to the left, agree=0.589, adj=0.153, (0 split)
## SalePriceCH < 1.875 to the left, agree=0.586, adj=0.147, (0 split)
## SalePriceMM < 1.84 to the left, agree=0.571, adj=0.118, (0 split)
##
## Node number 4: 423 observations
## predicted class=CH expected loss=0.09929078 P(node) =0.52875
## class counts: 381 42
## probabilities: 0.901 0.099
##
## Node number 5: 27 observations
## predicted class=MM expected loss=0.2962963 P(node) =0.03375
## class counts: 8 19
## probabilities: 0.296 0.704
##
## Node number 6: 180 observations, complexity param=0.02555911
## predicted class=MM expected loss=0.4222222 P(node) =0.225
## class counts: 76 104
## probabilities: 0.422 0.578
## left son=12 (106 obs) right son=13 (74 obs)
## Primary splits:
## PriceDiff < 0.05 to the right, improve=12.110850, (0 missing)
## SalePriceMM < 2.04 to the right, improve=11.572070, (0 missing)
## DiscMM < 0.25 to the left, improve= 5.760121, (0 missing)
## PctDiscMM < 0.1345485 to the left, improve= 5.760121, (0 missing)
## ListPriceDiff < 0.18 to the right, improve= 5.597236, (0 missing)
## Surrogate splits:
## SalePriceMM < 1.94 to the right, agree=0.933, adj=0.838, (0 split)
## DiscMM < 0.08 to the left, agree=0.822, adj=0.568, (0 split)
## PctDiscMM < 0.038887 to the left, agree=0.822, adj=0.568, (0 split)
## ListPriceDiff < 0.135 to the right, agree=0.800, adj=0.514, (0 split)
## PriceMM < 2.04 to the right, agree=0.783, adj=0.473, (0 split)
##
## Node number 7: 170 observations
## predicted class=MM expected loss=0.1294118 P(node) =0.2125
## class counts: 22 148
## probabilities: 0.129 0.871
##
## Node number 12: 106 observations, complexity param=0.01277955
## predicted class=CH expected loss=0.4245283 P(node) =0.1325
## class counts: 61 45
## probabilities: 0.575 0.425
## left son=24 (8 obs) right son=25 (98 obs)
## Primary splits:
## LoyalCH < 0.3084325 to the left, improve=3.118983, (0 missing)
## WeekofPurchase < 247.5 to the right, improve=2.489639, (0 missing)
## SpecialMM < 0.5 to the left, improve=2.454538, (0 missing)
## PriceCH < 1.755 to the right, improve=2.048863, (0 missing)
## PriceMM < 2.04 to the right, improve=1.514675, (0 missing)
##
## Node number 13: 74 observations
## predicted class=MM expected loss=0.2027027 P(node) =0.0925
## class counts: 15 59
## probabilities: 0.203 0.797
##
## Node number 24: 8 observations
## predicted class=CH expected loss=0 P(node) =0.01
## class counts: 8 0
## probabilities: 1.000 0.000
##
## Node number 25: 98 observations, complexity param=0.01277955
## predicted class=CH expected loss=0.4591837 P(node) =0.1225
## class counts: 53 45
## probabilities: 0.541 0.459
## left son=50 (46 obs) right son=51 (52 obs)
## Primary splits:
## LoyalCH < 0.442144 to the right, improve=3.071463, (0 missing)
## WeekofPurchase < 248.5 to the right, improve=2.208454, (0 missing)
## SpecialMM < 0.5 to the left, improve=2.011796, (0 missing)
## STORE < 0.5 to the left, improve=1.624324, (0 missing)
## StoreID < 5.5 to the right, improve=1.624324, (0 missing)
## Surrogate splits:
## WeekofPurchase < 255 to the left, agree=0.622, adj=0.196, (0 split)
## SalePriceCH < 1.755 to the right, agree=0.571, adj=0.087, (0 split)
## STORE < 2.5 to the right, agree=0.571, adj=0.087, (0 split)
## PriceMM < 2.205 to the right, agree=0.561, adj=0.065, (0 split)
## DiscCH < 0.115 to the left, agree=0.561, adj=0.065, (0 split)
##
## Node number 50: 46 observations
## predicted class=CH expected loss=0.326087 P(node) =0.0575
## class counts: 31 15
## probabilities: 0.674 0.326
##
## Node number 51: 52 observations, complexity param=0.01277955
## predicted class=MM expected loss=0.4230769 P(node) =0.065
## class counts: 22 30
## probabilities: 0.423 0.577
## left son=102 (8 obs) right son=103 (44 obs)
## Primary splits:
## SpecialCH < 0.5 to the right, improve=2.020979, (0 missing)
## STORE < 1.5 to the left, improve=1.724009, (0 missing)
## SpecialMM < 0.5 to the left, improve=1.680070, (0 missing)
## WeekofPurchase < 245 to the right, improve=1.384615, (0 missing)
## StoreID < 5.5 to the right, improve=1.319751, (0 missing)
## Surrogate splits:
## DiscCH < 0.27 to the right, agree=0.942, adj=0.625, (0 split)
## SalePriceCH < 1.54 to the left, agree=0.942, adj=0.625, (0 split)
## PctDiscCH < 0.149059 to the right, agree=0.942, adj=0.625, (0 split)
## SalePriceMM < 1.64 to the left, agree=0.923, adj=0.500, (0 split)
## DiscMM < 0.42 to the right, agree=0.904, adj=0.375, (0 split)
##
## Node number 102: 8 observations
## predicted class=CH expected loss=0.25 P(node) =0.01
## class counts: 6 2
## probabilities: 0.750 0.250
##
## Node number 103: 44 observations
## predicted class=MM expected loss=0.3636364 P(node) =0.055
## class counts: 16 28
## probabilities: 0.364 0.636Training Error Rate:
The training error rate is given by the expected loss at the root node:
Expected loss = 0.39125, which represents the misclassification rate in the training set.
Training accuracy = 1 - 0.39125 = 0.60875 (≈ 60.88%).
the following nodes are terminal:
Nodes: 4, 5, 7, 24, 25, 50, 51, 102, 103 (total: 9 terminal nodes)
tree_model## n= 800
##
## node), split, n, loss, yval, (yprob)
## * denotes terminal node
##
## 1) root 800 313 CH (0.60875000 0.39125000)
## 2) LoyalCH>=0.5036 450 61 CH (0.86444444 0.13555556)
## 4) PriceDiff>=-0.39 423 42 CH (0.90070922 0.09929078) *
## 5) PriceDiff< -0.39 27 8 MM (0.29629630 0.70370370) *
## 3) LoyalCH< 0.5036 350 98 MM (0.28000000 0.72000000)
## 6) LoyalCH>=0.2761415 180 76 MM (0.42222222 0.57777778)
## 12) PriceDiff>=0.05 106 45 CH (0.57547170 0.42452830)
## 24) LoyalCH< 0.3084325 8 0 CH (1.00000000 0.00000000) *
## 25) LoyalCH>=0.3084325 98 45 CH (0.54081633 0.45918367)
## 50) LoyalCH>=0.442144 46 15 CH (0.67391304 0.32608696) *
## 51) LoyalCH< 0.442144 52 22 MM (0.42307692 0.57692308)
## 102) SpecialCH>=0.5 8 2 CH (0.75000000 0.25000000) *
## 103) SpecialCH< 0.5 44 16 MM (0.36363636 0.63636364) *
## 13) PriceDiff< 0.05 74 15 MM (0.20270270 0.79729730) *
## 7) LoyalCH< 0.2761415 170 22 MM (0.12941176 0.87058824) *Terminal node 7:
Node Number: 7
Observations: 170
Predicted Class: MM (Minute Maid)
Expected Loss: 0.129 (≈ 12.9% misclassification rate)
Class Counts: 22 customers bought CH (Tropicana), while 148 chose MM
Probability Distribution: 12.9% CH, 87.1% MM
Interpretation:
This node represents a segment of consumers who overwhelmingly purchase Minute Maid over Tropicana.
The factors leading to this group could be low loyalty to CH, StoreID, special discounts, or pricing differences.
The expected loss (misclassification rate) suggests that most instances here are confidently classified as MM, with low uncertainty.
# Generate the tree plot
rpart.plot(tree_model, type = 3, extra = 104, box.palette = "auto", main = "Classification Tree for OJ Purchase")
Interpret the results:
Primary Predictor - Customer Loyalty (LoyalCH)
The first split happens at LoyalCH >= 0.5.
Customers loyal to CH are more likely to purchase Tropicana.
Those less loyal undergo further splits based on price and promotions.
Price Difference (PriceDiff) Impact
When PriceDiff >= -0.39, CH purchase likelihood increases to 90%.
If PriceDiff < -0.39, MM is preferred (70% probability).
Effect of Special Offers (SpecialCH)
If SpecialCH = 1 in certain segments, CH purchase probability increases.
Without a promotion, the likelihood shifts toward MM.
Terminal Nodes & Decision Paths
Each leaf node represents a final prediction based on conditions.
Some nodes predict 100% CH purchases, meaning those customers show absolute preference.
Others indicate 80%+ MM purchases, suggesting stronger Minute Maid loyalty.
# Predict class labels for the test set
pred_test <- predict(tree_model, newdata = test_set, type = "class")# Load required package for confusion matrix
library(caret)## Warning: package 'caret' was built under R version 4.4.2## Loading required package: lattice##
## Attaching package: 'caret'## The following object is masked from 'package:purrr':
##
## lift# Generate confusion matrix
conf_matrix <- confusionMatrix(pred_test, test_set$Purchase)
print(conf_matrix)## Confusion Matrix and Statistics
##
## Reference
## Prediction CH MM
## CH 141 24
## MM 25 80
##
## Accuracy : 0.8185
## 95% CI : (0.7673, 0.8626)
## No Information Rate : 0.6148
## P-Value [Acc > NIR] : 3.407e-13
##
## Kappa : 0.6175
##
## Mcnemar's Test P-Value : 1
##
## Sensitivity : 0.8494
## Specificity : 0.7692
## Pos Pred Value : 0.8545
## Neg Pred Value : 0.7619
## Prevalence : 0.6148
## Detection Rate : 0.5222
## Detection Prevalence : 0.6111
## Balanced Accuracy : 0.8093
##
## 'Positive' Class : CH
## # Extract misclassification rate
test_error <- 1 - conf_matrix$overall["Accuracy"]
print(paste("Test Error Rate:", round(test_error, 3)))## [1] "Test Error Rate: 0.181"library(tree)## Warning: package 'tree' was built under R version 4.4.3set.seed(123) # Ensures reproducibility
tree_model <- tree(Purchase ~ ., data = train_set)
# Summary of the tree
summary(tree_model)##
## Classification tree:
## tree(formula = Purchase ~ ., data = train_set)
## Variables actually used in tree construction:
## [1] "LoyalCH" "PriceDiff"
## Number of terminal nodes: 8
## Residual mean deviance: 0.7625 = 603.9 / 792
## Misclassification error rate: 0.165 = 132 / 800cv_results <- cv.tree(tree_model, FUN = prune.misclass)
# Print cross-validation results
print(cv_results)## $size
## [1] 8 5 3 2 1
##
## $dev
## [1] 139 139 157 167 313
##
## $k
## [1] -Inf 0 8 11 154
##
## $method
## [1] "misclass"
##
## attr(,"class")
## [1] "prune" "tree.sequence"cv_df <- data.frame(TreeSize = cv_results$size, CV_Error = cv_results$dev)
# Print dataframe to verify
print(cv_df)## TreeSize CV_Error
## 1 8 139
## 2 5 139
## 3 3 157
## 4 2 167
## 5 1 313library(ggplot2)
ggplot(cv_df, aes(x = TreeSize, y = CV_Error)) +
geom_line(color = "blue", size = 1) +
geom_point(color = "red", size = 2) +
labs(title = "Tree Size vs. Cross-Validated Classification Error",
x = "Tree Size (Number of Terminal Nodes)",
y = "Cross-Validation Error") +
theme_minimal()## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
The tree size corresponding to the lowest cross-validated classification error rate is 8 terminal nodes.
From the plot, the error rate decreases steadily as the tree size increases, reaching its minimum at tree size 8. This suggests that the optimal complexity for the model balances accuracy without overfitting.
cv_results <- cv.tree(tree_model, FUN = prune.misclass)
# Extract tree size with the lowest cross-validation error
optimal_size <- cv_results$size[which.min(cv_results$dev)]
print(paste("Optimal Tree Size:", optimal_size))## [1] "Optimal Tree Size: 8"pruned_tree <- prune.misclass(tree_model, best = optimal_size)
# Plot pruned tree
library(tree)
plot(pruned_tree)
text(pruned_tree, pretty = 0)
title("Pruned Tree - Optimal Size")
# Predict on training set using the full (unpruned) tree
pred_train_unpruned <- predict(tree_model, newdata = train_set, type = "class")
# Compute confusion matrix for unpruned tree
conf_matrix_unpruned <- table(Predicted = pred_train_unpruned, Actual = train_set$Purchase)
# Compute training error rate
train_error_unpruned <- 1 - sum(diag(conf_matrix_unpruned)) / sum(conf_matrix_unpruned)
print(paste("Training Error Rate (Unpruned Tree):", round(train_error_unpruned, 3)))## [1] "Training Error Rate (Unpruned Tree): 0.165"# Predict on training set using the pruned tree
pred_train_pruned <- predict(pruned_tree, newdata = train_set, type = "class")
# Compute confusion matrix for pruned tree
conf_matrix_pruned <- table(Predicted = pred_train_pruned, Actual = train_set$Purchase)
# Compute training error rate
train_error_pruned <- 1 - sum(diag(conf_matrix_pruned)) / sum(conf_matrix_pruned)
print(paste("Training Error Rate (Pruned Tree):", round(train_error_pruned, 3)))## [1] "Training Error Rate (Pruned Tree): 0.165"# Predict on test set using pruned tree
pred_pruned <- predict(pruned_tree, newdata = test_set, type = "class")
# Generate confusion matrix
conf_matrix_pruned <- confusionMatrix(pred_pruned, test_set$Purchase)
# Compute test error rate
test_error_pruned <- 1 - conf_matrix_pruned$overall["Accuracy"]
print(paste("Test Error Rate after Pruning:", round(test_error_pruned, 3)))## [1] "Test Error Rate after Pruning: 0.185"Unpruned tree test error = 0.181, therefore the pruned tree has a slightly higher test error rate (0.185 vs. 0.181).