Section A

The response is count and the predictors are season, holiday, workingday, temp, atemp, humidity, windspeed, casual, and registered.

# Build a tree model using function tree()
tr.model = tree(
  count ~ season + holiday + workingday + temp +
    atemp + humidity + windspeed + casual + registered,
  data = trainData
)

# Display the statistical summary of the tree
summary(tr.model)

## 
## Regression tree:
## tree(formula = count ~ season + holiday + workingday + temp + 
##     atemp + humidity + windspeed + casual + registered, data = trainData)
## Variables actually used in tree construction:
## [1] "registered" "casual"    
## Number of terminal nodes:  8 
## Residual mean deviance:  1875 = 13590000 / 7249 
## Distribution of residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -158.000  -19.290   -3.983    0.000   18.020  259.700

• The decision tree mainly depends on the registered and casual variables, which seem to be the most direct predictors of bike rental counts.
• With 8 terminal nodes, the model reflects a moderate level of complexity while still managing to avoid overfitting.
• The Residual Mean Deviance (1875) and the distribution of residuals give insight into how well the model performs. Although there are some notable outliers (like -158 and +259.7), the residuals generally cluster around zero.
• Overall, the tree offers a simplified yet understandable view of the data relationships. However, its predictive accuracy could potentially be enhanced through hyperparameter tuning or by exploring more advanced models such as random forests or gradient boosting.

Section B

Perform cross-validation to choose the best tree by calling cv.tree().

CV_Bike = cv.tree(tr.model)

Section C

Plot the model results of b) and determine the best size of the optimal tree.

plot(CV_Bike$size, CV_Bike$dev, type = "b",
     xlab = "Tree Size",
     ylab = "Deviance",
     main = "Cross Validation Results")

• The plot indicates that the ideal tree size is 4.
• At this point, the deviance reaches its lowest value before leveling off with larger trees. This suggests the model achieves a good balance—complex enough to capture key patterns without being overly complicated or prone to overfitting.

Section D

Prune the tree by calling prune.tree() function with the best size found in c).

prunedTree = prune.tree(tr.model, best = 4)

Section E

Plot the best tree model.

plot(prunedTree)
text(prunedTree, pretty = 0)

• The pruned decision tree, refined to 4 terminal nodes through cross-validation, forecasts a numeric target variable. It begins by splitting on the registered variable, with the initial split at registered < 200.5. Subsequent splits lead to four final predicted values: 40.02, 180.40, 332.80, and 553.20.
• This streamlined version of the tree is more interpretable, reduces the risk of overfitting, and maintains solid predictive power by emphasizing the most impactful decision points.

Section F

Compute the test error using the test data set.

tr.predictions = predict(prunedTree, newdata = testData)

#compute and display the MSE
tr.mse <- mean((tr.predictions - testData$count)^2)
tr.mse

## [1] 4762.857

• The Mean Squared Error (MSE) of 4762.857 gives a numerical indication of the average difference between the model’s predictions and the actual values in the test dataset.
• Whether this MSE is considered good or bad depends on the scale of the target variable count.
  • For example, if count values usually fall in the thousands, this level of error might be reasonable. But if the values are much smaller, the error could be considered quite large in comparison.

Section A

The response is count and the predictors are season, holiday, workingday, temp, atemp, humidity, windspeed, casual, and registered.

set.seed(123)  # Set seed for reproducibility
rf.model = randomForest(count ~ season + holiday + workingday + temp + 
                          atemp + humidity + windspeed + casual + registered,
                        data = trainData, importance = TRUE)

# Print the random forest model result
print(rf.model)

## 
## Call:
##  randomForest(formula = count ~ season + holiday + workingday +      temp + atemp + humidity + windspeed + casual + registered,      data = trainData, importance = TRUE) 
##                Type of random forest: regression
##                      Number of trees: 500
## No. of variables tried at each split: 3
## 
##           Mean of squared residuals: 130.3723
##                     % Var explained: 99.6

• The Random Forest model performs exceptionally well on the training data, accounting for 99.6% of the variance and showing a low residual error of 130.3723. However, this strong performance might suggest overfitting, so it’s important to validate the model on unseen test data.
• Using 500 trees and 3 variables at each split, the model strikes a good balance between precision and computational efficiency.
• To determine which features, have the greatest impact on predictions, tools like importance() or varImpPlot() can be used to assess variable importance.

Section B

Compute the test error using the test data set.

rf.predictions = predict(rf.model, newdata = testData)

# Calculate the mean squared error
rf.mse <- mean((rf.predictions - testData$count)^2)
rf.mse

## [1] 177.4572

• The test Mean Squared Error (MSE) of 177.457 reflects strong predictive accuracy on new data, although it’s slightly higher than the training error of 130.3723—indicating a minor degree of overfitting.
• Overall, the model demonstrates solid performance, with only a small drop in accuracy from training to testing, which is a normal and expected outcome in most modeling scenarios.

Section C

Extract variable importance measure using importance() function.

importance(rf.model)

##               %IncMSE IncNodePurity
## season      14.332261    2084868.26
## holiday      4.836423      77828.05
## workingday  27.288525    2036925.63
## temp        13.620187    6064088.53
## atemp       16.569101    7734568.86
## humidity    17.822489    5095570.13
## windspeed    7.787657    1215118.78
## casual      34.829490   58165642.18
## registered 117.913863  152055043.04

• The variable registered plays the most significant role in predicting the target variable count, with casual and workingday also contributing notably.
• In contrast, features such as holiday and windspeed have little influence on the model’s predictive capability.
• These findings can guide model simplification by removing less impact variables and highlight the main drivers behind the predictions.

Section D

Plot the variable importance using function varImpPlot(). Which are the top 2 important predictors in this model?

varImpPlot(rf.model)

• Registered and casual are the top predictors, playing a major role in improving the model’s accuracy and reducing variance.
• The model relies heavily on these two variables, indicating they are the primary factors affecting the target variable, count.