# Load necessary libraries
library(dbscan)
## 
## Attaching package: 'dbscan'
## The following object is masked from 'package:stats':
## 
##     as.dendrogram
library(NbClust)
library(ellipse)
## 
## Attaching package: 'ellipse'
## The following object is masked from 'package:graphics':
## 
##     pairs
library(mclust)
## Package 'mclust' version 6.1
## Type 'citation("mclust")' for citing this R package in publications.
# Function to simulate student data
simulate_student_data <- function(n = 200) {
  set.seed(789)
  student_interaction <- rnorm(n, mean = 70, sd = 15)
  student_scores <- rnorm(n, mean = 80, sd = 20)
  
  # Combine into a data frame
  data.frame(student_interaction = student_interaction, student_scores = student_scores)
}

# Simulate the data
student_data <- simulate_student_data(n = 200)

# Perform dimensionality reduction on the data using PCA
pca_result <- prcomp(student_data[, c("student_interaction", "student_scores")], scale = TRUE)

# Plot the PCA results
plot(pca_result$x, col = 'blue', pch = 19, cex = 0.6)

# Determine the optimal number of clusters using the silhouette method
nb <- NbClust(student_data, distance = "euclidean", min.nc = 2, max.nc = 15, method = "ward.D2")

## *** : The Hubert index is a graphical method of determining the number of clusters.
##                 In the plot of Hubert index, we seek a significant knee that corresponds to a 
##                 significant increase of the value of the measure i.e the significant peak in Hubert
##                 index second differences plot. 
## 

## *** : The D index is a graphical method of determining the number of clusters. 
##                 In the plot of D index, we seek a significant knee (the significant peak in Dindex
##                 second differences plot) that corresponds to a significant increase of the value of
##                 the measure. 
##  
## ******************************************************************* 
## * Among all indices:                                                
## * 5 proposed 2 as the best number of clusters 
## * 8 proposed 3 as the best number of clusters 
## * 3 proposed 5 as the best number of clusters 
## * 1 proposed 9 as the best number of clusters 
## * 2 proposed 12 as the best number of clusters 
## * 1 proposed 13 as the best number of clusters 
## * 1 proposed 14 as the best number of clusters 
## * 2 proposed 15 as the best number of clusters 
## 
##                    ***** Conclusion *****                            
##  
## * According to the majority rule, the best number of clusters is  3 
##  
##  
## *******************************************************************
# Cluster the data using DBSCAN
set.seed(789)
dbscan_result <- dbscan(student_data[, c("student_interaction", "student_scores")], eps = 0.3, minPts = 5)

# Add the cluster results to the data frame
student_data$cluster <- as.factor(dbscan_result$cluster)

# Plot the clusters
pairs(student_data, col = dbscan_result$cluster)
legend("topright", legend = levels(as.factor(dbscan_result$cluster)), fill = 1:max(dbscan_result$cluster))

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