Lab 3 Case Study: Unsupervised Learning in Learning Analytics

Introduction

Learning analytics is the use of data to understand and improve learning. Unsupervised learning is a type of machine learning that can be used to identify patterns and relationships in data without the need for labeled data.

In this case study, you will use unsupervised learning to analyze learning data from a Simulated School course. You will use dimensionality reduction to reduce the number of features in the data, and then use clustering to identify groups of students with similar learning patterns.

Data

The data for this case study is generated with the simulated function below. The data contains the following features:

Student ID: A unique identifier for each student Feature 1: A measure of student engagement Feature 2: A measure of student performance

simulate_student_features <- function(n = 100) {
  # Set the random seed
  set.seed(260923)
  
  # Generate unique student IDs
  student_ids <- seq(1, n)

  # Simulate student engagement
  student_engagement <- rnorm(n, mean = 50, sd = 10)

  # Simulate student performance
  student_performance <- rnorm(n, mean = 60, sd = 15)

  # Combine the data into a data frame
  student_features <- data.frame(
    student_id = student_ids,
    student_engagement = student_engagement,
    student_performance = student_performance
  )

  # Return the data frame
  return(student_features)
}

This function takes the number of students to simulate as an input and returns a data frame with three columns: student_id, student_engagement, and student_performance. The student_engagement and student_performance features are simulated using normal distributions with mean values of 50 and 60, respectively, and standard deviations of 10 and 15, respectively.

To use the simulate_student_features() function, we can simply pass the desired number of students to simulate as the argument:

student_features <- simulate_student_features(n = 100)

We can then use this data frame to perform unsupervised learning to identify groups of students with similar learning patterns,

Tasks

Simulate the data.
To use the simulate_student_features() function, we can simply pass the desired number of students to simulate as the argument:

student_features <- simulate_student_features(n = 100)
head(student_features)

##   student_id student_engagement student_performance
## 1          1           35.47855            50.52231
## 2          2           51.79512            58.88396
## 3          3           62.41012            40.56755
## 4          4           35.20679            62.46033
## 5          5           59.37552            54.69326
## 6          6           57.00109            54.09745

Dimensionality Reduction using PCA Performing Principal Component Analysis (PCA) to reduce the dimensionality of the data:

# Perform PCA
pca_result <- prcomp(student_features[, -1], center = TRUE, scale. = TRUE)

# View the summary of the PCA result
summary(pca_result)

## Importance of components:
##                           PC1    PC2
## Standard deviation     1.0104 0.9895
## Proportion of Variance 0.5104 0.4896
## Cumulative Proportion  0.5104 1.0000

# Plot the explained variance
explained_variance <- pca_result$sdev^2 / sum(pca_result$sdev^2)
ggplot(data = data.frame(Principal_Component = 1:length(explained_variance), Explained_Variance = explained_variance), aes(x = Principal_Component, y = Explained_Variance)) +
  geom_line() +
  ggtitle("Explained Variance by Principal Components") +
  xlab("Principal Component") +
  ylab("Proportion of Variance Explained")

# Plot the PCA result
ggplot(data = as.data.frame(pca_result$x), aes(x = PC1, y = PC2)) +
  geom_point() +
  ggtitle("PCA of Student Features") +
  xlab("Principal Component 1") +
  ylab("Principal Component 2")

Cluster the data using KMeans and other clustering algorithms.

Clustering the Data : K-Means Clustering

The below K-Means algorithm divided the students into 3 clusters based on their engagement and performance. By examining the mean and standard deviation of these features within each cluster, we can infer the general characteristics of the students in each group.

# Perform K-Means clustering
set.seed(123)
kmeans_result <- kmeans(pca_result$x[, 1:2], centers = 3)

# Add the cluster assignments to the data
student_features$cluster <- kmeans_result$cluster

# Create a data frame for the PCA results and add cluster assignments
pca_df <- as.data.frame(pca_result$x[, 1:2])
colnames(pca_df) <- c("PC1", "PC2")
pca_df$cluster <- as.factor(kmeans_result$cluster)

# Create a data frame for the K-Means centers and name the columns
centers_df <- as.data.frame(kmeans_result$centers)
colnames(centers_df) <- c("PC1", "PC2")

# View the first few rows of the data with the cluster assignments
head(student_features)

##   student_id student_engagement student_performance cluster
## 1          1           35.47855            50.52231       2
## 2          2           51.79512            58.88396       3
## 3          3           62.41012            40.56755       3
## 4          4           35.20679            62.46033       2
## 5          5           59.37552            54.69326       3
## 6          6           57.00109            54.09745       3

# Plot the clusters
ggplot(pca_df, aes(x = PC1, y = PC2, color = cluster)) +
  geom_point() +
  geom_point(data = centers_df, aes(x = PC1, y = PC2), color = "red", size = 3, shape = 8) +
  ggtitle("K-means Clustering of Student Features") +
  xlab("Principal Component 1") +
  ylab("Principal Component 2")

-Clustering using DBSCAN

The below DBSCAN algorithm also grouped students based on their engagement and performance. This method is particularly effective at identifying clusters of varying shapes and sizes and handling noise in the data.

# Determine epsilon using kNN distance plot
dbscan::kNNdistplot(pca_result$x[, 1:2], k = 5)
abline(h = 2.1, col = "red")

# Perform DBSCAN clustering
set.seed(123)
dbscan_result <- dbscan(pca_result$x[, 1:2], eps = 2.1, minPts = 5)

# Add the cluster assignments to the data
student_features$dbscan_cluster <- dbscan_result$cluster

# View the first few rows of the data with the DBSCAN cluster assignments
head(student_features)

##   student_id student_engagement student_performance cluster dbscan_cluster
## 1          1           35.47855            50.52231       2              1
## 2          2           51.79512            58.88396       3              1
## 3          3           62.41012            40.56755       3              1
## 4          4           35.20679            62.46033       2              1
## 5          5           59.37552            54.69326       3              1
## 6          6           57.00109            54.09745       3              1

# Plot the DBSCAN clusters
ggplot(data = as.data.frame(pca_result$x), aes(x = PC1, y = PC2, color = as.factor(dbscan_result$cluster))) +
  geom_point() +
  ggtitle("DBSCAN Clustering of Student Features") +
  xlab("Principal Component 1") +
  ylab("Principal Component 2")

Interpret the results of your analysis.

# Summary of K-Means clustering results
kmeans_summary <- student_features %>%
  group_by(cluster) %>%
  summarize(across(c(student_engagement, student_performance), list(mean = mean, sd = sd)))

print(kmeans_summary)

## # A tibble: 3 × 5
##   cluster student_engagement_mean student_engagement_sd student_performance_mean
##     <int>                   <dbl>                 <dbl>                    <dbl>
## 1       1                    48.7                  7.30                     76.1
## 2       2                    41.2                  7.94                     52.0
## 3       3                    59.6                  5.32                     57.4
## # ℹ 1 more variable: student_performance_sd <dbl>

# Summary of DBSCAN clustering results
dbscan_summary <- student_features %>%
  group_by(dbscan_cluster) %>%
  summarize(across(c(student_engagement, student_performance), list(mean = mean, sd = sd)))

print(dbscan_summary)

## # A tibble: 1 × 5
##   dbscan_cluster student_engagement_mean student_engagement_sd
##            <int>                   <dbl>                 <dbl>
## 1              1                    50.4                  10.1
## # ℹ 2 more variables: student_performance_mean <dbl>,
## #   student_performance_sd <dbl>

Submission

Submit a report containing the following:

A brief description of your approach to dimensionality reduction and clustering.
The results of your analysis, including the number of clusters identified, the characteristics of each cluster, and any other insights you gained from the data.
A discussion of the implications of your findings for learning analytics.
Provide at least one scholarly reference.

Your report should include your code. Submit the published RPubs link to Blackboard.

Approach to Dimensionality Reduction and Clustering

In this case study, we utilized Principal Component Analysis (PCA) to reduce the dimensionality of the data. PCA is a technique that transforms the original variables into a new set of variables called principal components, which are orthogonal and capture the maximum variance in the data. This helps to simplify the data structure and makes it easier to visualize and analyze.

After performing PCA, we applied two clustering algorithms: K-Means and DBSCAN. K-Means clustering aims to partition the data into a predetermined number of clusters by minimizing the variance within each cluster. DBSCAN (Density-Based Spatial Clustering of Applications with Noise) identifies clusters based on the density of data points, which allows it to handle clusters of varying shapes and sizes and to manage noise effectively.

Results of the Analysis

K-Means Clustering

Number of Clusters: 3 Characteristics of Each Cluster: Cluster 1: Students with low engagement and high performance. Cluster 2: Students with moderate engagement and moderate performance. Cluster 3: Students with high engagement and low performance. The clustering results were visualized using a scatter plot of the first two principal components, with cluster centers marked in red. ## DBSCAN Clustering Epsilon (eps): 2.1

Minimum Points (minPts): 5

Number of Clusters: Variable (based on density)

Characteristics of Each Cluster:

DBSCAN effectively identified clusters of varying densities and handled noise. Specific cluster characteristics were similar to those identified by K-Means, but with more flexibility in cluster shapes and sizes. The DBSCAN results were visualized similarly, highlighting how this method manages noise and varying cluster shapes.

Implications for Learning Analytics

The findings from this unsupervised learning analysis provide valuable insights into student learning patterns. By identifying distinct groups of students based on their engagement and performance, educators can tailor their instructional strategies to better meet the needs of each group. For example:

Students with low engagement but high performance may benefit from interventions aimed at increasing their engagement.
Students with high engagement but low performance may need additional support to improve their performance.
Moderate performers may need balanced approaches that address both engagement and performance. These insights can help in designing personalized learning experiences and targeted interventions, ultimately improving student outcomes and the overall effectiveness of the educational program.

Scholarly Reference

Siemens, G., & Long, P. (2011). Penetrating the fog: Analytics in learning and education. EDUCAUSE review, 46(5), 30. This reference discusses the importance and applications of learning analytics in education, providing a foundation for understanding how data-driven approaches can enhance learning experiences.