# Clustering Analysis for Classifying Student Academic Performance in Higher Education
## 3 Methodology
### Clustering Model Evaluation  
In the clustering analysis phase, the accuracy or quality of clustering results will be determined and confirmed. It is an important measurement in determining which algorithm achieved the best performance by using input data for the study. Clustering evaluation is a stand-alone process and is not included during the clustering process. It is always carried out after the final output of the clustering is produced [38]. There are two methods practiced in measuring the quality of clustering results: internal validation and external validation. Internal validation is the process of evaluating clustering that is compared to the results of the clustering itself, namely the relationship between the structures of clusters that have been formed. This is more realistic and efficient in solving problems involving educational datasets with increasing daily sizes and dimensions. This study used three types of internal validation methods that are often used in recent clustering studies: (1) the Davies-Bouldin index (DB), (2) the silhouette coefficient index, and (3) the Calinski-Harabasz index (CH). Important notations to be used in mathematical formulas for grouping assessment measurements are as follows: D is the input data set, n is the number of data points in D, g is the midpoint for the entire D data set, P is the dimension number of D, NC is the number of the group, C\({}_{i}\) is the \(i\)-th group, n\({}_{i}\) is the number of data points in \(C_{i}\), \(C_{i}\) is the midpoint for the C\({}_{i}\) group, and \(d\)(\(x\),\(y\)) is the distance between points x and y [40]. **Davies-Bouldin. ** The Davies-Bouldin (DB) metric is a method that has long been introduced but is still widely used in internal validation measurements. DB uses intra-group variance and inter-group midpoint distance to determine the worst group pairs. Thus, the reduction in DB index value provides the optimum group number. The mathematical formula for DB is defined as Equation (3) [40]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \
\begin{tabular}{c} **Clustering** \\ **Model** \\ \end{tabular} & **Parameter** & **Tested Tuning Value** &
\begin{tabular}{c} **Optimum Tuning** \\ **Value** \\ \end{tabular} \\ \hline \multirow{4}{*}{k-means} & \(k\) & 2 to 10 & 5 \\  & \(n\_iter\) & 5–10 in steps of 1 & 10 \\  & \(max\_iter\) & 100–500 in steps of 100 & 300 \\  & \(Distance\) metric & & Euclidean \\ \hline \multirow{4}{*}{BIRCH} & \(n\_clusters\) & 2 to 10 & 5 \\  & \(threshold\) & 0.1–0.10 in steps of 0.1 & 0.5 \\  & \(branching\_factor\) & 10–100 in steps of 10 & 50 \\ \hline \multirow{4}{*}{DBSCAN} & \(epsilon\) & 0.1–3.0 in steps 0.5 & 0.5 (Model C) and 1.0 \\  & \(metric\) & Euclidean & (Model A and B) \\  & \(min\_samples\) & 1000–2000 in steps of 20 & 1000 (Model B and C) \\ \cline{1-1}  & \(1200\) (Model A) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of parameter tuning. \[DB=\frac{1}{NC}\sum_{i}\max_{j\
eq i}\frac{\frac{1}{n_{i}}\sum_{x\in C_{i}}d(x,c_{i} )+\frac{1}{n_{j}}\sum_{x\in C_{j}}d\left(x,c_{j}\right)}{d\left(c_{i},c_{j} \right)} \tag{3}\]  
**Silhouette Coefficient Index. ** The silhouette coefficient index is used to evaluate the quality and strength of a group. The high silhouette coefficient value indicates a model with a better batch and signals that an object is well matched to its batch and does not match the adjacent batches. The equation for calculating the value of the silhouette coefficient of a single sample is as Equation (4):  
\[s=\frac{1}{NC}\sum_{i}\left(\frac{1}{n_{i}}\sum_{x\in C_{i}}\frac{b(x)-a(x)}{ max[b(x),a(x)]}\right) \tag{4}\]  
where \(a(x)=\frac{1}{n_{i}-1}\sum_{y\in C_{i}y\
eq x}d(x,y)\), and \(b(x)=min_{j\
eq i}\left[\frac{1}{n_{j}}\sum_{y\in C_{j}}d(x,y)\right]\). \(S\) does not consider \(c_{i}\) or \(g\) and uses pairwise distances between all objects in the clusters to calculate the density \(a(x)\). At the same time, \(b(x)\) measures separation, the average distance of objects to alternative groups or the nearest second group.