One of the primary disadvantages of any clustering technique is that it is difficult to evaluate its performance. To tackle this problem, the metric of V-Measure was developed. The calculation of the V-Measure first requires the calculation of two terms:-
- Homogeneity: A perfectly homogeneous clustering is one where each cluster has data-points belonging to the same class label. Homogeneity describes the closeness of the clustering algorithm to this perfection.
- Completeness: A perfectly complete clustering is one where all data-points belonging to the same class are clustered into the same cluster. Completeness describes the closeness of the clustering algorithm to this perfection.
Trivial Homogeneity: It is the case when the number of clusters is equal to the number of data points and each point is in exactly one cluster. It is the extreme case when homogeneity is highest while completeness is minimum. 
Trivial Completeness: It is the case when all the data points are clustered into one cluster. It is the extreme case when homogeneity is minimum and completeness is maximum. 
Assume that each data point in the above diagrams is of the different class label for Trivial Homogeneity and Trivial Completeness. Note: The term homogeneous is different from completeness in the sense that while talking about homogeneity, the base concept is of the respective cluster which we check whether in each cluster does each data point is of the same class label. While talking about completeness, the base concept is of the respective class label which we check whether data points of each class label is in the same cluster. 
In the above diagram, the clustering is perfectly homogeneous since in each cluster the data points of are of the same class label but it is not complete because not all data points of the same class label belong to the same class label. 
In the above diagram, the clustering is perfectly complete because all data points of the same class label belong to the same cluster but it is not homogeneous because the 1st cluster contains data points of many class labels. Let us assume that there are N data samples, C different class labels, K clusters and 
number of data-points belonging to the class c and cluster k. Then the homogeneity h is given by the following:- 
where 
and 
The completeness c is given by the following:- 
where 
and 
Thus the weighted V-Measure 
is given by the following:- 
The factor 
can be adjusted to favour either the homogeneity or the completeness of the clustering algorithm. The primary advantage of this evaluation metric is that it is independent of the number of class labels, the number of clusters, the size of the data and the clustering algorithm used and is a very reliable metric. The following code will demonstrate how to compute the V-Measure of a clustering algorithm. The data used is the Detection of Credit Card Fraud which can be downloaded from Kaggle. The clustering algorithm used is the Variational Bayesian Inference for Gaussian Mixture Model. Step 1: Importing the required libraries
Python3
import pandas as pdimport matplotlib.pyplot as pltfrom sklearn.cluster import KMeansfrom sklearn.metrics import v_measure_score |
Step 2: Loading and Cleaning the data
Python3
# Changing the working location to the location of the filecd C:\Users\Dev\Desktop\Kaggle\Credit Card Fraud# Loading the datadf = pd.read_csv('creditcard.csv')# Separating the dependent and independent variablesy = df['Class']X = df.drop('Class', axis = 1)X.head() |
Step 3: Building different clustering models and comparing their V-Measure scores In this step, 5 different K-Means Clustering Models will be built with each model clustering the data into a different number of clusters.
Python3
# List of V-Measure Scores for different modelsv_scores = []# List of different types of covariance parametersN_Clusters = [2, 3, 4, 5, 6] |
a) n_clusters = 2
Python3
# Building the clustering modelkmeans2 = KMeans(n_clusters = 2)# Training the clustering modelkmeans2.fit(X)# Storing the predicted Clustering labelslabels2 = kmeans2.predict(X)# Evaluating the performancev_scores.append(v_measure_score(y, labels2)) |
b) n_clusters = 3
Python3
# Building the clustering modelkmeans3 = KMeans(n_clusters = 3)# Training the clustering modelkmeans3.fit(X)# Storing the predicted Clustering labelslabels3 = kmeans3.predict(X)# Evaluating the performancev_scores.append(v_measure_score(y, labels3)) |
c) n_clusters = 4
Python3
# Building the clustering modelkmeans4 = KMeans(n_clusters = 4)# Training the clustering modelkmeans4.fit(X)# Storing the predicted Clustering labelslabels4 = kmeans4.predict(X)# Evaluating the performancev_scores.append(v_measure_score(y, labels4)) |
d) n_clusters = 5
Python3
# Building the clustering modelkmeans5 = KMeans(n_clusters = 5)# Training the clustering modelkmeans5.fit(X)# Storing the predicted Clustering labelslabels5 = kmeans5.predict(X)# Evaluating the performancev_scores.append(v_measure_score(y, labels5)) |
e) n_clusters = 6
Python3
# Building the clustering modelkmeans6 = KMeans(n_clusters = 6)# Training the clustering modelkmeans6.fit(X)# Storing the predicted Clustering labelslabels6 = kmeans6.predict(X)# Evaluating the performancev_scores.append(v_measure_score(y, labels6)) |
Step 4: Visualizing the results and comparing the performances
Python3
# Plotting a Bar Graph to compare the modelsplt.bar(N_Clusters, v_scores)plt.xlabel('Number of Clusters')plt.ylabel('V-Measure Score')plt.title('Comparison of different Clustering Models')plt.show() |
