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Perform Agglomerative Hierarchical Clustering Using AGNES Algorithm

Perform Agglomerative Hierarchical Clustering Using AGNES Algorithm

AGNES clustering algo

AGNES (Agglomerative Nesting) is one of the most popular hierarchical clustering algorithms  used in data mining. You might be familiar with some basic clustering algorithms such as K-Means, K-Medoids and CLARANS clustering techniques. If not, check out the following links to our previous related articles!

This article focuses on a clustering algorithm approach that works somewhat differently from the above-mentioned ones. But before going into its details, have a quick look at the term ‘hierarchical clustering’ means.

Hierarchical clustering in data mining is a cluster formation and analysis technique that builds groups of similar objects by forming a hierarchy of clusters. That hierarchy forms a tree-like structure which is known as a dendrogram.

What AGNES algorithm does?

AGNES dendrogram

Image source

AGNES algorithm uses a “bottom-up” approach for hierarchical clustering. As shown in the above figure, the algorithm forms singleton clusters of each of the data points. It then groups them from bottom to top in the tree-like structure (called dendrogram) until all the similar points form a single cluster (depicted by the root of the dendrogram). Since it collects or in other words, agglomerates data points for clusters’ formation, AGNES is said to be an “agglomerative” hierarchical clustering technique and hence the name “AGglomerative NESting” – collective merging of clusters.

Practical implementation of AGNES

Here’s a demonstration of implementing the AGNES algorithm on a custom-created dataset. The code has been implemented using Google colab with Python version 3.7.10. Step-wise explanation of the code is as follows:

  1. Import required Python modules (math for computations to create clusters and pylab for clusters’ visualization)
 import math
 import pylab as pl 
  1. Initialize a collection of numerical points, which will be used to form coordinates of data points to be clustered.
 data_points = """
 1,0.527,0.63,2,0.753,0.434,4,0.672,0.724,4,0.573,0.373,2,0.846,0.283,
 6,0.763,0.172,7,0.941,0.782,8,0.542,0.541,3,0.666,0.078,10,0.121,0.372,
 12,0.256,0.927,10,0.891,0.039,10,0.682,0.122,15,0.671,0.878,12,0.67,0.51,
 16,0.91,0.892,13,0.777,0.893,17,0.139,0.748,16,0.373,0.781,22,0.672,0.289,
 20,0.778,0.892,21,0.743,0.367,22,0.453,0.367,23,0.488,0.567,24,0.573,0.319, 24,0.341,0.449,25,0.545,0.372,26,0.474,0.936,27,0.445,0.725,32,0.436,0.468"""  
  1. Split the above-defined comma-separated collection to extract coordinates of data points.

x = data_points.split(',') 

Form the data points by pairing successive points as x and y coordinates.

dataset = [(float(x[i]), float(x[i+1])) for i in range(1, len(x)-1, 3)]
  1. Define a function to compute distance between two data points.
 def distance(x, y):
     return math.sqrt(math.pow(x[0]-y[0], 2)+math.pow(x[1]-y[1], 2)) 

The function will calculate distance using the distance between points (a1,b1) and (a2,b2) using the distance formula as follows:                                    

AGNES distance formula

The following 3 steps define functions for different ways of computing distance/similarity between two clusters. We can find distance between closest points of two clusters; less the distance, more is the similarity between the clusters. The function min_distance() for such calculation has been defined in step (5) below. It takes one-one point from two clusters and computes distance between them using above eq(i). The process is repeated for each point in the first cluster taken with each of the points of the other cluster. The minimum of the computed distances is then returned. 

The same procedure, except min() replaced with max(), is followed if we want to compute the distance between two clusters based on the distance between their farthest points. The function max_distance() defined in step (6) performs such calculation. 

Instead of considering one-one points from both clusters to compute the similarity between two clusters, step (7) defines a function avg_distance() which takes all points of both the clusters into consideration. 

  1. Define min_distance() function.
 def min_distance(Ci, Cj):
      return min(distance(i, j) for i in Ci for j in Cj) 

Minimum distance between clusters Ci and Cj is calculated as:

See Also

AGNES min distance eq

6. Define max_distance function

 def max_distance(Ci, Cj):
      return max(distance(i, j) for i in Ci for j in Cj) 

Maximum distance between clusters Ci and Cj is given as:

AGNES max distance eq
  1. Define avg_distance() function.
 def avg_distance(Ci, Cj):
      return sum(distance(i, j) for i in Ci for j in Cj)/(len(Ci)*len(Cj)) 

Average distance between clusters Ci and Cj is given by the formula:

AGNES avg distance eq
  1. Define a function to calculate minimum distance between two clusters.
 def minimum(A):
      min = 1000
      a = 0; b = 0 
      for x in range(len(A)):
             for y in range(len(A[x])):
              if x != y and A[x][y] < min:
                  min = A[x][y];a = x; b = y
     return (a, b, min) 
  1. Define a function to form the actual clusters.
 def AGNES(ds, distance, k):
 “””
The parameters will be a tuple as (dataset, min/max/avg distance between clusters, number of clusters to be formed i.e. k)

Initialize empty arrays to store clusters (array C) and distances between clusters (array M) (this distance between clusters will be computed depending on which of the three methods from those defined in steps (5),(6) and (7) you use).
 “””
     C = [];M = []
     for i in ds:  #For each point in the dataset
         Ci = []   
         Ci.append(i)   #Add point i to ith cluster division
         C.append(Ci)   #Append Ci to the array of clusters C
     for i in C:  #For each point in C
         Mi = []  #Initialize an array for ith point
         for j in C: #For every point j in C (including itself)
 “””
Compute distance between i and j points. Min_distance, max_distance or avg_distance whichever distance calculation method’s name will be provided as an argument to AGNES(), accordingly that method will be used when distance(i,j) will be called in the following step.
 “””
 #Append distances of i from all points j to array Mi
     Mi.append(distance(i, j))  
 #Append the distances calculated for each point i to array M
         M.append(Mi)
 “””
 A variable ‘q’ will record the number of clusters formed at any point of time. It is initialized as the length of the dataset because initially there will be singleton-clusters i.e. each point assigned to a different cluster. As we merge the clusters, the value of q will be decremented.
 “””
     q = len(ds)  #Initialized to length of dataset 
 #Merge clusters till value of q exceeds the number of required clusters ‘k’
     while q > k:
 “”” 
Call the minimum() function defined in step (8) to know the clusters with minimum distance
 “””
         x, y, min = minimum(M)
       #Add the cluster C[y] to the cluster group of C[x] (merging of clusters)
         C[x].extend(C[y])
 “””
Remove the cluster C[y] from the array C. The reason behind this removal is -  suppose, we grouped clusters 1 and 2. We remove cluster 2 from array C. So now distance between cluster 1 and clusters other than 2 should be computed for further merging. Since cluster 2 has already been merged with 1, it’s distance from 1 need not be computed so cluster 2 is removed from array C.
 “””
         C.remove(C[y])
“””
Initialize array M, compute its distance from other clusters and store it in M. If while loop’s condition holds true, again the process of finding minimum distant clusters will be repeated based on the distance values stored in array M.
“””
         M = []
         for i in C:
             Mi = []
             for j in C:
                 Mi.append(distance(i, j))
             M.append(Mi)
 “””
Decrement number of clusters formed i.e. ‘q’ which will again be compared with ‘k’ to know if further merging of clusters is required.
 “””
         q -= 1
   #Return the final group of clusters C=[C1, C2,...,Ck]
   return C 
  1. Define a function to plot the final clusters.
 def plot(C):
 “””
Initialize an array to define colors for marking the data points.
Color values as per matplotlib.colors are: ‘r’ for red, ‘y’ for yellow, ‘g’ for green, ‘b’ for blue, ‘c’ for cyan, ‘k’ for black and ‘m’ for magenta.
 “””
     color_Value = ['r', 'y', 'g', 'b', 'c', 'k', 'm']
 #For each cluster in the clusters’ division
     for i in range(len(C)):
 #Initialize arrays for x and y coordinates of a datapoint
         coordinate_X = []    
         coordinate_Y = []    
 #Add x and y coordinates to relevant arrays
         for j in range(len(C[i])):
             coordinate_X.append(C[i][j][0])
             coordinate_Y.append(C[i][j][1])
      #Scatter plot of the data points
         pl.scatter(coordinate_X, coordinate_Y, marker='x', 
        color=color_Value[i%len(color_Value)], label=i)
 “””
Suppose, i=0 i.e. 1st cluster from the cluster division C. Then, i%len(color_Value) will be equal to 0%7=0 so 1st color from the color_Values array i.e. ‘r’ (red) will be assigned to the first cluster in the visualization plot.
 “””
 #Define the location of displaying legend of the plit
     pl.legend(loc='upper right')
 #Display the plot
     pl.show() 
  1. Now, try applying the AGNES algorithm to the dataset with different clusters and methods to calculate the distance between clusters.
 #With max_distance() used for distance computation and 4 clusters to be formed
 C = AGNES(dataset, max_distance, 4)
 #Visualize the clusters
 plot(C) 

Output:

 #With min_distance() and 5 clusters to be formed
 C = AGNES(dataset, min_distance, 5)
 #Visualize the clusters
 plot(C) 

Output:

 #With avg_distance() and 6 clusters to be formed
 C = AGNES(dataset, avg_distance, 6)
 #Visualize the clusters
 plot(C) 

Output:


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