In [14]:
from matplotlib import pyplot as plt
from scipy.cluster.hierarchy import dendrogram, linkage, fcluster
import numpy as np
%matplotlib inline
np.set_printoptions(precision=5, suppress=True)
In [15]:
np.random.seed(42)
a = np.random.multivariate_normal([10, 0], [[3, 1], [1, 3]], size=[50,])
b = np.random.multivariate_normal([0, 20], [[3, 1], [1, 3]], size=[500,])
X = np.concatenate((a, b),)
print(X.shape) # 100 samples with 2 dimensions
plt.scatter(X[:,0], X[:,1])
plt.show()
In [16]:
Z = linkage(X, 'single')
print(Z.shape)
Z[i] will tell us which clusters were merged in the i-th iteration. The format is [idx1, idx2, dist, sample_count]. For example:
In [17]:
print(Z[0])
In the first iteration, the algorithm decide to merge points with indics 66 and 91, as they only had a distance of 0.02128. This created a cluster with a total of 2 samples.
Let's see the two points in the figure
In [18]:
idxs = [66, 91]
# plt.figure(figsize=(10, 8))
plt.scatter(X[:,0], X[:,1]) # plot all points
plt.scatter(X[idxs,0], X[idxs,1], c='r') # plot interesting points in red again
plt.show()
As you can see, the two red points are quite close to each other.
You can try the rest iterations by yourself.
In [19]:
plt.figure(figsize=(25, 10))
plt.title('Hierarchical Clustering Dendrogram')
plt.xlabel('sample index')
plt.ylabel('distance')
dendrogram(
Z,
leaf_rotation=90.,
leaf_font_size=9.,
)
plt.show()
Double click the figure to zoom in.
y-axis tells the distance in each merge (that's why the blue line is "tall")
We can also plot a truncated dendrogram, which only shows part of the tree. For more details please refer to the scipy page.
In [20]:
plt.title('Hierarchical Clustering Dendrogram (truncated)')
plt.xlabel('sample index')
plt.ylabel('distance')
dendrogram(
Z,
truncate_mode='lastp', # show only the last p merged clusters,
p=12, # show only the last p merged clusters
show_leaf_counts=True, # numbers with brackets are counts, without brackets are indics
leaf_rotation=90.,
leaf_font_size=12.,
)
plt.show()
There are two ways to get the clusters, either given the number of clusters or given the distance threshold.
In [21]:
# given number of clusters k (in our example, k = 2)
k=2
fcluster(Z, k, criterion='maxclust')
Out[21]:
In [22]:
# given the distance threshold max_d (it is quite clear from the Dendrogram that 6 is a good threshold)
max_d = 6
clusters = fcluster(Z, max_d, criterion='distance')
clusters
Out[22]:
In [23]:
Z_complete = linkage(X, 'complete')
In [24]:
plt.figure(figsize=(25, 10))
plt.title('Hierarchical Clustering Dendrogram')
plt.xlabel('sample index')
plt.ylabel('distance')
dendrogram(
Z_complete,
leaf_rotation=90.,
leaf_font_size=9.,
)
plt.show()