Examples¶

In this section we present some usage examples for MultiCons. We replicate the examples presented in the Thesis of Atheer Al-Najdi (A closed patterns-based approach to the consensus clustering problem).

Cassini dataset¶

The dataset consists of 1000 instances, each represents a point in a 2D space, forming a structure of three clusters.

As in the Thesis of Atheer Al-Najdi, we will try to cluster the dataset with 8 different clustering algorithms and then compute and visualize the consensus clustering candidates using the MultiCons and ConsTree methods.

Let's get a first visual of the dataset and our base clusterings:

In [1]:

from os import path

import numpy as np
import pandas as pd
from fcmeans import FCM
from kmedoids import KMedoids
from matplotlib import pyplot as plt
from sklearn.cluster import (
    DBSCAN,
    AgglomerativeClustering,
    Birch,
    KMeans,
    SpectralClustering,
)
from sklearn.mixture import GaussianMixture

from multicons import MultiCons
from multicons.utils import jaccard_similarity

np.set_printoptions(threshold=100)

from os import path import numpy as np import pandas as pd from fcmeans import FCM from kmedoids import KMedoids from matplotlib import pyplot as plt from sklearn.cluster import ( DBSCAN, AgglomerativeClustering, Birch, KMeans, SpectralClustering, ) from sklearn.mixture import GaussianMixture from multicons import MultiCons from multicons.utils import jaccard_similarity np.set_printoptions(threshold=100)

In [2]:

# Load the data
file_prefix = "" if path.exists("cassini.csv") else "docs/"
file_name = f"{file_prefix}cassini.csv"
cassini = pd.read_csv(file_name)
# Remove the class labels
cassini_train_data = cassini.drop(['class'], axis=1)

# Load the data file_prefix = "" if path.exists("cassini.csv") else "docs/" file_name = f"{file_prefix}cassini.csv" cassini = pd.read_csv(file_name) # Remove the class labels cassini_train_data = cassini.drop(['class'], axis=1)

In [3]:

# Setup the plot axes
fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(18, 12), sharex=True, sharey=True
)
# Common plot arguments
common_kwargs = {"x": "x", "y": "y", "colorbar": False, "colormap": "Paired"}
# Our collection of base clusterings
base_clusterings = []

# Cassini
cassini.plot.scatter(c="class", title="Cassini", ax=axes[0, 0], **common_kwargs)

# K-means
base_clusterings.append(KMeans(n_clusters=3).fit_predict(cassini_train_data))
cassini_train_data.plot.scatter(
    title="K-means", ax=axes[0, 1], c=base_clusterings[-1], **common_kwargs
)

# Average linkage
base_clusterings.append(
    AgglomerativeClustering(n_clusters=3).fit_predict(cassini_train_data)
)
cassini_train_data.plot.scatter(
    title="Average linkage", ax=axes[0, 2], c=base_clusterings[-1], **common_kwargs
)

# Gaussian model
base_clusterings.append(
    GaussianMixture(n_components=3, random_state=5).fit_predict(cassini_train_data)
)
cassini_train_data.plot.scatter(
    title="Gaussian model", ax=axes[1, 0], c=base_clusterings[-1], **common_kwargs
)

# C-means
fcm = FCM(n_clusters=3, max_iter=5, m=5)
fcm.fit(cassini_train_data.values)
base_clusterings.append(fcm.predict(cassini_train_data.values))
cassini_train_data.plot.scatter(
    title="C-means", ax=axes[1, 1], c=base_clusterings[-1], **common_kwargs
)

# PAM
base_clusterings.append(
    KMedoids(3, metric="euclidean", method="pam")
    .fit_predict(cassini_train_data.to_numpy())
)
cassini_train_data.plot.scatter(
    title="PAM", ax=axes[1, 2], c=base_clusterings[-1], **common_kwargs
)

# BIRCH
birch = Birch(n_clusters=3, threshold=0.5)
base_clusterings.append(birch.fit_predict(np.ascontiguousarray(cassini_train_data)))
cassini_train_data.plot.scatter(
    title="BIRCH", ax=axes[2, 0], c=base_clusterings[-1], **common_kwargs
)

# Spectral
base_clusterings.append(
    SpectralClustering(n_clusters=3).fit_predict(cassini_train_data)
)
cassini_train_data.plot.scatter(
    title="Spectral", ax=axes[2, 1], c=base_clusterings[-1], **common_kwargs
)

# DBSCAN
base_clusterings.append(DBSCAN(eps=0.2).fit_predict(cassini_train_data))
cassini_train_data.plot.scatter(
    title="DBSCAN", ax=axes[2, 2], c=base_clusterings[-1], **common_kwargs
)

fig.show()

# Setup the plot axes fig, axes = plt.subplots( nrows=3, ncols=3, figsize=(18, 12), sharex=True, sharey=True ) # Common plot arguments common_kwargs = {"x": "x", "y": "y", "colorbar": False, "colormap": "Paired"} # Our collection of base clusterings base_clusterings = [] # Cassini cassini.plot.scatter(c="class", title="Cassini", ax=axes[0, 0], **common_kwargs) # K-means base_clusterings.append(KMeans(n_clusters=3).fit_predict(cassini_train_data)) cassini_train_data.plot.scatter( title="K-means", ax=axes[0, 1], c=base_clusterings[-1], **common_kwargs ) # Average linkage base_clusterings.append( AgglomerativeClustering(n_clusters=3).fit_predict(cassini_train_data) ) cassini_train_data.plot.scatter( title="Average linkage", ax=axes[0, 2], c=base_clusterings[-1], **common_kwargs ) # Gaussian model base_clusterings.append( GaussianMixture(n_components=3, random_state=5).fit_predict(cassini_train_data) ) cassini_train_data.plot.scatter( title="Gaussian model", ax=axes[1, 0], c=base_clusterings[-1], **common_kwargs ) # C-means fcm = FCM(n_clusters=3, max_iter=5, m=5) fcm.fit(cassini_train_data.values) base_clusterings.append(fcm.predict(cassini_train_data.values)) cassini_train_data.plot.scatter( title="C-means", ax=axes[1, 1], c=base_clusterings[-1], **common_kwargs ) # PAM base_clusterings.append( KMedoids(3, metric="euclidean", method="pam") .fit_predict(cassini_train_data.to_numpy()) ) cassini_train_data.plot.scatter( title="PAM", ax=axes[1, 2], c=base_clusterings[-1], **common_kwargs ) # BIRCH birch = Birch(n_clusters=3, threshold=0.5) base_clusterings.append(birch.fit_predict(np.ascontiguousarray(cassini_train_data))) cassini_train_data.plot.scatter( title="BIRCH", ax=axes[2, 0], c=base_clusterings[-1], **common_kwargs ) # Spectral base_clusterings.append( SpectralClustering(n_clusters=3).fit_predict(cassini_train_data) ) cassini_train_data.plot.scatter( title="Spectral", ax=axes[2, 1], c=base_clusterings[-1], **common_kwargs ) # DBSCAN base_clusterings.append(DBSCAN(eps=0.2).fit_predict(cassini_train_data)) cassini_train_data.plot.scatter( title="DBSCAN", ax=axes[2, 2], c=base_clusterings[-1], **common_kwargs ) fig.show()

At this point, base_clusterings now contains all clustering candidates in a list (of lists):

In [4]:

np.array(base_clusterings)

np.array(base_clusterings)

Out[4]:

array([[0., 2., 2., ..., 0., 1., 0.],
       [1., 1., 1., ..., 2., 2., 2.],
       [0., 0., 0., ..., 1., 1., 1.],
       ...,
       [0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 2., 0.],
       [0., 0., 0., ..., 2., 2., 2.]])

Note: This MultiCons implementation requires the clustering labels to be numerical!

Now, let's compute the consensus candidates with MultiCons:

In [5]:

# MultiCons implementation aims to follow scikit-learn conventions.
consensus = MultiCons().fit(base_clusterings)
consensus

# MultiCons implementation aims to follow scikit-learn conventions. consensus = MultiCons().fit(base_clusterings) consensus

Out[5]:

MultiCons(consensus_function=<function consensus_function_10 at 0x7f10893cf6a0>,
          similarity_measure=<function jaccard_similarity at 0x7f108943c400>)

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

In [6]:

# The `consensus_vectors` attribute is a python list containing the
# consensus candidates.
# We transform it to a numpy array to better visualize it here.
np.array(consensus.consensus_vectors)

# The `consensus_vectors` attribute is a python list containing the # consensus candidates. # We transform it to a numpy array to better visualize it here. np.array(consensus.consensus_vectors)

Out[6]:

array([[ 0,  0,  0, ...,  0,  0,  0],
       [ 1,  1,  1, ...,  1,  1,  1],
       [ 2,  2,  2, ...,  0,  0,  0],
       [25, 24, 24, ..., 21, 19, 21]])

In [7]:

# The `decision_thresholds` attribute contains a list of decision thresholds
# for each consensus vector.
consensus.decision_thresholds

# The `decision_thresholds` attribute contains a list of decision thresholds # for each consensus vector. consensus.decision_thresholds

Out[7]:

[5, 6, 7, 8]

In [8]:

# The `recommended` attribute contains the index of the recommended consensus
# vector
consensus.recommended

# The `recommended` attribute contains the index of the recommended consensus # vector consensus.recommended

Out[8]:

2

In [9]:

# The `labels_` attribute contains the recommended consensus vector
consensus.labels_

# The `labels_` attribute contains the recommended consensus vector consensus.labels_

Out[9]:

array([2, 2, 2, ..., 0, 0, 0])

In [10]:

# Plot the recommended consensus clustering solution
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(6, 4))
cassini_train_data.plot.scatter(
    title="MultiCons", ax=axes, c=consensus.labels_, **common_kwargs
)
fig.show()

# Plot the recommended consensus clustering solution fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(6, 4)) cassini_train_data.plot.scatter( title="MultiCons", ax=axes, c=consensus.labels_, **common_kwargs ) fig.show()

In [11]:

# The `stability` attribute contains a list of stability values
# for each consensus vector.
consensus.stability

# The `stability` attribute contains a list of stability values # for each consensus vector. consensus.stability

Out[11]:

[5, 1, 1, 1]

In [12]:

# The `tree_quality` member contains a measure of the tree quality.
# The measure ranges between 0 and 1. Higher is better.
consensus.tree_quality

# The `tree_quality` member contains a measure of the tree quality. # The measure ranges between 0 and 1. Higher is better. consensus.tree_quality

Out[12]:

0.5

In [13]:

# The `ensemble_similarity` contains a list of ensemble similarity measures
# for each consensus vector.
# They are between 0 and 1. Higher is better.
consensus.ensemble_similarity

# The `ensemble_similarity` contains a list of ensemble similarity measures # for each consensus vector. # They are between 0 and 1. Higher is better. consensus.ensemble_similarity

Out[13]:

array([0.37943769, 0.65271836, 0.73920413, 0.3633146 ])

Finally, let's visualize the consenus candidates using the ConsTree method:

In [14]:

cons_tree = consensus.cons_tree()
cons_tree

cons_tree = consensus.cons_tree() cons_tree

Out[14]:

In [15]:

# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}CassiniConsTree.svg", cleanup=True)

# Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}CassiniConsTree.svg", cleanup=True)

Out[15]:

'docs/CassiniConsTree.svg'

View ConsTree graph in full size: CassiniConsTree.svg

5 Overlapping Gaussian distributions¶

Replicating the example with a synthetic dataset used in the thesis of Atheer that consist of:

• generating 5 overlapping Gaussian distributed points in a 2D features space
• appying 6 different clustering algorithms with random choices for K values (in the range [2, 9])
• comparing the results of 5 different MultiCons consensus solutions (by altering the consensus functions)

Let's start by generating the dataset:

In [16]:

cov = np.array([[0.3, 0.1], [0.1, 0.3]])
gaussian_distributions = pd.DataFrame(
    np.concatenate(
        (
            np.concatenate(
                (
                    np.random.multivariate_normal([-0.25, -2], cov, 400),
                    np.random.multivariate_normal([-1.75, -0.5], cov, 400),
                    np.random.multivariate_normal([-0.5, 2], cov, 400),
                    np.random.multivariate_normal([1.75, 2], cov, 400),
                    np.random.multivariate_normal([2, -0.5], cov, 400),
                ),
            ),
            np.concatenate(
                (
                    np.repeat([[1]], 400, 0),
                    np.repeat([[2]], 400, 0),
                    np.repeat([[3]], 400, 0),
                    np.repeat([[4]], 400, 0),
                    np.repeat([[5]], 400, 0),
                )
            )
        ),
        axis=1
    ),
    columns=["x", "y", "class"],
)
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(6, 4))
gaussian_distributions.plot.scatter(
    title="5 Overlapping Gaussians", ax=axes, c="class", **common_kwargs
)
fig.show()

cov = np.array([[0.3, 0.1], [0.1, 0.3]]) gaussian_distributions = pd.DataFrame( np.concatenate( ( np.concatenate( ( np.random.multivariate_normal([-0.25, -2], cov, 400), np.random.multivariate_normal([-1.75, -0.5], cov, 400), np.random.multivariate_normal([-0.5, 2], cov, 400), np.random.multivariate_normal([1.75, 2], cov, 400), np.random.multivariate_normal([2, -0.5], cov, 400), ), ), np.concatenate( ( np.repeat([[1]], 400, 0), np.repeat([[2]], 400, 0), np.repeat([[3]], 400, 0), np.repeat([[4]], 400, 0), np.repeat([[5]], 400, 0), ) ) ), axis=1 ), columns=["x", "y", "class"], ) fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(6, 4)) gaussian_distributions.plot.scatter( title="5 Overlapping Gaussians", ax=axes, c="class", **common_kwargs ) fig.show()

In [17]:

# Remove the class labels
gaussian_train_data = gaussian_distributions.drop(['class'], axis=1)

# Remove the class labels gaussian_train_data = gaussian_distributions.drop(['class'], axis=1)

Next, let's compute the base clusterings and visualize their outcome:

In [18]:

# Setup the plot axes
fig, axes = plt.subplots(
    nrows=3, ncols=2, figsize=(12, 12), sharex=True, sharey=True
)
# Our collection of base clusterings
base_clusterings = []

# K-means (4 clusters)
base_clusterings.append(KMeans(n_clusters=4).fit_predict(gaussian_train_data))
gaussian_train_data.plot.scatter(
    title="K-means (4 clusters)",
    ax=axes[0, 0],
    c=base_clusterings[-1],
    **common_kwargs
)

# Average linkage (9 clusters)
base_clusterings.append(
    AgglomerativeClustering(n_clusters=9, linkage="average").fit_predict(
        gaussian_train_data
    )
)
gaussian_train_data.plot.scatter(
    title="Average linkage (9 clusters)",
    ax=axes[0, 1],
    c=base_clusterings[-1],
    **common_kwargs
)

# Gaussian model (8 clusters)
base_clusterings.append(
    GaussianMixture(n_components=8, random_state=2, reg_covar=0.2).fit_predict(
        gaussian_train_data
    )
)
gaussian_train_data.plot.scatter(
    title="Gaussian model (8 clusters)",
    ax=axes[1, 0],
    c=base_clusterings[-1],
    **common_kwargs
)

# C-means (2 clusters)
fcm = FCM(n_clusters=2, max_iter=5, m=5)
fcm.fit(gaussian_train_data.values)
base_clusterings.append(fcm.predict(gaussian_train_data.values))
gaussian_train_data.plot.scatter(
    title="C-means (2 clusters)",
    ax=axes[1, 1],
    c=base_clusterings[-1],
    **common_kwargs
)

# PAM (3 clusters)
base_clusterings.append(
    KMedoids(3, metric="euclidean", method="pam")
    .fit_predict(gaussian_train_data.to_numpy())
)
gaussian_train_data.plot.scatter(
    title="PAM (3 clusters)", ax=axes[2, 0], c=base_clusterings[-1], **common_kwargs
)

# BIRCH (5 clusters)
birch = Birch(n_clusters=6, threshold=0.5)
base_clusterings.append(birch.fit_predict(gaussian_train_data))
gaussian_train_data.plot.scatter(
    title="BIRCH (6 clusters)",
    ax=axes[2, 1],
    c=base_clusterings[-1],
    **common_kwargs
)

fig.show()

# Setup the plot axes fig, axes = plt.subplots( nrows=3, ncols=2, figsize=(12, 12), sharex=True, sharey=True ) # Our collection of base clusterings base_clusterings = [] # K-means (4 clusters) base_clusterings.append(KMeans(n_clusters=4).fit_predict(gaussian_train_data)) gaussian_train_data.plot.scatter( title="K-means (4 clusters)", ax=axes[0, 0], c=base_clusterings[-1], **common_kwargs ) # Average linkage (9 clusters) base_clusterings.append( AgglomerativeClustering(n_clusters=9, linkage="average").fit_predict( gaussian_train_data ) ) gaussian_train_data.plot.scatter( title="Average linkage (9 clusters)", ax=axes[0, 1], c=base_clusterings[-1], **common_kwargs ) # Gaussian model (8 clusters) base_clusterings.append( GaussianMixture(n_components=8, random_state=2, reg_covar=0.2).fit_predict( gaussian_train_data ) ) gaussian_train_data.plot.scatter( title="Gaussian model (8 clusters)", ax=axes[1, 0], c=base_clusterings[-1], **common_kwargs ) # C-means (2 clusters) fcm = FCM(n_clusters=2, max_iter=5, m=5) fcm.fit(gaussian_train_data.values) base_clusterings.append(fcm.predict(gaussian_train_data.values)) gaussian_train_data.plot.scatter( title="C-means (2 clusters)", ax=axes[1, 1], c=base_clusterings[-1], **common_kwargs ) # PAM (3 clusters) base_clusterings.append( KMedoids(3, metric="euclidean", method="pam") .fit_predict(gaussian_train_data.to_numpy()) ) gaussian_train_data.plot.scatter( title="PAM (3 clusters)", ax=axes[2, 0], c=base_clusterings[-1], **common_kwargs ) # BIRCH (5 clusters) birch = Birch(n_clusters=6, threshold=0.5) base_clusterings.append(birch.fit_predict(gaussian_train_data)) gaussian_train_data.plot.scatter( title="BIRCH (6 clusters)", ax=axes[2, 1], c=base_clusterings[-1], **common_kwargs ) fig.show()

Now, let's compute the consensus candidates with MultiCons and visualize their outcome:

In [19]:

consensus_1 = MultiCons()
consensus_2 = MultiCons(consensus_function="consensus_function_12")
consensus_3 = MultiCons(consensus_function="consensus_function_13")
consensus_4 = MultiCons(consensus_function="consensus_function_14")
consensus_5 = MultiCons(consensus_function="consensus_function_15")

consensus_1.fit(base_clusterings)
consensus_2.fit(base_clusterings)
consensus_3.fit(base_clusterings)
consensus_4.fit(base_clusterings)
consensus_5.fit(base_clusterings)

# Plot the recommended consensus clustering solutions
fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(12, 12), sharex=True, sharey=True)
gaussian_train_data.plot.scatter(
    title="MultiCons Approach 1",
    ax=axes[0, 0],
    c=consensus_1.labels_,
    **common_kwargs
)
gaussian_train_data.plot.scatter(
    title="MultiCons Approach 2",
    ax=axes[0, 1],
    c=consensus_2.labels_,
    **common_kwargs
)
gaussian_train_data.plot.scatter(
    title="MultiCons Approach 3",
    ax=axes[1, 0],
    c=consensus_3.labels_,
    **common_kwargs
)
gaussian_train_data.plot.scatter(
    title="MultiCons Approach 4",
    ax=axes[1, 1],
    c=consensus_4.labels_,
    **common_kwargs
)
gaussian_train_data.plot.scatter(
    title="MultiCons Approach 5",
    ax=axes[2, 1],
    c=consensus_5.labels_,
    **common_kwargs
)
fig.show()

consensus_1 = MultiCons() consensus_2 = MultiCons(consensus_function="consensus_function_12") consensus_3 = MultiCons(consensus_function="consensus_function_13") consensus_4 = MultiCons(consensus_function="consensus_function_14") consensus_5 = MultiCons(consensus_function="consensus_function_15") consensus_1.fit(base_clusterings) consensus_2.fit(base_clusterings) consensus_3.fit(base_clusterings) consensus_4.fit(base_clusterings) consensus_5.fit(base_clusterings) # Plot the recommended consensus clustering solutions fig, axes = plt.subplots(nrows=3, ncols=2, figsize=(12, 12), sharex=True, sharey=True) gaussian_train_data.plot.scatter( title="MultiCons Approach 1", ax=axes[0, 0], c=consensus_1.labels_, **common_kwargs ) gaussian_train_data.plot.scatter( title="MultiCons Approach 2", ax=axes[0, 1], c=consensus_2.labels_, **common_kwargs ) gaussian_train_data.plot.scatter( title="MultiCons Approach 3", ax=axes[1, 0], c=consensus_3.labels_, **common_kwargs ) gaussian_train_data.plot.scatter( title="MultiCons Approach 4", ax=axes[1, 1], c=consensus_4.labels_, **common_kwargs ) gaussian_train_data.plot.scatter( title="MultiCons Approach 5", ax=axes[2, 1], c=consensus_5.labels_, **common_kwargs ) fig.show()

Also, let's visualize the ConsTrees:

In [20]:

cons_tree = consensus_1.cons_tree()
# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}GaussianConsTree1.svg", cleanup=True)
cons_tree

cons_tree = consensus_1.cons_tree() # Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}GaussianConsTree1.svg", cleanup=True) cons_tree

Out[20]:

View ConsTree graph 1 in full size: GaussianConsTree1.svg

In [21]:

cons_tree = consensus_2.cons_tree()
# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}GaussianConsTree2.svg", cleanup=True)
cons_tree

cons_tree = consensus_2.cons_tree() # Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}GaussianConsTree2.svg", cleanup=True) cons_tree

Out[21]:

View ConsTree graph 2 in full size: GaussianConsTree2.svg

In [22]:

cons_tree = consensus_3.cons_tree()
# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}GaussianConsTree3.svg", cleanup=True)
cons_tree

cons_tree = consensus_3.cons_tree() # Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}GaussianConsTree3.svg", cleanup=True) cons_tree

Out[22]:

View ConsTree graph 1 in full size: GaussianConsTree3.svg

In [23]:

cons_tree = consensus_4.cons_tree()
# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}GaussianConsTree4.svg", cleanup=True)
cons_tree

cons_tree = consensus_4.cons_tree() # Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}GaussianConsTree4.svg", cleanup=True) cons_tree

Out[23]:

View ConsTree graph 4 in full size: GaussianConsTree4.svg

In [24]:

cons_tree = consensus_5.cons_tree()
# Save the ConsTree graph to a file
cons_tree.render(outfile=f"{file_prefix}GaussianConsTree5.svg", cleanup=True)
cons_tree

cons_tree = consensus_5.cons_tree() # Save the ConsTree graph to a file cons_tree.render(outfile=f"{file_prefix}GaussianConsTree5.svg", cleanup=True) cons_tree

Out[24]:

View ConsTree graph 5 in full size: GaussianConsTree5.svg

Finally, let's compare the clustering results:

In [25]:

true_labels = gaussian_distributions["class"].to_numpy()
# We compare the results using the pair-wise Jaccard Similarity Measure
pd.DataFrame(
    [
        ["K-means", jaccard_similarity(base_clusterings[0], true_labels)],
        ["Average linkage", jaccard_similarity(base_clusterings[1], true_labels)],
        ["Gaussian model", jaccard_similarity(base_clusterings[2], true_labels)],
        ["C-means", jaccard_similarity(base_clusterings[3], true_labels)],
        ["PAM", jaccard_similarity(base_clusterings[4], true_labels)],
        ["BIRCH", jaccard_similarity(base_clusterings[5], true_labels)],
        ["MultiCons_1", jaccard_similarity(consensus_1.labels_, true_labels)],
        ["MultiCons_2", jaccard_similarity(consensus_2.labels_, true_labels)],
        ["MultiCons_3", jaccard_similarity(consensus_3.labels_, true_labels)],
        ["MultiCons_4", jaccard_similarity(consensus_4.labels_, true_labels)],
        ["MultiCons_5", jaccard_similarity(consensus_5.labels_, true_labels)],
    ],
    columns=["Algorithm", "Jaccard"]
)

true_labels = gaussian_distributions["class"].to_numpy() # We compare the results using the pair-wise Jaccard Similarity Measure pd.DataFrame( [ ["K-means", jaccard_similarity(base_clusterings[0], true_labels)], ["Average linkage", jaccard_similarity(base_clusterings[1], true_labels)], ["Gaussian model", jaccard_similarity(base_clusterings[2], true_labels)], ["C-means", jaccard_similarity(base_clusterings[3], true_labels)], ["PAM", jaccard_similarity(base_clusterings[4], true_labels)], ["BIRCH", jaccard_similarity(base_clusterings[5], true_labels)], ["MultiCons_1", jaccard_similarity(consensus_1.labels_, true_labels)], ["MultiCons_2", jaccard_similarity(consensus_2.labels_, true_labels)], ["MultiCons_3", jaccard_similarity(consensus_3.labels_, true_labels)], ["MultiCons_4", jaccard_similarity(consensus_4.labels_, true_labels)], ["MultiCons_5", jaccard_similarity(consensus_5.labels_, true_labels)], ], columns=["Algorithm", "Jaccard"] )

Out[25]:

	Algorithm	Jaccard
0	K-means	0.610621
1	Average linkage	0.798826
2	Gaussian model	0.795417
3	C-means	0.339772
4	PAM	0.517568
5	BIRCH	0.803973
6	MultiCons_1	0.541964
7	MultiCons_2	0.894097
8	MultiCons_3	0.778591
9	MultiCons_4	0.894080
10	MultiCons_5	0.863604