Adjustment for chance in clustering performance evaluation¶

The following plots demonstrate the impact of the number of clusters and number of samples on various clustering performance evaluation metrics.

Non-adjusted measures such as the V-Measure show a dependency between the number of clusters and the number of samples: the mean V-Measure of random labeling increases signicantly as the number of clusters is closer to the total number of samples used to compute the measure.

Adjusted for chance measure such as ARI display some random variations centered around a mean score of 0.0 for any number of samples and clusters.

Only adjusted measures can hence safely be used as a consensus index to evaluate the average stability of clustering algorithms for a given value of k on various overlapping sub-samples of the dataset.

Script output:

Computing adjusted_rand_score for 10 values of n_clusters and n_samples=100
done in 0.176s
Computing v_measure_score for 10 values of n_clusters and n_samples=100
done in 1.618s
Computing adjusted_mutual_info_score for 10 values of n_clusters and n_samples=100
done in 5.827s
Computing mutual_info_score for 10 values of n_clusters and n_samples=100
done in 0.023s
Computing adjusted_rand_score for 10 values of n_clusters and n_samples=1000
done in 0.254s
Computing v_measure_score for 10 values of n_clusters and n_samples=1000
done in 0.650s
Computing adjusted_mutual_info_score for 10 values of n_clusters and n_samples=1000
done in 15.673s
Computing mutual_info_score for 10 values of n_clusters and n_samples=1000
done in 0.109s

Python source code: plot_adjusted_for_chance_measures.py

print __doc__

# Author: Olivier Grisel <olivier.grisel@ensta.org>
# License: Simplified BSD

import numpy as np
import pylab as pl
from time import time
from sklearn import metrics


def uniform_labelings_scores(score_func, n_samples, n_clusters_range,
                             fixed_n_classes=None, n_runs=5, seed=42):
    """Compute score for 2 random uniform cluster labelings.

    Both random labelings have the same number of clusters for each value
    possible value in ``n_clusters_range``.

    When fixed_n_classes is not None the first labeling is considered a ground
    truth class assignement with fixed number of classes.
    """
    random_labels = np.random.RandomState(seed).random_integers
    scores = np.zeros((len(n_clusters_range), n_runs))

    if fixed_n_classes is not None:
        labels_a = random_labels(low=0, high=fixed_n_classes - 1,
                                 size=n_samples)

    for i, k in enumerate(n_clusters_range):
        for j in range(n_runs):
            if fixed_n_classes is None:
                labels_a = random_labels(low=0, high=k - 1, size=n_samples)
            labels_b = random_labels(low=0, high=k - 1, size=n_samples)
            scores[i, j] = score_func(labels_a, labels_b)
    return scores

score_funcs = [
    metrics.adjusted_rand_score,
    metrics.v_measure_score,
    metrics.adjusted_mutual_info_score,
    metrics.mutual_info_score,
]

# 2 independent random clusterings with equal cluster number

n_samples = 100
n_clusters_range = np.linspace(2, n_samples, 10).astype(np.int)

pl.figure(1)

plots = []
names = []
for score_func in score_funcs:
    print "Computing %s for %d values of n_clusters and n_samples=%d" % (
        score_func.__name__, len(n_clusters_range), n_samples)

    t0 = time()
    scores = uniform_labelings_scores(score_func, n_samples, n_clusters_range)
    print "done in %0.3fs" % (time() - t0)
    plots.append(pl.errorbar(
    #    n_clusters_range, scores.mean(axis=1), scores.std(axis=1)))
        n_clusters_range, np.median(scores, axis=1), scores.std(axis=1)))
    names.append(score_func.__name__)

pl.title("Clustering measures for 2 random uniform labelings\n"
         "with equal number of clusters")
pl.xlabel('Number of clusters (Number of samples is fixed to %d)' % n_samples)
pl.ylabel('Score value')
pl.legend(plots, names)
pl.ylim(ymin=-0.05, ymax=1.05)


# Random labeling with varying n_clusters against ground class labels
# with fixed number of clusters

n_samples = 1000
n_clusters_range = np.linspace(2, 100, 10).astype(np.int)
n_classes = 10

pl.figure(2)

plots = []
names = []
for score_func in score_funcs:
    print "Computing %s for %d values of n_clusters and n_samples=%d" % (
        score_func.__name__, len(n_clusters_range), n_samples)

    t0 = time()
    scores = uniform_labelings_scores(score_func, n_samples, n_clusters_range,
                                      fixed_n_classes=n_classes)
    print "done in %0.3fs" % (time() - t0)
    plots.append(pl.errorbar(
        n_clusters_range, scores.mean(axis=1), scores.std(axis=1)))
    names.append(score_func.__name__)

pl.title("Clustering measures for random uniform labeling\n"
         "against reference assignement with %d classes" % n_classes)
pl.xlabel('Number of clusters (Number of samples is fixed to %d)' % n_samples)
pl.ylabel('Score value')
pl.ylim(ymin=-0.05, ymax=1.05)
pl.legend(plots, names)
pl.show()

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Adjustment for chance in clustering performance evaluation¶