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X-means

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2.5.3

X-means

Last updated 2026-03-04 Read time 3 min
Summary
  • X-means extends K-means by estimating the number of clusters using split decisions evaluated with BIC/MDL criteria.
  • Minimum and maximum cluster settings define the search range and prevent uncontrolled over-splitting.
  • Comparing fixed-k K-means and X-means clarifies how model selection affects cluster interpretation.

Intuition #

X-means follows a split-and-test strategy: start with coarse clusters, then split only when a statistical criterion indicates improvement. This reduces manual trial-and-error over k while keeping the optimization tractable.

Detailed Explanation #

<p>X-means is a type of clustering algorithm that automatically determines the number of clusters as the clustering proceeds. This page compares the results of k-means++ and X-means.</p>

pyclustering.cluster.xmeans.xmeans Class Reference

Pelleg, Dan, and Andrew W. Moore. “X-means: Extending k-means with efficient estimation of the number of clusters.” Icml. Vol. 1. 2000.

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import numpy as np
import matplotlib.pyplot as plt

from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs

When k is pre-specified in k-means #

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def plot_by_kmeans(X, k=5):
    y_pred = KMeans(n_clusters=k, random_state=random_state, init="random").fit_predict(
        X
    )

    plt.scatter(X[:, 0], X[:, 1], c=y_pred, marker="x")
    plt.title(f"k-means, n_clusters={k}")

# Create sample data
n_samples = 1000
random_state = 117117
X, _ = make_blobs(
    n_samples=n_samples, random_state=random_state, cluster_std=1, centers=10
)

# Run k-means++.
plot_by_kmeans(X)

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Run without specifying the number of clusters in x-mean #

BIC(bayesian information criterion) #

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from pyclustering.cluster.xmeans import xmeans
from pyclustering.cluster.center_initializer import kmeans_plusplus_initializer

BAYESIAN_INFORMATION_CRITERION = 0
MINIMUM_NOISELESS_DESCRIPTION_LENGTH = 1

def plot_by_xmeans(
    X, c_min=3, c_max=10, criterion=BAYESIAN_INFORMATION_CRITERION, tolerance=0.025
):
    initial_centers = kmeans_plusplus_initializer(X, c_min).initialize()
    xmeans_instance = xmeans(
        X, initial_centers, c_max, criterion=criterion, tolerance=tolerance
    )
    xmeans_instance.process()

    # Create data for plots
    clusters = xmeans_instance.get_clusters()
    n_samples = X.shape[0]
    c = []
    for i, cluster_i in enumerate(clusters):
        X_ci = X[cluster_i]
        color_ci = [i for _ in cluster_i]
        plt.scatter(X_ci[:, 0], X_ci[:, 1], marker="x")
    plt.title("x-means")

# Run x-means
plot_by_xmeans(X, c_min=3, c_max=10, criterion=BAYESIAN_INFORMATION_CRITERION)

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MINIMUM_NOISELESS_DESCRIPTION_LENGTH #

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plot_by_xmeans(X, c_min=3, c_max=10, criterion=MINIMUM_NOISELESS_DESCRIPTION_LENGTH)

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Influence of tolerance parameter #

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X, _ = make_blobs(
    n_samples=2000,
    random_state=random_state,
    cluster_std=0.4,
    centers=10,
)

plt.figure(figsize=(25, 5))
for i, ti in enumerate(np.linspace(0.0001, 1, 5)):
    ti = np.round(ti, 4)
    plt.subplot(1, 10, i + 1)
    plot_by_xmeans(
        X, c_min=3, c_max=10, criterion=BAYESIAN_INFORMATION_CRITERION, tolerance=ti
    )
    plt.title(f"tol={ti}")

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Compare k-means and x-means for various data #

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for i in range(5):
    X, _ = make_blobs(
        n_samples=n_samples,
        random_state=random_state,
        cluster_std=0.7,
        centers=5 + i * 5,
    )
    plt.figure(figsize=(10, 5))
    plt.subplot(1, 2, 1)
    plot_by_kmeans(X)
    plt.subplot(1, 2, 2)
    plot_by_xmeans(X, c_min=3, c_max=20)
    plt.show()

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