一个关于 K-Means 聚类在手写数字数据上的演示#

在这个例子中，我们比较了 K-means 的各种初始化策略在运行时间和结果质量方面的表现。

由于这里已知真实情况，我们还应用了不同的聚类质量指标来判断聚类标签与真实情况的拟合程度。

评估的聚类质量指标（有关指标的定义和讨论，请参见聚类性能评估）

简写	全称
homo	同质性得分
compl	完整性得分
v-meas	V 测度
ARI	调整后的 Rand 指数
AMI	调整后的互信息
silhouette	轮廓系数

加载数据集#

我们将从加载 digits 数据集开始。该数据集包含从 0 到 9 的手写数字。在聚类的背景下，人们希望将图像分组，使得图像上的手写数字相同。

import numpy as np

from sklearn.datasets import load_digits

data, labels = load_digits(return_X_y=True)
(n_samples, n_features), n_digits = data.shape, np.unique(labels).size

print(f"# digits: {n_digits}; # samples: {n_samples}; # features {n_features}")

# digits: 10; # samples: 1797; # features 64

定义我们的评估基准#

我们首先定义我们的评估基准。在这个基准测试中，我们打算比较 KMeans 的不同初始化方法。我们的基准将

创建一个管道，使用 StandardScaler 对数据进行缩放；
训练和计时管道拟合；
通过不同的指标衡量聚类的性能。

from time import time

from sklearn import metrics
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler


def bench_k_means(kmeans, name, data, labels):
    """Benchmark to evaluate the KMeans initialization methods.

    Parameters
    ----------
    kmeans : KMeans instance
        A :class:`~sklearn.cluster.KMeans` instance with the initialization
        already set.
    name : str
        Name given to the strategy. It will be used to show the results in a
        table.
    data : ndarray of shape (n_samples, n_features)
        The data to cluster.
    labels : ndarray of shape (n_samples,)
        The labels used to compute the clustering metrics which requires some
        supervision.
    """
    t0 = time()
    estimator = make_pipeline(StandardScaler(), kmeans).fit(data)
    fit_time = time() - t0
    results = [name, fit_time, estimator[-1].inertia_]

    # Define the metrics which require only the true labels and estimator
    # labels
    clustering_metrics = [
        metrics.homogeneity_score,
        metrics.completeness_score,
        metrics.v_measure_score,
        metrics.adjusted_rand_score,
        metrics.adjusted_mutual_info_score,
    ]
    results += [m(labels, estimator[-1].labels_) for m in clustering_metrics]

    # The silhouette score requires the full dataset
    results += [
        metrics.silhouette_score(
            data,
            estimator[-1].labels_,
            metric="euclidean",
            sample_size=300,
        )
    ]

    # Show the results
    formatter_result = (
        "{:9s}\t{:.3f}s\t{:.0f}\t{:.3f}\t{:.3f}\t{:.3f}\t{:.3f}\t{:.3f}\t{:.3f}"
    )
    print(formatter_result.format(*results))

运行基准#

我们将比较三种方法

使用 k-means++ 的初始化。此方法是随机的，我们将运行初始化 4 次；
随机初始化。此方法也是随机的，我们将运行初始化 4 次；
基于 PCA 投影的初始化。实际上，我们将使用 PCA 的分量来初始化 KMeans。此方法是确定性的，只需要一次初始化。

from sklearn.cluster import KMeans
from sklearn.decomposition import PCA

print(82 * "_")
print("init\t\ttime\tinertia\thomo\tcompl\tv-meas\tARI\tAMI\tsilhouette")

kmeans = KMeans(init="k-means++", n_clusters=n_digits, n_init=4, random_state=0)
bench_k_means(kmeans=kmeans, name="k-means++", data=data, labels=labels)

kmeans = KMeans(init="random", n_clusters=n_digits, n_init=4, random_state=0)
bench_k_means(kmeans=kmeans, name="random", data=data, labels=labels)

pca = PCA(n_components=n_digits).fit(data)
kmeans = KMeans(init=pca.components_, n_clusters=n_digits, n_init=1)
bench_k_means(kmeans=kmeans, name="PCA-based", data=data, labels=labels)

print(82 * "_")

__________________________________________________________________________________
init            time    inertia homo    compl   v-meas  ARI     AMI     silhouette
k-means++       0.050s  69545   0.598   0.645   0.621   0.469   0.617   0.152
random          0.064s  69735   0.681   0.723   0.701   0.574   0.698   0.170
PCA-based       0.017s  69513   0.600   0.647   0.622   0.468   0.618   0.162
__________________________________________________________________________________

在 PCA 降维数据上可视化结果#

PCA 允许将数据从原始的 64 维空间投影到更低维的空间。随后，我们可以使用 PCA 投影到二维空间，并在该新空间中绘制数据和聚类。

import matplotlib.pyplot as plt

reduced_data = PCA(n_components=2).fit_transform(data)
kmeans = KMeans(init="k-means++", n_clusters=n_digits, n_init=4)
kmeans.fit(reduced_data)

# Step size of the mesh. Decrease to increase the quality of the VQ.
h = 0.02  # point in the mesh [x_min, x_max]x[y_min, y_max].

# Plot the decision boundary. For that, we will assign a color to each
x_min, x_max = reduced_data[:, 0].min() - 1, reduced_data[:, 0].max() + 1
y_min, y_max = reduced_data[:, 1].min() - 1, reduced_data[:, 1].max() + 1
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))

# Obtain labels for each point in mesh. Use last trained model.
Z = kmeans.predict(np.c_[xx.ravel(), yy.ravel()])

# Put the result into a color plot
Z = Z.reshape(xx.shape)
plt.figure(1)
plt.clf()
plt.imshow(
    Z,
    interpolation="nearest",
    extent=(xx.min(), xx.max(), yy.min(), yy.max()),
    cmap=plt.cm.Paired,
    aspect="auto",
    origin="lower",
)

plt.plot(reduced_data[:, 0], reduced_data[:, 1], "k.", markersize=2)
# Plot the centroids as a white X
centroids = kmeans.cluster_centers_
plt.scatter(
    centroids[:, 0],
    centroids[:, 1],
    marker="x",
    s=169,
    linewidths=3,
    color="w",
    zorder=10,
)
plt.title(
    "K-means clustering on the digits dataset (PCA-reduced data)\n"
    "Centroids are marked with white cross"
)
plt.xlim(x_min, x_max)
plt.ylim(y_min, y_max)
plt.xticks(())
plt.yticks(())
plt.show()

K-means clustering on the digits dataset (PCA-reduced data) Centroids are marked with white cross

脚本的总运行时间：（0 分钟 2.318 秒）