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# A demo of K-Means clustering on the handwritten digits data

In this example we compare the various initialization strategies for K-means in terms of runtime and quality of the results.

As the ground truth is known here, we also apply different cluster quality metrics to judge the goodness of fit of the cluster labels to the ground truth.

Cluster quality metrics evaluated (see Clustering performance evaluation for definitions and discussions of the metrics):

Shorthand | full name
--- | ---
homo | homogeneity score
compl | completeness score
v-meas | V measure
ARI | adjusted Rand index
AMI | adjusted mutual information
silhouette | silhouette coefficient

## Load the dataset

We will start by loading the `digits` dataset. This dataset contains handwritten digits from 0 to 9. In the context of clustering, one would like to group images such that the handwritten digits on the image are the same.

    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

## Define our evaluation benchmark

We will first our evaluation benchmark. During this benchmark, we intend to compare different initialization methods for KMeans. Our benchmark will:

- create a pipeline which will scale the data using a `StandardScaler`;

- train and time the pipeline fitting;

- measure the performance of the clustering obtained via different metrics.

    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))

## Run the benchmark

We will compare three approaches:

- an initialization using `k-means++`. This method is stochastic and we will run the initialization 4 times;

- a random initialization. This method is stochastic as well and we will run the initialization 4 times;

- an initialization based on a `PCA` projection. Indeed, we will use the components of the `PCA` to initialize KMeans. This method is deterministic and a single initialization suffice.

    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++       0.173s  69545   0.598   0.645   0.621   0.469   0.617   0.152
    random          0.077s  69735   0.681   0.723   0.701   0.574   0.698   0.170
    PCA-based       0.072s  72686   0.636   0.658   0.647   0.521   0.643   0.142
    __________________________________________________________________________________

## Visualize the results on PCA-reduced data

`PCA` allows to project the data from the original 64-dimensional space into a lower dimensional space. Subsequently, we can use `PCA` to project into a 2-dimensional space and plot the data and the clusters in this new space.

    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()

Total running time of the script: (0 minutes 0.682 seconds)

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