""" Unsupervised evaluation metrics. """ # Authors: Robert Layton # # License: BSD Style. import numpy as np from ...utils import check_random_state from ..pairwise import pairwise_distances def silhouette_score(X, labels, metric='euclidean', sample_size=None, random_state=None, **kwds): """Compute the mean Silhouette Coefficient of all samples. The Silhouette Coefficient is calculated using the mean intra-cluster distance (``a``) and the mean nearest-cluster distance (``b``) for each sample. The Silhouette Coefficient for a sample is ``(b - a) / max(a, b)``. To clarify, ``b`` is the distance between a sample and the nearest cluster that the sample is not a part of. This function returns the mean Silhouette Coefficient over all samples. To obtain the values for each sample, use :func:`silhouette_samples`. The best value is 1 and the worst value is -1. Values near 0 indicate overlapping clusters. Negative values generally indicate that a sample has been assigned to the wrong cluster, as a different cluster is more similar. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise distances between samples, or a feature array. labels : array, shape = [n_samples] label values for each sample metric : string, or callable The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by :func:`metrics.pairwise.pairwise_distances `. If X is the distance array itself, use ``metric="precomputed"``. sample_size : int or None The size of the sample to use when computing the Silhouette Coefficient. If ``sample_size is None``, no sampling is used. random_state : integer or numpy.RandomState, optional The generator used to initialize the centers. If an integer is given, it fixes the seed. Defaults to the global numpy random number generator. `**kwds` : optional keyword parameters Any further parameters are passed directly to the distance function. If using a scipy.spatial.distance metric, the parameters are still metric dependent. See the scipy docs for usage examples. Returns ------- silhouette : float Mean Silhouette Coefficient for all samples. References ---------- .. [1] `Peter J. Rousseeuw (1987). "Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis". Computational and Applied Mathematics 20: 53-65. `_ .. [2] `Wikipedia entry on the Silhouette Coefficient `_ """ if sample_size is not None: random_state = check_random_state(random_state) indices = random_state.permutation(X.shape[0])[:sample_size] if metric == "precomputed": X, labels = X[indices].T[indices].T, labels[indices] else: X, labels = X[indices], labels[indices] return np.mean(silhouette_samples(X, labels, metric=metric, **kwds)) def silhouette_samples(X, labels, metric='euclidean', **kwds): """Compute the Silhouette Coefficient for each sample. The Silhoeutte Coefficient is a measure of how well samples are clustered with samples that are similar to themselves. Clustering models with a high Silhouette Coefficient are said to be dense, where samples in the same cluster are similar to each other, and well separated, where samples in different clusters are not very similar to each other. The Silhouette Coefficient is calculated using the mean intra-cluster distance (``a``) and the mean nearest-cluster distance (``b``) for each sample. The Silhouette Coefficient for a sample is ``(b - a) / max(a, b)``. This function returns the Silhouette Coefficient for each sample. The best value is 1 and the worst value is -1. Values near 0 indicate overlapping clusters. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise distances between samples, or a feature array. labels : array, shape = [n_samples] label values for each sample metric : string, or callable The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by :func:`sklearn.metrics.pairwise.pairwise_distances`. If X is the distance array itself, use "precomputed" as the metric. `**kwds` : optional keyword parameters Any further parameters are passed directly to the distance function. If using a ``scipy.spatial.distance`` metric, the parameters are still metric dependent. See the scipy docs for usage examples. Returns ------- silhouette : array, shape = [n_samples] Silhouette Coefficient for each samples. References ---------- .. [1] `Peter J. Rousseeuw (1987). "Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis". Computational and Applied Mathematics 20: 53-65. `_ .. [2] `Wikipedia entry on the Silhouette Coefficient `_ """ distances = pairwise_distances(X, metric=metric, **kwds) n = labels.shape[0] A = np.array([_intra_cluster_distance(distances[i], labels, i) for i in range(n)]) B = np.array([_nearest_cluster_distance(distances[i], labels, i) for i in range(n)]) sil_samples = (B - A) / np.maximum(A, B) # nan values are for clusters of size 1, and should be 0 return np.nan_to_num(sil_samples) def _intra_cluster_distance(distances_row, labels, i): """Calculate the mean intra-cluster distance for sample i. Parameters ---------- distances_row : array, shape = [n_samples] Pairwise distance matrix between sample i and each sample. labels : array, shape = [n_samples] label values for each sample i : int Sample index being calculated. It is excluded from calculation and used to determine the current label Returns ------- a : float Mean intra-cluster distance for sample i """ mask = labels == labels[i] mask[i] = False a = np.mean(distances_row[mask]) return a def _nearest_cluster_distance(distances_row, labels, i): """Calculate the mean nearest-cluster distance for sample i. Parameters ---------- distances_row : array, shape = [n_samples] Pairwise distance matrix between sample i and each sample. labels : array, shape = [n_samples] label values for each sample i : int Sample index being calculated. It is used to determine the current label. Returns ------- b : float Mean nearest-cluster distance for sample i """ label = labels[i] b = np.min([np.mean(distances_row[labels == cur_label]) for cur_label in set(labels) if not cur_label == label]) return b