"""Unsupervised evaluation metrics.""" # Authors: Robert Layton # Arnaud Fouchet # Thierry Guillemot # License: BSD 3 clause import functools import numpy as np from ...utils import check_random_state from ...utils import check_X_y from ...utils import _safe_indexing from ..pairwise import pairwise_distances_chunked from ..pairwise import pairwise_distances from ...preprocessing import LabelEncoder def check_number_of_labels(n_labels, n_samples): """Check that number of labels are valid. Parameters ---------- n_labels : int Number of labels. n_samples : int Number of samples. """ if not 1 < n_labels < n_samples: raise ValueError( "Number of labels is %d. Valid values are 2 to n_samples - 1 (inclusive)" % n_labels ) 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. Note that Silhouette Coefficient is only defined if number of labels is ``2 <= n_labels <= n_samples - 1``. 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. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_a, n_samples_a) if metric == \ "precomputed" or (n_samples_a, n_features) otherwise An array of pairwise distances between samples, or a feature array. labels : array-like of shape (n_samples,) Predicted labels for each sample. metric : str or callable, default='euclidean' 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, default=None The size of the sample to use when computing the Silhouette Coefficient on a random subset of the data. If ``sample_size is None``, no sampling is used. random_state : int, RandomState instance or None, default=None Determines random number generation for selecting a subset of samples. Used when ``sample_size is not None``. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. **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: X, labels = check_X_y(X, labels, accept_sparse=["csc", "csr"]) 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_reduce(D_chunk, start, labels, label_freqs): """Accumulate silhouette statistics for vertical chunk of X. Parameters ---------- D_chunk : array-like of shape (n_chunk_samples, n_samples) Precomputed distances for a chunk. start : int First index in the chunk. labels : array-like of shape (n_samples,) Corresponding cluster labels, encoded as {0, ..., n_clusters-1}. label_freqs : array-like Distribution of cluster labels in ``labels``. """ # accumulate distances from each sample to each cluster clust_dists = np.zeros((len(D_chunk), len(label_freqs)), dtype=D_chunk.dtype) for i in range(len(D_chunk)): clust_dists[i] += np.bincount( labels, weights=D_chunk[i], minlength=len(label_freqs) ) # intra_index selects intra-cluster distances within clust_dists intra_index = (np.arange(len(D_chunk)), labels[start : start + len(D_chunk)]) # intra_clust_dists are averaged over cluster size outside this function intra_clust_dists = clust_dists[intra_index] # of the remaining distances we normalise and extract the minimum clust_dists[intra_index] = np.inf clust_dists /= label_freqs inter_clust_dists = clust_dists.min(axis=1) return intra_clust_dists, inter_clust_dists def silhouette_samples(X, labels, *, metric="euclidean", **kwds): """Compute the Silhouette Coefficient for each sample. The Silhouette 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)``. Note that Silhouette Coefficient is only defined if number of labels is 2 ``<= n_labels <= n_samples - 1``. 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. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_a, n_samples_a) if metric == \ "precomputed" or (n_samples_a, n_features) otherwise An array of pairwise distances between samples, or a feature array. labels : array-like of shape (n_samples,) Label values for each sample. metric : str or callable, default='euclidean' 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. Precomputed distance matrices must have 0 along the diagonal. **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-like of shape (n_samples,) Silhouette Coefficients for each sample. 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 `_ """ X, labels = check_X_y(X, labels, accept_sparse=["csc", "csr"]) # Check for non-zero diagonal entries in precomputed distance matrix if metric == "precomputed": error_msg = ValueError( "The precomputed distance matrix contains non-zero " "elements on the diagonal. Use np.fill_diagonal(X, 0)." ) if X.dtype.kind == "f": atol = np.finfo(X.dtype).eps * 100 if np.any(np.abs(np.diagonal(X)) > atol): raise ValueError(error_msg) elif np.any(np.diagonal(X) != 0): # integral dtype raise ValueError(error_msg) le = LabelEncoder() labels = le.fit_transform(labels) n_samples = len(labels) label_freqs = np.bincount(labels) check_number_of_labels(len(le.classes_), n_samples) kwds["metric"] = metric reduce_func = functools.partial( _silhouette_reduce, labels=labels, label_freqs=label_freqs ) results = zip(*pairwise_distances_chunked(X, reduce_func=reduce_func, **kwds)) intra_clust_dists, inter_clust_dists = results intra_clust_dists = np.concatenate(intra_clust_dists) inter_clust_dists = np.concatenate(inter_clust_dists) denom = (label_freqs - 1).take(labels, mode="clip") with np.errstate(divide="ignore", invalid="ignore"): intra_clust_dists /= denom sil_samples = inter_clust_dists - intra_clust_dists with np.errstate(divide="ignore", invalid="ignore"): sil_samples /= np.maximum(intra_clust_dists, inter_clust_dists) # nan values are for clusters of size 1, and should be 0 return np.nan_to_num(sil_samples) def calinski_harabasz_score(X, labels): """Compute the Calinski and Harabasz score. It is also known as the Variance Ratio Criterion. The score is defined as ratio of the sum of between-cluster dispersion and of within-cluster dispersion. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples, n_features) A list of ``n_features``-dimensional data points. Each row corresponds to a single data point. labels : array-like of shape (n_samples,) Predicted labels for each sample. Returns ------- score : float The resulting Calinski-Harabasz score. References ---------- .. [1] `T. Calinski and J. Harabasz, 1974. "A dendrite method for cluster analysis". Communications in Statistics `_ """ X, labels = check_X_y(X, labels) le = LabelEncoder() labels = le.fit_transform(labels) n_samples, _ = X.shape n_labels = len(le.classes_) check_number_of_labels(n_labels, n_samples) extra_disp, intra_disp = 0.0, 0.0 mean = np.mean(X, axis=0) for k in range(n_labels): cluster_k = X[labels == k] mean_k = np.mean(cluster_k, axis=0) extra_disp += len(cluster_k) * np.sum((mean_k - mean) ** 2) intra_disp += np.sum((cluster_k - mean_k) ** 2) return ( 1.0 if intra_disp == 0.0 else extra_disp * (n_samples - n_labels) / (intra_disp * (n_labels - 1.0)) ) def davies_bouldin_score(X, labels): """Compute the Davies-Bouldin score. The score is defined as the average similarity measure of each cluster with its most similar cluster, where similarity is the ratio of within-cluster distances to between-cluster distances. Thus, clusters which are farther apart and less dispersed will result in a better score. The minimum score is zero, with lower values indicating better clustering. Read more in the :ref:`User Guide `. .. versionadded:: 0.20 Parameters ---------- X : array-like of shape (n_samples, n_features) A list of ``n_features``-dimensional data points. Each row corresponds to a single data point. labels : array-like of shape (n_samples,) Predicted labels for each sample. Returns ------- score: float The resulting Davies-Bouldin score. References ---------- .. [1] Davies, David L.; Bouldin, Donald W. (1979). `"A Cluster Separation Measure" `__. IEEE Transactions on Pattern Analysis and Machine Intelligence. PAMI-1 (2): 224-227 """ X, labels = check_X_y(X, labels) le = LabelEncoder() labels = le.fit_transform(labels) n_samples, _ = X.shape n_labels = len(le.classes_) check_number_of_labels(n_labels, n_samples) intra_dists = np.zeros(n_labels) centroids = np.zeros((n_labels, len(X[0])), dtype=float) for k in range(n_labels): cluster_k = _safe_indexing(X, labels == k) centroid = cluster_k.mean(axis=0) centroids[k] = centroid intra_dists[k] = np.average(pairwise_distances(cluster_k, [centroid])) centroid_distances = pairwise_distances(centroids) if np.allclose(intra_dists, 0) or np.allclose(centroid_distances, 0): return 0.0 centroid_distances[centroid_distances == 0] = np.inf combined_intra_dists = intra_dists[:, None] + intra_dists scores = np.max(combined_intra_dists / centroid_distances, axis=1) return np.mean(scores)