"""Unsupervised evaluation metrics."""
# Authors: Robert Layton <robertlayton@gmail.com>
# Arnaud Fouchet <foucheta@gmail.com>
# Thierry Guillemot <thierry.guillemot.work@gmail.com>
# License: BSD 3 clause
import functools
from numbers import Integral
import numpy as np
from scipy.sparse import issparse
from ...preprocessing import LabelEncoder
from ...utils import _safe_indexing, check_random_state, check_X_y
from ...utils._array_api import _atol_for_type
from ...utils._param_validation import (
Interval,
StrOptions,
validate_params,
)
from ..pairwise import _VALID_METRICS, pairwise_distances, pairwise_distances_chunked
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
)
@validate_params(
{
"X": ["array-like", "sparse matrix"],
"labels": ["array-like"],
"metric": [StrOptions(set(_VALID_METRICS) | {"precomputed"}), callable],
"sample_size": [Interval(Integral, 1, None, closed="left"), None],
"random_state": ["random_state"],
},
prefer_skip_nested_validation=True,
)
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 <silhouette_coefficient>`.
Parameters
----------
X : {array-like, sparse matrix} 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:`~sklearn.metrics.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 <random_state>`.
**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.
<https://www.sciencedirect.com/science/article/pii/0377042787901257>`_
.. [2] `Wikipedia entry on the Silhouette Coefficient
<https://en.wikipedia.org/wiki/Silhouette_(clustering)>`_
Examples
--------
>>> from sklearn.datasets import make_blobs
>>> from sklearn.cluster import KMeans
>>> from sklearn.metrics import silhouette_score
>>> X, y = make_blobs(random_state=42)
>>> kmeans = KMeans(n_clusters=2, random_state=42)
>>> silhouette_score(X, kmeans.fit_predict(X))
0.49...
"""
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, sparse matrix} of shape (n_chunk_samples, n_samples)
Precomputed distances for a chunk. If a sparse matrix is provided,
only CSR format is accepted.
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``.
"""
n_chunk_samples = D_chunk.shape[0]
# accumulate distances from each sample to each cluster
cluster_distances = np.zeros(
(n_chunk_samples, len(label_freqs)), dtype=D_chunk.dtype
)
if issparse(D_chunk):
if D_chunk.format != "csr":
raise TypeError(
"Expected CSR matrix. Please pass sparse matrix in CSR format."
)
for i in range(n_chunk_samples):
indptr = D_chunk.indptr
indices = D_chunk.indices[indptr[i] : indptr[i + 1]]
sample_weights = D_chunk.data[indptr[i] : indptr[i + 1]]
sample_labels = np.take(labels, indices)
cluster_distances[i] += np.bincount(
sample_labels, weights=sample_weights, minlength=len(label_freqs)
)
else:
for i in range(n_chunk_samples):
sample_weights = D_chunk[i]
sample_labels = labels
cluster_distances[i] += np.bincount(
sample_labels, weights=sample_weights, minlength=len(label_freqs)
)
# intra_index selects intra-cluster distances within cluster_distances
end = start + n_chunk_samples
intra_index = (np.arange(n_chunk_samples), labels[start:end])
# intra_cluster_distances are averaged over cluster size outside this function
intra_cluster_distances = cluster_distances[intra_index]
# of the remaining distances we normalise and extract the minimum
cluster_distances[intra_index] = np.inf
cluster_distances /= label_freqs
inter_cluster_distances = cluster_distances.min(axis=1)
return intra_cluster_distances, inter_cluster_distances
@validate_params(
{
"X": ["array-like", "sparse matrix"],
"labels": ["array-like"],
"metric": [StrOptions(set(_VALID_METRICS) | {"precomputed"}), callable],
},
prefer_skip_nested_validation=True,
)
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 <silhouette_coefficient>`.
Parameters
----------
X : {array-like, sparse matrix} 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. If
a sparse matrix is provided, CSR format should be favoured avoiding
an additional copy.
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_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.
<https://www.sciencedirect.com/science/article/pii/0377042787901257>`_
.. [2] `Wikipedia entry on the Silhouette Coefficient
<https://en.wikipedia.org/wiki/Silhouette_(clustering)>`_
Examples
--------
>>> from sklearn.metrics import silhouette_samples
>>> from sklearn.datasets import make_blobs
>>> from sklearn.cluster import KMeans
>>> X, y = make_blobs(n_samples=50, random_state=42)
>>> kmeans = KMeans(n_clusters=3, random_state=42)
>>> labels = kmeans.fit_predict(X)
>>> silhouette_samples(X, labels)
array([...])
"""
X, labels = check_X_y(X, labels, accept_sparse=["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 = _atol_for_type(X.dtype)
if np.any(np.abs(X.diagonal()) > atol):
raise error_msg
elif np.any(X.diagonal() != 0): # integral dtype
raise 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)
@validate_params(
{
"X": ["array-like"],
"labels": ["array-like"],
},
prefer_skip_nested_validation=True,
)
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 <calinski_harabasz_index>`.
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
<https://www.tandfonline.com/doi/abs/10.1080/03610927408827101>`_
Examples
--------
>>> from sklearn.datasets import make_blobs
>>> from sklearn.cluster import KMeans
>>> from sklearn.metrics import calinski_harabasz_score
>>> X, _ = make_blobs(random_state=0)
>>> kmeans = KMeans(n_clusters=3, random_state=0,).fit(X)
>>> calinski_harabasz_score(X, kmeans.labels_)
114.8...
"""
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))
)
@validate_params(
{
"X": ["array-like"],
"labels": ["array-like"],
},
prefer_skip_nested_validation=True,
)
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 <davies-bouldin_index>`.
.. 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"
<https://ieeexplore.ieee.org/document/4766909>`__.
IEEE Transactions on Pattern Analysis and Machine Intelligence.
PAMI-1 (2): 224-227
Examples
--------
>>> from sklearn.metrics import davies_bouldin_score
>>> X = [[0, 1], [1, 1], [3, 4]]
>>> labels = [0, 0, 1]
>>> davies_bouldin_score(X, labels)
0.12...
"""
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)