# Source code for sklearn.cluster.k_means_

```
"""K-means clustering"""
# Authors: Gael Varoquaux <gael.varoquaux@normalesup.org>
# Thomas Rueckstiess <ruecksti@in.tum.de>
# James Bergstra <james.bergstra@umontreal.ca>
# Jan Schlueter <scikit-learn@jan-schlueter.de>
# Nelle Varoquaux
# Peter Prettenhofer <peter.prettenhofer@gmail.com>
# Olivier Grisel <olivier.grisel@ensta.org>
# Mathieu Blondel <mathieu@mblondel.org>
# Robert Layton <robertlayton@gmail.com>
# License: BSD 3 clause
import warnings
import numpy as np
import scipy.sparse as sp
from ..base import BaseEstimator, ClusterMixin, TransformerMixin
from ..metrics.pairwise import euclidean_distances
from ..metrics.pairwise import pairwise_distances_argmin_min
from ..utils.extmath import row_norms, squared_norm, stable_cumsum
from ..utils.sparsefuncs_fast import assign_rows_csr
from ..utils.sparsefuncs import mean_variance_axis
from ..utils import check_array
from ..utils import check_random_state
from ..utils import as_float_array
from ..utils import gen_batches
from ..utils.validation import check_is_fitted
from ..utils.validation import FLOAT_DTYPES
from ..externals.joblib import Parallel
from ..externals.joblib import delayed
from ..externals.six import string_types
from . import _k_means
from ._k_means_elkan import k_means_elkan
###############################################################################
# Initialization heuristic
def _k_init(X, n_clusters, x_squared_norms, random_state, n_local_trials=None):
"""Init n_clusters seeds according to k-means++
Parameters
-----------
X : array or sparse matrix, shape (n_samples, n_features)
The data to pick seeds for. To avoid memory copy, the input data
should be double precision (dtype=np.float64).
n_clusters : integer
The number of seeds to choose
x_squared_norms : array, shape (n_samples,)
Squared Euclidean norm of each data point.
random_state : numpy.RandomState
The generator used to initialize the centers.
n_local_trials : integer, optional
The number of seeding trials for each center (except the first),
of which the one reducing inertia the most is greedily chosen.
Set to None to make the number of trials depend logarithmically
on the number of seeds (2+log(k)); this is the default.
Notes
-----
Selects initial cluster centers for k-mean clustering in a smart way
to speed up convergence. see: Arthur, D. and Vassilvitskii, S.
"k-means++: the advantages of careful seeding". ACM-SIAM symposium
on Discrete algorithms. 2007
Version ported from http://www.stanford.edu/~darthur/kMeansppTest.zip,
which is the implementation used in the aforementioned paper.
"""
n_samples, n_features = X.shape
centers = np.empty((n_clusters, n_features), dtype=X.dtype)
assert x_squared_norms is not None, 'x_squared_norms None in _k_init'
# Set the number of local seeding trials if none is given
if n_local_trials is None:
# This is what Arthur/Vassilvitskii tried, but did not report
# specific results for other than mentioning in the conclusion
# that it helped.
n_local_trials = 2 + int(np.log(n_clusters))
# Pick first center randomly
center_id = random_state.randint(n_samples)
if sp.issparse(X):
centers[0] = X[center_id].toarray()
else:
centers[0] = X[center_id]
# Initialize list of closest distances and calculate current potential
closest_dist_sq = euclidean_distances(
centers[0, np.newaxis], X, Y_norm_squared=x_squared_norms,
squared=True)
current_pot = closest_dist_sq.sum()
# Pick the remaining n_clusters-1 points
for c in range(1, n_clusters):
# Choose center candidates by sampling with probability proportional
# to the squared distance to the closest existing center
rand_vals = random_state.random_sample(n_local_trials) * current_pot
candidate_ids = np.searchsorted(stable_cumsum(closest_dist_sq),
rand_vals)
# Compute distances to center candidates
distance_to_candidates = euclidean_distances(
X[candidate_ids], X, Y_norm_squared=x_squared_norms, squared=True)
# Decide which candidate is the best
best_candidate = None
best_pot = None
best_dist_sq = None
for trial in range(n_local_trials):
# Compute potential when including center candidate
new_dist_sq = np.minimum(closest_dist_sq,
distance_to_candidates[trial])
new_pot = new_dist_sq.sum()
# Store result if it is the best local trial so far
if (best_candidate is None) or (new_pot < best_pot):
best_candidate = candidate_ids[trial]
best_pot = new_pot
best_dist_sq = new_dist_sq
# Permanently add best center candidate found in local tries
if sp.issparse(X):
centers[c] = X[best_candidate].toarray()
else:
centers[c] = X[best_candidate]
current_pot = best_pot
closest_dist_sq = best_dist_sq
return centers
###############################################################################
# K-means batch estimation by EM (expectation maximization)
def _validate_center_shape(X, n_centers, centers):
"""Check if centers is compatible with X and n_centers"""
if len(centers) != n_centers:
raise ValueError('The shape of the initial centers (%s) '
'does not match the number of clusters %i'
% (centers.shape, n_centers))
if centers.shape[1] != X.shape[1]:
raise ValueError(
"The number of features of the initial centers %s "
"does not match the number of features of the data %s."
% (centers.shape[1], X.shape[1]))
def _tolerance(X, tol):
"""Return a tolerance which is independent of the dataset"""
if sp.issparse(X):
variances = mean_variance_axis(X, axis=0)[1]
else:
variances = np.var(X, axis=0)
return np.mean(variances) * tol
def k_means(X, n_clusters, init='k-means++', precompute_distances='auto',
n_init=10, max_iter=300, verbose=False,
tol=1e-4, random_state=None, copy_x=True, n_jobs=1,
algorithm="auto", return_n_iter=False):
"""K-means clustering algorithm.
Read more in the :ref:`User Guide <k_means>`.
Parameters
----------
X : array-like or sparse matrix, shape (n_samples, n_features)
The observations to cluster.
n_clusters : int
The number of clusters to form as well as the number of
centroids to generate.
init : {'k-means++', 'random', or ndarray, or a callable}, optional
Method for initialization, default to 'k-means++':
'k-means++' : selects initial cluster centers for k-mean
clustering in a smart way to speed up convergence. See section
Notes in k_init for more details.
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
If an ndarray is passed, it should be of shape (n_clusters, n_features)
and gives the initial centers.
If a callable is passed, it should take arguments X, k and
and a random state and return an initialization.
precompute_distances : {'auto', True, False}
Precompute distances (faster but takes more memory).
'auto' : do not precompute distances if n_samples * n_clusters > 12
million. This corresponds to about 100MB overhead per job using
double precision.
True : always precompute distances
False : never precompute distances
n_init : int, optional, default: 10
Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of
n_init consecutive runs in terms of inertia.
max_iter : int, optional, default 300
Maximum number of iterations of the k-means algorithm to run.
verbose : boolean, optional
Verbosity mode.
tol : float, optional
The relative increment in the results before declaring convergence.
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
copy_x : boolean, optional
When pre-computing distances it is more numerically accurate to center
the data first. If copy_x is True, then the original data is not
modified. If False, the original data is modified, and put back before
the function returns, but small numerical differences may be introduced
by subtracting and then adding the data mean.
n_jobs : int
The number of jobs to use for the computation. This works by computing
each of the n_init runs in parallel.
If -1 all CPUs are used. If 1 is given, no parallel computing code is
used at all, which is useful for debugging. For n_jobs below -1,
(n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
are used.
algorithm : "auto", "full" or "elkan", default="auto"
K-means algorithm to use. The classical EM-style algorithm is "full".
The "elkan" variation is more efficient by using the triangle
inequality, but currently doesn't support sparse data. "auto" chooses
"elkan" for dense data and "full" for sparse data.
return_n_iter : bool, optional
Whether or not to return the number of iterations.
Returns
-------
centroid : float ndarray with shape (k, n_features)
Centroids found at the last iteration of k-means.
label : integer ndarray with shape (n_samples,)
label[i] is the code or index of the centroid the
i'th observation is closest to.
inertia : float
The final value of the inertia criterion (sum of squared distances to
the closest centroid for all observations in the training set).
best_n_iter : int
Number of iterations corresponding to the best results.
Returned only if `return_n_iter` is set to True.
"""
if n_init <= 0:
raise ValueError("Invalid number of initializations."
" n_init=%d must be bigger than zero." % n_init)
random_state = check_random_state(random_state)
if max_iter <= 0:
raise ValueError('Number of iterations should be a positive number,'
' got %d instead' % max_iter)
X = as_float_array(X, copy=copy_x)
tol = _tolerance(X, tol)
# If the distances are precomputed every job will create a matrix of shape
# (n_clusters, n_samples). To stop KMeans from eating up memory we only
# activate this if the created matrix is guaranteed to be under 100MB. 12
# million entries consume a little under 100MB if they are of type double.
if precompute_distances == 'auto':
n_samples = X.shape[0]
precompute_distances = (n_clusters * n_samples) < 12e6
elif isinstance(precompute_distances, bool):
pass
else:
raise ValueError("precompute_distances should be 'auto' or True/False"
", but a value of %r was passed" %
precompute_distances)
# Validate init array
if hasattr(init, '__array__'):
init = check_array(init, dtype=X.dtype.type, copy=True)
_validate_center_shape(X, n_clusters, init)
if n_init != 1:
warnings.warn(
'Explicit initial center position passed: '
'performing only one init in k-means instead of n_init=%d'
% n_init, RuntimeWarning, stacklevel=2)
n_init = 1
# subtract of mean of x for more accurate distance computations
if not sp.issparse(X):
X_mean = X.mean(axis=0)
# The copy was already done above
X -= X_mean
if hasattr(init, '__array__'):
init -= X_mean
# precompute squared norms of data points
x_squared_norms = row_norms(X, squared=True)
best_labels, best_inertia, best_centers = None, None, None
if n_clusters == 1:
# elkan doesn't make sense for a single cluster, full will produce
# the right result.
algorithm = "full"
if algorithm == "auto":
algorithm = "full" if sp.issparse(X) else 'elkan'
if algorithm == "full":
kmeans_single = _kmeans_single_lloyd
elif algorithm == "elkan":
kmeans_single = _kmeans_single_elkan
else:
raise ValueError("Algorithm must be 'auto', 'full' or 'elkan', got"
" %s" % str(algorithm))
if n_jobs == 1:
# For a single thread, less memory is needed if we just store one set
# of the best results (as opposed to one set per run per thread).
for it in range(n_init):
# run a k-means once
labels, inertia, centers, n_iter_ = kmeans_single(
X, n_clusters, max_iter=max_iter, init=init, verbose=verbose,
precompute_distances=precompute_distances, tol=tol,
x_squared_norms=x_squared_norms, random_state=random_state)
# determine if these results are the best so far
if best_inertia is None or inertia < best_inertia:
best_labels = labels.copy()
best_centers = centers.copy()
best_inertia = inertia
best_n_iter = n_iter_
else:
# parallelisation of k-means runs
seeds = random_state.randint(np.iinfo(np.int32).max, size=n_init)
results = Parallel(n_jobs=n_jobs, verbose=0)(
delayed(kmeans_single)(X, n_clusters, max_iter=max_iter, init=init,
verbose=verbose, tol=tol,
precompute_distances=precompute_distances,
x_squared_norms=x_squared_norms,
# Change seed to ensure variety
random_state=seed)
for seed in seeds)
# Get results with the lowest inertia
labels, inertia, centers, n_iters = zip(*results)
best = np.argmin(inertia)
best_labels = labels[best]
best_inertia = inertia[best]
best_centers = centers[best]
best_n_iter = n_iters[best]
if not sp.issparse(X):
if not copy_x:
X += X_mean
best_centers += X_mean
if return_n_iter:
return best_centers, best_labels, best_inertia, best_n_iter
else:
return best_centers, best_labels, best_inertia
def _kmeans_single_elkan(X, n_clusters, max_iter=300, init='k-means++',
verbose=False, x_squared_norms=None,
random_state=None, tol=1e-4,
precompute_distances=True):
if sp.issparse(X):
raise ValueError("algorithm='elkan' not supported for sparse input X")
X = check_array(X, order="C")
random_state = check_random_state(random_state)
if x_squared_norms is None:
x_squared_norms = row_norms(X, squared=True)
# init
centers = _init_centroids(X, n_clusters, init, random_state=random_state,
x_squared_norms=x_squared_norms)
centers = np.ascontiguousarray(centers)
if verbose:
print('Initialization complete')
centers, labels, n_iter = k_means_elkan(X, n_clusters, centers, tol=tol,
max_iter=max_iter, verbose=verbose)
inertia = np.sum((X - centers[labels]) ** 2, dtype=np.float64)
return labels, inertia, centers, n_iter
def _kmeans_single_lloyd(X, n_clusters, max_iter=300, init='k-means++',
verbose=False, x_squared_norms=None,
random_state=None, tol=1e-4,
precompute_distances=True):
"""A single run of k-means, assumes preparation completed prior.
Parameters
----------
X : array-like of floats, shape (n_samples, n_features)
The observations to cluster.
n_clusters : int
The number of clusters to form as well as the number of
centroids to generate.
max_iter : int, optional, default 300
Maximum number of iterations of the k-means algorithm to run.
init : {'k-means++', 'random', or ndarray, or a callable}, optional
Method for initialization, default to 'k-means++':
'k-means++' : selects initial cluster centers for k-mean
clustering in a smart way to speed up convergence. See section
Notes in k_init for more details.
'random': generate k centroids from a Gaussian with mean and
variance estimated from the data.
If an ndarray is passed, it should be of shape (k, p) and gives
the initial centers.
If a callable is passed, it should take arguments X, k and
and a random state and return an initialization.
tol : float, optional
The relative increment in the results before declaring convergence.
verbose : boolean, optional
Verbosity mode
x_squared_norms : array
Precomputed x_squared_norms.
precompute_distances : boolean, default: True
Precompute distances (faster but takes more memory).
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
Returns
-------
centroid : float ndarray with shape (k, n_features)
Centroids found at the last iteration of k-means.
label : integer ndarray with shape (n_samples,)
label[i] is the code or index of the centroid the
i'th observation is closest to.
inertia : float
The final value of the inertia criterion (sum of squared distances to
the closest centroid for all observations in the training set).
n_iter : int
Number of iterations run.
"""
random_state = check_random_state(random_state)
best_labels, best_inertia, best_centers = None, None, None
# init
centers = _init_centroids(X, n_clusters, init, random_state=random_state,
x_squared_norms=x_squared_norms)
if verbose:
print("Initialization complete")
# Allocate memory to store the distances for each sample to its
# closer center for reallocation in case of ties
distances = np.zeros(shape=(X.shape[0],), dtype=X.dtype)
# iterations
for i in range(max_iter):
centers_old = centers.copy()
# labels assignment is also called the E-step of EM
labels, inertia = \
_labels_inertia(X, x_squared_norms, centers,
precompute_distances=precompute_distances,
distances=distances)
# computation of the means is also called the M-step of EM
if sp.issparse(X):
centers = _k_means._centers_sparse(X, labels, n_clusters,
distances)
else:
centers = _k_means._centers_dense(X, labels, n_clusters, distances)
if verbose:
print("Iteration %2d, inertia %.3f" % (i, inertia))
if best_inertia is None or inertia < best_inertia:
best_labels = labels.copy()
best_centers = centers.copy()
best_inertia = inertia
center_shift_total = squared_norm(centers_old - centers)
if center_shift_total <= tol:
if verbose:
print("Converged at iteration %d: "
"center shift %e within tolerance %e"
% (i, center_shift_total, tol))
break
if center_shift_total > 0:
# rerun E-step in case of non-convergence so that predicted labels
# match cluster centers
best_labels, best_inertia = \
_labels_inertia(X, x_squared_norms, best_centers,
precompute_distances=precompute_distances,
distances=distances)
return best_labels, best_inertia, best_centers, i + 1
def _labels_inertia_precompute_dense(X, x_squared_norms, centers, distances):
"""Compute labels and inertia using a full distance matrix.
This will overwrite the 'distances' array in-place.
Parameters
----------
X : numpy array, shape (n_sample, n_features)
Input data.
x_squared_norms : numpy array, shape (n_samples,)
Precomputed squared norms of X.
centers : numpy array, shape (n_clusters, n_features)
Cluster centers which data is assigned to.
distances : numpy array, shape (n_samples,)
Pre-allocated array in which distances are stored.
Returns
-------
labels : numpy array, dtype=np.int, shape (n_samples,)
Indices of clusters that samples are assigned to.
inertia : float
Sum of squared distances of samples to their closest cluster center.
"""
n_samples = X.shape[0]
# Breakup nearest neighbor distance computation into batches to prevent
# memory blowup in the case of a large number of samples and clusters.
# TODO: Once PR #7383 is merged use check_inputs=False in metric_kwargs.
labels, mindist = pairwise_distances_argmin_min(
X=X, Y=centers, metric='euclidean', metric_kwargs={'squared': True})
# cython k-means code assumes int32 inputs
labels = labels.astype(np.int32)
if n_samples == distances.shape[0]:
# distances will be changed in-place
distances[:] = mindist
inertia = mindist.sum()
return labels, inertia
def _labels_inertia(X, x_squared_norms, centers,
precompute_distances=True, distances=None):
"""E step of the K-means EM algorithm.
Compute the labels and the inertia of the given samples and centers.
This will compute the distances in-place.
Parameters
----------
X : float64 array-like or CSR sparse matrix, shape (n_samples, n_features)
The input samples to assign to the labels.
x_squared_norms : array, shape (n_samples,)
Precomputed squared euclidean norm of each data point, to speed up
computations.
centers : float array, shape (k, n_features)
The cluster centers.
precompute_distances : boolean, default: True
Precompute distances (faster but takes more memory).
distances : float array, shape (n_samples,)
Pre-allocated array to be filled in with each sample's distance
to the closest center.
Returns
-------
labels : int array of shape(n)
The resulting assignment
inertia : float
Sum of squared distances of samples to their closest cluster center.
"""
n_samples = X.shape[0]
# set the default value of centers to -1 to be able to detect any anomaly
# easily
labels = -np.ones(n_samples, np.int32)
if distances is None:
distances = np.zeros(shape=(0,), dtype=X.dtype)
# distances will be changed in-place
if sp.issparse(X):
inertia = _k_means._assign_labels_csr(
X, x_squared_norms, centers, labels, distances=distances)
else:
if precompute_distances:
return _labels_inertia_precompute_dense(X, x_squared_norms,
centers, distances)
inertia = _k_means._assign_labels_array(
X, x_squared_norms, centers, labels, distances=distances)
return labels, inertia
def _init_centroids(X, k, init, random_state=None, x_squared_norms=None,
init_size=None):
"""Compute the initial centroids
Parameters
----------
X : array, shape (n_samples, n_features)
k : int
number of centroids
init : {'k-means++', 'random' or ndarray or callable} optional
Method for initialization
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
x_squared_norms : array, shape (n_samples,), optional
Squared euclidean norm of each data point. Pass it if you have it at
hands already to avoid it being recomputed here. Default: None
init_size : int, optional
Number of samples to randomly sample for speeding up the
initialization (sometimes at the expense of accuracy): the
only algorithm is initialized by running a batch KMeans on a
random subset of the data. This needs to be larger than k.
Returns
-------
centers : array, shape(k, n_features)
"""
random_state = check_random_state(random_state)
n_samples = X.shape[0]
if x_squared_norms is None:
x_squared_norms = row_norms(X, squared=True)
if init_size is not None and init_size < n_samples:
if init_size < k:
warnings.warn(
"init_size=%d should be larger than k=%d. "
"Setting it to 3*k" % (init_size, k),
RuntimeWarning, stacklevel=2)
init_size = 3 * k
init_indices = random_state.randint(0, n_samples, init_size)
X = X[init_indices]
x_squared_norms = x_squared_norms[init_indices]
n_samples = X.shape[0]
elif n_samples < k:
raise ValueError(
"n_samples=%d should be larger than k=%d" % (n_samples, k))
if isinstance(init, string_types) and init == 'k-means++':
centers = _k_init(X, k, random_state=random_state,
x_squared_norms=x_squared_norms)
elif isinstance(init, string_types) and init == 'random':
seeds = random_state.permutation(n_samples)[:k]
centers = X[seeds]
elif hasattr(init, '__array__'):
# ensure that the centers have the same dtype as X
# this is a requirement of fused types of cython
centers = np.array(init, dtype=X.dtype)
elif callable(init):
centers = init(X, k, random_state=random_state)
centers = np.asarray(centers, dtype=X.dtype)
else:
raise ValueError("the init parameter for the k-means should "
"be 'k-means++' or 'random' or an ndarray, "
"'%s' (type '%s') was passed." % (init, type(init)))
if sp.issparse(centers):
centers = centers.toarray()
_validate_center_shape(X, k, centers)
return centers
class KMeans(BaseEstimator, ClusterMixin, TransformerMixin):
"""K-Means clustering
Read more in the :ref:`User Guide <k_means>`.
Parameters
----------
n_clusters : int, optional, default: 8
The number of clusters to form as well as the number of
centroids to generate.
init : {'k-means++', 'random' or an ndarray}
Method for initialization, defaults to 'k-means++':
'k-means++' : selects initial cluster centers for k-mean
clustering in a smart way to speed up convergence. See section
Notes in k_init for more details.
'random': choose k observations (rows) at random from data for
the initial centroids.
If an ndarray is passed, it should be of shape (n_clusters, n_features)
and gives the initial centers.
n_init : int, default: 10
Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of
n_init consecutive runs in terms of inertia.
max_iter : int, default: 300
Maximum number of iterations of the k-means algorithm for a
single run.
tol : float, default: 1e-4
Relative tolerance with regards to inertia to declare convergence
precompute_distances : {'auto', True, False}
Precompute distances (faster but takes more memory).
'auto' : do not precompute distances if n_samples * n_clusters > 12
million. This corresponds to about 100MB overhead per job using
double precision.
True : always precompute distances
False : never precompute distances
verbose : int, default 0
Verbosity mode.
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
copy_x : boolean, default True
When pre-computing distances it is more numerically accurate to center
the data first. If copy_x is True, then the original data is not
modified. If False, the original data is modified, and put back before
the function returns, but small numerical differences may be introduced
by subtracting and then adding the data mean.
n_jobs : int
The number of jobs to use for the computation. This works by computing
each of the n_init runs in parallel.
If -1 all CPUs are used. If 1 is given, no parallel computing code is
used at all, which is useful for debugging. For n_jobs below -1,
(n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
are used.
algorithm : "auto", "full" or "elkan", default="auto"
K-means algorithm to use. The classical EM-style algorithm is "full".
The "elkan" variation is more efficient by using the triangle
inequality, but currently doesn't support sparse data. "auto" chooses
"elkan" for dense data and "full" for sparse data.
Attributes
----------
cluster_centers_ : array, [n_clusters, n_features]
Coordinates of cluster centers
labels_ :
Labels of each point
inertia_ : float
Sum of squared distances of samples to their closest cluster center.
Examples
--------
>>> from sklearn.cluster import KMeans
>>> import numpy as np
>>> X = np.array([[1, 2], [1, 4], [1, 0],
... [4, 2], [4, 4], [4, 0]])
>>> kmeans = KMeans(n_clusters=2, random_state=0).fit(X)
>>> kmeans.labels_
array([0, 0, 0, 1, 1, 1], dtype=int32)
>>> kmeans.predict([[0, 0], [4, 4]])
array([0, 1], dtype=int32)
>>> kmeans.cluster_centers_
array([[ 1., 2.],
[ 4., 2.]])
See also
--------
MiniBatchKMeans
Alternative online implementation that does incremental updates
of the centers positions using mini-batches.
For large scale learning (say n_samples > 10k) MiniBatchKMeans is
probably much faster than the default batch implementation.
Notes
------
The k-means problem is solved using Lloyd's algorithm.
The average complexity is given by O(k n T), were n is the number of
samples and T is the number of iteration.
The worst case complexity is given by O(n^(k+2/p)) with
n = n_samples, p = n_features. (D. Arthur and S. Vassilvitskii,
'How slow is the k-means method?' SoCG2006)
In practice, the k-means algorithm is very fast (one of the fastest
clustering algorithms available), but it falls in local minima. That's why
it can be useful to restart it several times.
"""
def __init__(self, n_clusters=8, init='k-means++', n_init=10,
max_iter=300, tol=1e-4, precompute_distances='auto',
verbose=0, random_state=None, copy_x=True,
n_jobs=1, algorithm='auto'):
self.n_clusters = n_clusters
self.init = init
self.max_iter = max_iter
self.tol = tol
self.precompute_distances = precompute_distances
self.n_init = n_init
self.verbose = verbose
self.random_state = random_state
self.copy_x = copy_x
self.n_jobs = n_jobs
self.algorithm = algorithm
def _check_fit_data(self, X):
"""Verify that the number of samples given is larger than k"""
X = check_array(X, accept_sparse='csr', dtype=[np.float64, np.float32])
if X.shape[0] < self.n_clusters:
raise ValueError("n_samples=%d should be >= n_clusters=%d" % (
X.shape[0], self.n_clusters))
return X
def _check_test_data(self, X):
X = check_array(X, accept_sparse='csr', dtype=FLOAT_DTYPES)
n_samples, n_features = X.shape
expected_n_features = self.cluster_centers_.shape[1]
if not n_features == expected_n_features:
raise ValueError("Incorrect number of features. "
"Got %d features, expected %d" % (
n_features, expected_n_features))
return X
[docs] def fit(self, X, y=None):
"""Compute k-means clustering.
Parameters
----------
X : array-like or sparse matrix, shape=(n_samples, n_features)
Training instances to cluster.
y : Ignored
"""
random_state = check_random_state(self.random_state)
X = self._check_fit_data(X)
self.cluster_centers_, self.labels_, self.inertia_, self.n_iter_ = \
k_means(
X, n_clusters=self.n_clusters, init=self.init,
n_init=self.n_init, max_iter=self.max_iter, verbose=self.verbose,
precompute_distances=self.precompute_distances,
tol=self.tol, random_state=random_state, copy_x=self.copy_x,
n_jobs=self.n_jobs, algorithm=self.algorithm,
return_n_iter=True)
return self
[docs] def fit_predict(self, X, y=None):
"""Compute cluster centers and predict cluster index for each sample.
Convenience method; equivalent to calling fit(X) followed by
predict(X).
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data to transform.
u : Ignored
Returns
-------
labels : array, shape [n_samples,]
Index of the cluster each sample belongs to.
"""
return self.fit(X).labels_
[docs] def fit_transform(self, X, y=None):
"""Compute clustering and transform X to cluster-distance space.
Equivalent to fit(X).transform(X), but more efficiently implemented.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data to transform.
y : Ignored
Returns
-------
X_new : array, shape [n_samples, k]
X transformed in the new space.
"""
# Currently, this just skips a copy of the data if it is not in
# np.array or CSR format already.
# XXX This skips _check_test_data, which may change the dtype;
# we should refactor the input validation.
X = self._check_fit_data(X)
return self.fit(X)._transform(X)
[docs] def transform(self, X):
"""Transform X to a cluster-distance space.
In the new space, each dimension is the distance to the cluster
centers. Note that even if X is sparse, the array returned by
`transform` will typically be dense.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data to transform.
Returns
-------
X_new : array, shape [n_samples, k]
X transformed in the new space.
"""
check_is_fitted(self, 'cluster_centers_')
X = self._check_test_data(X)
return self._transform(X)
def _transform(self, X):
"""guts of transform method; no input validation"""
return euclidean_distances(X, self.cluster_centers_)
[docs] def predict(self, X):
"""Predict the closest cluster each sample in X belongs to.
In the vector quantization literature, `cluster_centers_` is called
the code book and each value returned by `predict` is the index of
the closest code in the code book.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data to predict.
Returns
-------
labels : array, shape [n_samples,]
Index of the cluster each sample belongs to.
"""
check_is_fitted(self, 'cluster_centers_')
X = self._check_test_data(X)
x_squared_norms = row_norms(X, squared=True)
return _labels_inertia(X, x_squared_norms, self.cluster_centers_)[0]
[docs] def score(self, X, y=None):
"""Opposite of the value of X on the K-means objective.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data.
y : Ignored
Returns
-------
score : float
Opposite of the value of X on the K-means objective.
"""
check_is_fitted(self, 'cluster_centers_')
X = self._check_test_data(X)
x_squared_norms = row_norms(X, squared=True)
return -_labels_inertia(X, x_squared_norms, self.cluster_centers_)[1]
def _mini_batch_step(X, x_squared_norms, centers, counts,
old_center_buffer, compute_squared_diff,
distances, random_reassign=False,
random_state=None, reassignment_ratio=.01,
verbose=False):
"""Incremental update of the centers for the Minibatch K-Means algorithm.
Parameters
----------
X : array, shape (n_samples, n_features)
The original data array.
x_squared_norms : array, shape (n_samples,)
Squared euclidean norm of each data point.
centers : array, shape (k, n_features)
The cluster centers. This array is MODIFIED IN PLACE
counts : array, shape (k,)
The vector in which we keep track of the numbers of elements in a
cluster. This array is MODIFIED IN PLACE
distances : array, dtype float, shape (n_samples), optional
If not None, should be a pre-allocated array that will be used to store
the distances of each sample to its closest center.
May not be None when random_reassign is True.
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
random_reassign : boolean, optional
If True, centers with very low counts are randomly reassigned
to observations.
reassignment_ratio : float, optional
Control the fraction of the maximum number of counts for a
center to be reassigned. A higher value means that low count
centers are more likely to be reassigned, which means that the
model will take longer to converge, but should converge in a
better clustering.
verbose : bool, optional, default False
Controls the verbosity.
compute_squared_diff : bool
If set to False, the squared diff computation is skipped.
old_center_buffer : int
Copy of old centers for monitoring convergence.
Returns
-------
inertia : float
Sum of squared distances of samples to their closest cluster center.
squared_diff : numpy array, shape (n_clusters,)
Squared distances between previous and updated cluster centers.
"""
# Perform label assignment to nearest centers
nearest_center, inertia = _labels_inertia(X, x_squared_norms, centers,
distances=distances)
if random_reassign and reassignment_ratio > 0:
random_state = check_random_state(random_state)
# Reassign clusters that have very low counts
to_reassign = counts < reassignment_ratio * counts.max()
# pick at most .5 * batch_size samples as new centers
if to_reassign.sum() > .5 * X.shape[0]:
indices_dont_reassign = np.argsort(counts)[int(.5 * X.shape[0]):]
to_reassign[indices_dont_reassign] = False
n_reassigns = to_reassign.sum()
if n_reassigns:
# Pick new clusters amongst observations with uniform probability
new_centers = random_state.choice(X.shape[0], replace=False,
size=n_reassigns)
if verbose:
print("[MiniBatchKMeans] Reassigning %i cluster centers."
% n_reassigns)
if sp.issparse(X) and not sp.issparse(centers):
assign_rows_csr(X, new_centers.astype(np.intp),
np.where(to_reassign)[0].astype(np.intp),
centers)
else:
centers[to_reassign] = X[new_centers]
# reset counts of reassigned centers, but don't reset them too small
# to avoid instant reassignment. This is a pretty dirty hack as it
# also modifies the learning rates.
counts[to_reassign] = np.min(counts[~to_reassign])
# implementation for the sparse CSR representation completely written in
# cython
if sp.issparse(X):
return inertia, _k_means._mini_batch_update_csr(
X, x_squared_norms, centers, counts, nearest_center,
old_center_buffer, compute_squared_diff)
# dense variant in mostly numpy (not as memory efficient though)
k = centers.shape[0]
squared_diff = 0.0
for center_idx in range(k):
# find points from minibatch that are assigned to this center
center_mask = nearest_center == center_idx
count = center_mask.sum()
if count > 0:
if compute_squared_diff:
old_center_buffer[:] = centers[center_idx]
# inplace remove previous count scaling
centers[center_idx] *= counts[center_idx]
# inplace sum with new points members of this cluster
centers[center_idx] += np.sum(X[center_mask], axis=0)
# update the count statistics for this center
counts[center_idx] += count
# inplace rescale to compute mean of all points (old and new)
# Note: numpy >= 1.10 does not support '/=' for the following
# expression for a mixture of int and float (see numpy issue #6464)
centers[center_idx] = centers[center_idx] / counts[center_idx]
# update the squared diff if necessary
if compute_squared_diff:
diff = centers[center_idx].ravel() - old_center_buffer.ravel()
squared_diff += np.dot(diff, diff)
return inertia, squared_diff
def _mini_batch_convergence(model, iteration_idx, n_iter, tol,
n_samples, centers_squared_diff, batch_inertia,
context, verbose=0):
"""Helper function to encapsulate the early stopping logic"""
# Normalize inertia to be able to compare values when
# batch_size changes
batch_inertia /= model.batch_size
centers_squared_diff /= model.batch_size
# Compute an Exponentially Weighted Average of the squared
# diff to monitor the convergence while discarding
# minibatch-local stochastic variability:
# https://en.wikipedia.org/wiki/Moving_average
ewa_diff = context.get('ewa_diff')
ewa_inertia = context.get('ewa_inertia')
if ewa_diff is None:
ewa_diff = centers_squared_diff
ewa_inertia = batch_inertia
else:
alpha = float(model.batch_size) * 2.0 / (n_samples + 1)
alpha = 1.0 if alpha > 1.0 else alpha
ewa_diff = ewa_diff * (1 - alpha) + centers_squared_diff * alpha
ewa_inertia = ewa_inertia * (1 - alpha) + batch_inertia * alpha
# Log progress to be able to monitor convergence
if verbose:
progress_msg = (
'Minibatch iteration %d/%d:'
' mean batch inertia: %f, ewa inertia: %f ' % (
iteration_idx + 1, n_iter, batch_inertia,
ewa_inertia))
print(progress_msg)
# Early stopping based on absolute tolerance on squared change of
# centers position (using EWA smoothing)
if tol > 0.0 and ewa_diff <= tol:
if verbose:
print('Converged (small centers change) at iteration %d/%d'
% (iteration_idx + 1, n_iter))
return True
# Early stopping heuristic due to lack of improvement on smoothed inertia
ewa_inertia_min = context.get('ewa_inertia_min')
no_improvement = context.get('no_improvement', 0)
if ewa_inertia_min is None or ewa_inertia < ewa_inertia_min:
no_improvement = 0
ewa_inertia_min = ewa_inertia
else:
no_improvement += 1
if (model.max_no_improvement is not None
and no_improvement >= model.max_no_improvement):
if verbose:
print('Converged (lack of improvement in inertia)'
' at iteration %d/%d'
% (iteration_idx + 1, n_iter))
return True
# update the convergence context to maintain state across successive calls:
context['ewa_diff'] = ewa_diff
context['ewa_inertia'] = ewa_inertia
context['ewa_inertia_min'] = ewa_inertia_min
context['no_improvement'] = no_improvement
return False
class MiniBatchKMeans(KMeans):
"""Mini-Batch K-Means clustering
Read more in the :ref:`User Guide <mini_batch_kmeans>`.
Parameters
----------
n_clusters : int, optional, default: 8
The number of clusters to form as well as the number of
centroids to generate.
init : {'k-means++', 'random' or an ndarray}, default: 'k-means++'
Method for initialization, defaults to 'k-means++':
'k-means++' : selects initial cluster centers for k-mean
clustering in a smart way to speed up convergence. See section
Notes in k_init for more details.
'random': choose k observations (rows) at random from data for
the initial centroids.
If an ndarray is passed, it should be of shape (n_clusters, n_features)
and gives the initial centers.
max_iter : int, optional
Maximum number of iterations over the complete dataset before
stopping independently of any early stopping criterion heuristics.
batch_size : int, optional, default: 100
Size of the mini batches.
verbose : boolean, optional
Verbosity mode.
compute_labels : boolean, default=True
Compute label assignment and inertia for the complete dataset
once the minibatch optimization has converged in fit.
random_state : int, RandomState instance or None, optional, default: None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
tol : float, default: 0.0
Control early stopping based on the relative center changes as
measured by a smoothed, variance-normalized of the mean center
squared position changes. This early stopping heuristics is
closer to the one used for the batch variant of the algorithms
but induces a slight computational and memory overhead over the
inertia heuristic.
To disable convergence detection based on normalized center
change, set tol to 0.0 (default).
max_no_improvement : int, default: 10
Control early stopping based on the consecutive number of mini
batches that does not yield an improvement on the smoothed inertia.
To disable convergence detection based on inertia, set
max_no_improvement to None.
init_size : int, optional, default: 3 * batch_size
Number of samples to randomly sample for speeding up the
initialization (sometimes at the expense of accuracy): the
only algorithm is initialized by running a batch KMeans on a
random subset of the data. This needs to be larger than n_clusters.
n_init : int, default=3
Number of random initializations that are tried.
In contrast to KMeans, the algorithm is only run once, using the
best of the ``n_init`` initializations as measured by inertia.
reassignment_ratio : float, default: 0.01
Control the fraction of the maximum number of counts for a
center to be reassigned. A higher value means that low count
centers are more easily reassigned, which means that the
model will take longer to converge, but should converge in a
better clustering.
Attributes
----------
cluster_centers_ : array, [n_clusters, n_features]
Coordinates of cluster centers
labels_ :
Labels of each point (if compute_labels is set to True).
inertia_ : float
The value of the inertia criterion associated with the chosen
partition (if compute_labels is set to True). The inertia is
defined as the sum of square distances of samples to their nearest
neighbor.
See also
--------
KMeans
The classic implementation of the clustering method based on the
Lloyd's algorithm. It consumes the whole set of input data at each
iteration.
Notes
-----
See http://www.eecs.tufts.edu/~dsculley/papers/fastkmeans.pdf
"""
def __init__(self, n_clusters=8, init='k-means++', max_iter=100,
batch_size=100, verbose=0, compute_labels=True,
random_state=None, tol=0.0, max_no_improvement=10,
init_size=None, n_init=3, reassignment_ratio=0.01):
super(MiniBatchKMeans, self).__init__(
n_clusters=n_clusters, init=init, max_iter=max_iter,
verbose=verbose, random_state=random_state, tol=tol, n_init=n_init)
self.max_no_improvement = max_no_improvement
self.batch_size = batch_size
self.compute_labels = compute_labels
self.init_size = init_size
self.reassignment_ratio = reassignment_ratio
[docs] def fit(self, X, y=None):
"""Compute the centroids on X by chunking it into mini-batches.
Parameters
----------
X : array-like or sparse matrix, shape=(n_samples, n_features)
Training instances to cluster.
y : Ignored
"""
random_state = check_random_state(self.random_state)
X = check_array(X, accept_sparse="csr", order='C',
dtype=[np.float64, np.float32])
n_samples, n_features = X.shape
if n_samples < self.n_clusters:
raise ValueError("Number of samples smaller than number "
"of clusters.")
n_init = self.n_init
if hasattr(self.init, '__array__'):
self.init = np.ascontiguousarray(self.init, dtype=X.dtype)
if n_init != 1:
warnings.warn(
'Explicit initial center position passed: '
'performing only one init in MiniBatchKMeans instead of '
'n_init=%d'
% self.n_init, RuntimeWarning, stacklevel=2)
n_init = 1
x_squared_norms = row_norms(X, squared=True)
if self.tol > 0.0:
tol = _tolerance(X, self.tol)
# using tol-based early stopping needs the allocation of a
# dedicated before which can be expensive for high dim data:
# hence we allocate it outside of the main loop
old_center_buffer = np.zeros(n_features, dtype=X.dtype)
else:
tol = 0.0
# no need for the center buffer if tol-based early stopping is
# disabled
old_center_buffer = np.zeros(0, dtype=X.dtype)
distances = np.zeros(self.batch_size, dtype=X.dtype)
n_batches = int(np.ceil(float(n_samples) / self.batch_size))
n_iter = int(self.max_iter * n_batches)
init_size = self.init_size
if init_size is None:
init_size = 3 * self.batch_size
if init_size > n_samples:
init_size = n_samples
self.init_size_ = init_size
validation_indices = random_state.randint(0, n_samples, init_size)
X_valid = X[validation_indices]
x_squared_norms_valid = x_squared_norms[validation_indices]
# perform several inits with random sub-sets
best_inertia = None
for init_idx in range(n_init):
if self.verbose:
print("Init %d/%d with method: %s"
% (init_idx + 1, n_init, self.init))
counts = np.zeros(self.n_clusters, dtype=np.int32)
# TODO: once the `k_means` function works with sparse input we
# should refactor the following init to use it instead.
# Initialize the centers using only a fraction of the data as we
# expect n_samples to be very large when using MiniBatchKMeans
cluster_centers = _init_centroids(
X, self.n_clusters, self.init,
random_state=random_state,
x_squared_norms=x_squared_norms,
init_size=init_size)
# Compute the label assignment on the init dataset
batch_inertia, centers_squared_diff = _mini_batch_step(
X_valid, x_squared_norms[validation_indices],
cluster_centers, counts, old_center_buffer, False,
distances=None, verbose=self.verbose)
# Keep only the best cluster centers across independent inits on
# the common validation set
_, inertia = _labels_inertia(X_valid, x_squared_norms_valid,
cluster_centers)
if self.verbose:
print("Inertia for init %d/%d: %f"
% (init_idx + 1, n_init, inertia))
if best_inertia is None or inertia < best_inertia:
self.cluster_centers_ = cluster_centers
self.counts_ = counts
best_inertia = inertia
# Empty context to be used inplace by the convergence check routine
convergence_context = {}
# Perform the iterative optimization until the final convergence
# criterion
for iteration_idx in range(n_iter):
# Sample a minibatch from the full dataset
minibatch_indices = random_state.randint(
0, n_samples, self.batch_size)
# Perform the actual update step on the minibatch data
batch_inertia, centers_squared_diff = _mini_batch_step(
X[minibatch_indices], x_squared_norms[minibatch_indices],
self.cluster_centers_, self.counts_,
old_center_buffer, tol > 0.0, distances=distances,
# Here we randomly choose whether to perform
# random reassignment: the choice is done as a function
# of the iteration index, and the minimum number of
# counts, in order to force this reassignment to happen
# every once in a while
random_reassign=((iteration_idx + 1)
% (10 + self.counts_.min()) == 0),
random_state=random_state,
reassignment_ratio=self.reassignment_ratio,
verbose=self.verbose)
# Monitor convergence and do early stopping if necessary
if _mini_batch_convergence(
self, iteration_idx, n_iter, tol, n_samples,
centers_squared_diff, batch_inertia, convergence_context,
verbose=self.verbose):
break
self.n_iter_ = iteration_idx + 1
if self.compute_labels:
self.labels_, self.inertia_ = self._labels_inertia_minibatch(X)
return self
def _labels_inertia_minibatch(self, X):
"""Compute labels and inertia using mini batches.
This is slightly slower than doing everything at once but preventes
memory errors / segfaults.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Input data.
Returns
-------
labels : array, shap (n_samples,)
Cluster labels for each point.
inertia : float
Sum of squared distances of points to nearest cluster.
"""
if self.verbose:
print('Computing label assignment and total inertia')
x_squared_norms = row_norms(X, squared=True)
slices = gen_batches(X.shape[0], self.batch_size)
results = [_labels_inertia(X[s], x_squared_norms[s],
self.cluster_centers_) for s in slices]
labels, inertia = zip(*results)
return np.hstack(labels), np.sum(inertia)
[docs] def partial_fit(self, X, y=None):
"""Update k means estimate on a single mini-batch X.
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Coordinates of the data points to cluster.
y : Ignored
"""
X = check_array(X, accept_sparse="csr")
n_samples, n_features = X.shape
if hasattr(self.init, '__array__'):
self.init = np.ascontiguousarray(self.init, dtype=X.dtype)
if n_samples == 0:
return self
x_squared_norms = row_norms(X, squared=True)
self.random_state_ = getattr(self, "random_state_",
check_random_state(self.random_state))
if (not hasattr(self, 'counts_')
or not hasattr(self, 'cluster_centers_')):
# this is the first call partial_fit on this object:
# initialize the cluster centers
self.cluster_centers_ = _init_centroids(
X, self.n_clusters, self.init,
random_state=self.random_state_,
x_squared_norms=x_squared_norms, init_size=self.init_size)
self.counts_ = np.zeros(self.n_clusters, dtype=np.int32)
random_reassign = False
distances = None
else:
# The lower the minimum count is, the more we do random
# reassignment, however, we don't want to do random
# reassignment too often, to allow for building up counts
random_reassign = self.random_state_.randint(
10 * (1 + self.counts_.min())) == 0
distances = np.zeros(X.shape[0], dtype=X.dtype)
_mini_batch_step(X, x_squared_norms, self.cluster_centers_,
self.counts_, np.zeros(0, dtype=X.dtype), 0,
random_reassign=random_reassign, distances=distances,
random_state=self.random_state_,
reassignment_ratio=self.reassignment_ratio,
verbose=self.verbose)
if self.compute_labels:
self.labels_, self.inertia_ = _labels_inertia(
X, x_squared_norms, self.cluster_centers_)
return self
[docs] def predict(self, X):
"""Predict the closest cluster each sample in X belongs to.
In the vector quantization literature, `cluster_centers_` is called
the code book and each value returned by `predict` is the index of
the closest code in the code book.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
New data to predict.
Returns
-------
labels : array, shape [n_samples,]
Index of the cluster each sample belongs to.
"""
check_is_fitted(self, 'cluster_centers_')
X = self._check_test_data(X)
return self._labels_inertia_minibatch(X)[0]
```