796 lines
28 KiB
Python
796 lines
28 KiB
Python
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"""
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K-means clustering and vector quantization (:mod:`scipy.cluster.vq`)
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====================================================================
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Provides routines for k-means clustering, generating code books
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from k-means models and quantizing vectors by comparing them with
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centroids in a code book.
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.. autosummary::
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:toctree: generated/
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whiten -- Normalize a group of observations so each feature has unit variance
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vq -- Calculate code book membership of a set of observation vectors
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kmeans -- Perform k-means on a set of observation vectors forming k clusters
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kmeans2 -- A different implementation of k-means with more methods
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-- for initializing centroids
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Background information
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----------------------
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The k-means algorithm takes as input the number of clusters to
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generate, k, and a set of observation vectors to cluster. It
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returns a set of centroids, one for each of the k clusters. An
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observation vector is classified with the cluster number or
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centroid index of the centroid closest to it.
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A vector v belongs to cluster i if it is closer to centroid i than
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any other centroid. If v belongs to i, we say centroid i is the
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dominating centroid of v. The k-means algorithm tries to
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minimize distortion, which is defined as the sum of the squared distances
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between each observation vector and its dominating centroid.
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The minimization is achieved by iteratively reclassifying
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the observations into clusters and recalculating the centroids until
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a configuration is reached in which the centroids are stable. One can
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also define a maximum number of iterations.
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Since vector quantization is a natural application for k-means,
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information theory terminology is often used. The centroid index
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or cluster index is also referred to as a "code" and the table
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mapping codes to centroids and, vice versa, is often referred to as a
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"code book". The result of k-means, a set of centroids, can be
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used to quantize vectors. Quantization aims to find an encoding of
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vectors that reduces the expected distortion.
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All routines expect obs to be an M by N array, where the rows are
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the observation vectors. The codebook is a k by N array, where the
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ith row is the centroid of code word i. The observation vectors
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and centroids have the same feature dimension.
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As an example, suppose we wish to compress a 24-bit color image
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(each pixel is represented by one byte for red, one for blue, and
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one for green) before sending it over the web. By using a smaller
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8-bit encoding, we can reduce the amount of data by two
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thirds. Ideally, the colors for each of the 256 possible 8-bit
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encoding values should be chosen to minimize distortion of the
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color. Running k-means with k=256 generates a code book of 256
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codes, which fills up all possible 8-bit sequences. Instead of
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sending a 3-byte value for each pixel, the 8-bit centroid index
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(or code word) of the dominating centroid is transmitted. The code
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book is also sent over the wire so each 8-bit code can be
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translated back to a 24-bit pixel value representation. If the
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image of interest was of an ocean, we would expect many 24-bit
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blues to be represented by 8-bit codes. If it was an image of a
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human face, more flesh-tone colors would be represented in the
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code book.
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"""
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import warnings
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import numpy as np
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from collections import deque
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from scipy._lib._util import _asarray_validated, check_random_state,\
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rng_integers
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from scipy.spatial.distance import cdist
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from . import _vq
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__docformat__ = 'restructuredtext'
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__all__ = ['whiten', 'vq', 'kmeans', 'kmeans2']
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class ClusterError(Exception):
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pass
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def whiten(obs, check_finite=True):
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"""
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Normalize a group of observations on a per feature basis.
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Before running k-means, it is beneficial to rescale each feature
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dimension of the observation set by its standard deviation (i.e. "whiten"
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it - as in "white noise" where each frequency has equal power).
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Each feature is divided by its standard deviation across all observations
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to give it unit variance.
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Parameters
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----------
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obs : ndarray
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Each row of the array is an observation. The
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columns are the features seen during each observation.
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>>> # f0 f1 f2
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>>> obs = [[ 1., 1., 1.], #o0
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... [ 2., 2., 2.], #o1
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... [ 3., 3., 3.], #o2
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... [ 4., 4., 4.]] #o3
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Default: True
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Returns
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-------
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result : ndarray
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Contains the values in `obs` scaled by the standard deviation
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of each column.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.cluster.vq import whiten
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>>> features = np.array([[1.9, 2.3, 1.7],
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... [1.5, 2.5, 2.2],
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... [0.8, 0.6, 1.7,]])
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>>> whiten(features)
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array([[ 4.17944278, 2.69811351, 7.21248917],
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[ 3.29956009, 2.93273208, 9.33380951],
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[ 1.75976538, 0.7038557 , 7.21248917]])
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"""
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obs = _asarray_validated(obs, check_finite=check_finite)
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std_dev = obs.std(axis=0)
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zero_std_mask = std_dev == 0
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if zero_std_mask.any():
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std_dev[zero_std_mask] = 1.0
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warnings.warn("Some columns have standard deviation zero. "
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"The values of these columns will not change.",
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RuntimeWarning)
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return obs / std_dev
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def vq(obs, code_book, check_finite=True):
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"""
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Assign codes from a code book to observations.
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Assigns a code from a code book to each observation. Each
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observation vector in the 'M' by 'N' `obs` array is compared with the
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centroids in the code book and assigned the code of the closest
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centroid.
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The features in `obs` should have unit variance, which can be
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achieved by passing them through the whiten function. The code
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book can be created with the k-means algorithm or a different
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encoding algorithm.
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Parameters
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----------
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obs : ndarray
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Each row of the 'M' x 'N' array is an observation. The columns are
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the "features" seen during each observation. The features must be
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whitened first using the whiten function or something equivalent.
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code_book : ndarray
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The code book is usually generated using the k-means algorithm.
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Each row of the array holds a different code, and the columns are
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the features of the code.
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>>> # f0 f1 f2 f3
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>>> code_book = [
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... [ 1., 2., 3., 4.], #c0
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... [ 1., 2., 3., 4.], #c1
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... [ 1., 2., 3., 4.]] #c2
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Default: True
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Returns
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-------
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code : ndarray
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A length M array holding the code book index for each observation.
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dist : ndarray
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The distortion (distance) between the observation and its nearest
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code.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.cluster.vq import vq
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>>> code_book = np.array([[1.,1.,1.],
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... [2.,2.,2.]])
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>>> features = np.array([[ 1.9,2.3,1.7],
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... [ 1.5,2.5,2.2],
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... [ 0.8,0.6,1.7]])
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>>> vq(features,code_book)
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(array([1, 1, 0],'i'), array([ 0.43588989, 0.73484692, 0.83066239]))
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"""
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obs = _asarray_validated(obs, check_finite=check_finite)
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code_book = _asarray_validated(code_book, check_finite=check_finite)
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ct = np.common_type(obs, code_book)
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c_obs = obs.astype(ct, copy=False)
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c_code_book = code_book.astype(ct, copy=False)
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if np.issubdtype(ct, np.float64) or np.issubdtype(ct, np.float32):
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return _vq.vq(c_obs, c_code_book)
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return py_vq(obs, code_book, check_finite=False)
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def py_vq(obs, code_book, check_finite=True):
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""" Python version of vq algorithm.
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The algorithm computes the Euclidean distance between each
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observation and every frame in the code_book.
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Parameters
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----------
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obs : ndarray
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Expects a rank 2 array. Each row is one observation.
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code_book : ndarray
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Code book to use. Same format than obs. Should have same number of
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features (e.g., columns) than obs.
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Default: True
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Returns
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-------
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code : ndarray
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code[i] gives the label of the ith obversation; its code is
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code_book[code[i]].
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mind_dist : ndarray
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min_dist[i] gives the distance between the ith observation and its
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corresponding code.
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Notes
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-----
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This function is slower than the C version but works for
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all input types. If the inputs have the wrong types for the
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C versions of the function, this one is called as a last resort.
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It is about 20 times slower than the C version.
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"""
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obs = _asarray_validated(obs, check_finite=check_finite)
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code_book = _asarray_validated(code_book, check_finite=check_finite)
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if obs.ndim != code_book.ndim:
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raise ValueError("Observation and code_book should have the same rank")
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if obs.ndim == 1:
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obs = obs[:, np.newaxis]
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code_book = code_book[:, np.newaxis]
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dist = cdist(obs, code_book)
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code = dist.argmin(axis=1)
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min_dist = dist[np.arange(len(code)), code]
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return code, min_dist
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def _kmeans(obs, guess, thresh=1e-5):
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""" "raw" version of k-means.
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Returns
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-------
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code_book
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The lowest distortion codebook found.
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avg_dist
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The average distance a observation is from a code in the book.
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Lower means the code_book matches the data better.
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See Also
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--------
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kmeans : wrapper around k-means
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Examples
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--------
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Note: not whitened in this example.
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>>> import numpy as np
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>>> from scipy.cluster.vq import _kmeans
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>>> features = np.array([[ 1.9,2.3],
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... [ 1.5,2.5],
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... [ 0.8,0.6],
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... [ 0.4,1.8],
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... [ 1.0,1.0]])
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>>> book = np.array((features[0],features[2]))
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>>> _kmeans(features,book)
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(array([[ 1.7 , 2.4 ],
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[ 0.73333333, 1.13333333]]), 0.40563916697728591)
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"""
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code_book = np.asarray(guess)
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diff = np.inf
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prev_avg_dists = deque([diff], maxlen=2)
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while diff > thresh:
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# compute membership and distances between obs and code_book
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obs_code, distort = vq(obs, code_book, check_finite=False)
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prev_avg_dists.append(distort.mean(axis=-1))
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# recalc code_book as centroids of associated obs
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code_book, has_members = _vq.update_cluster_means(obs, obs_code,
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code_book.shape[0])
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code_book = code_book[has_members]
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diff = prev_avg_dists[0] - prev_avg_dists[1]
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return code_book, prev_avg_dists[1]
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def kmeans(obs, k_or_guess, iter=20, thresh=1e-5, check_finite=True,
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*, seed=None):
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"""
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Performs k-means on a set of observation vectors forming k clusters.
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The k-means algorithm adjusts the classification of the observations
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into clusters and updates the cluster centroids until the position of
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the centroids is stable over successive iterations. In this
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implementation of the algorithm, the stability of the centroids is
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determined by comparing the absolute value of the change in the average
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Euclidean distance between the observations and their corresponding
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centroids against a threshold. This yields
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a code book mapping centroids to codes and vice versa.
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Parameters
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----------
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obs : ndarray
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Each row of the M by N array is an observation vector. The
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columns are the features seen during each observation.
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The features must be whitened first with the `whiten` function.
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k_or_guess : int or ndarray
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The number of centroids to generate. A code is assigned to
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each centroid, which is also the row index of the centroid
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in the code_book matrix generated.
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The initial k centroids are chosen by randomly selecting
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observations from the observation matrix. Alternatively,
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passing a k by N array specifies the initial k centroids.
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iter : int, optional
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The number of times to run k-means, returning the codebook
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with the lowest distortion. This argument is ignored if
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initial centroids are specified with an array for the
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``k_or_guess`` parameter. This parameter does not represent the
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number of iterations of the k-means algorithm.
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thresh : float, optional
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Terminates the k-means algorithm if the change in
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distortion since the last k-means iteration is less than
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or equal to threshold.
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Default: True
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seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
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|
|
Seed for initializing the pseudo-random number generator.
|
||
|
|
If `seed` is None (or `numpy.random`), the `numpy.random.RandomState`
|
||
|
|
singleton is used.
|
||
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `seed`.
|
||
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
||
|
|
that instance is used.
|
||
|
|
The default is None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
codebook : ndarray
|
||
|
|
A k by N array of k centroids. The ith centroid
|
||
|
|
codebook[i] is represented with the code i. The centroids
|
||
|
|
and codes generated represent the lowest distortion seen,
|
||
|
|
not necessarily the globally minimal distortion.
|
||
|
|
Note that the number of centroids is not necessarily the same as the
|
||
|
|
``k_or_guess`` parameter, because centroids assigned to no observations
|
||
|
|
are removed during iterations.
|
||
|
|
|
||
|
|
distortion : float
|
||
|
|
The mean (non-squared) Euclidean distance between the observations
|
||
|
|
passed and the centroids generated. Note the difference to the standard
|
||
|
|
definition of distortion in the context of the k-means algorithm, which
|
||
|
|
is the sum of the squared distances.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
kmeans2 : a different implementation of k-means clustering
|
||
|
|
with more methods for generating initial centroids but without
|
||
|
|
using a distortion change threshold as a stopping criterion.
|
||
|
|
|
||
|
|
whiten : must be called prior to passing an observation matrix
|
||
|
|
to kmeans.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
For more functionalities or optimal performance, you can use
|
||
|
|
`sklearn.cluster.KMeans <https://scikit-learn.org/stable/modules/generated/sklearn.cluster.KMeans.html>`_.
|
||
|
|
`This <https://hdbscan.readthedocs.io/en/latest/performance_and_scalability.html#comparison-of-high-performance-implementations>`_
|
||
|
|
is a benchmark result of several implementations.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy.cluster.vq import vq, kmeans, whiten
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> features = np.array([[ 1.9,2.3],
|
||
|
|
... [ 1.5,2.5],
|
||
|
|
... [ 0.8,0.6],
|
||
|
|
... [ 0.4,1.8],
|
||
|
|
... [ 0.1,0.1],
|
||
|
|
... [ 0.2,1.8],
|
||
|
|
... [ 2.0,0.5],
|
||
|
|
... [ 0.3,1.5],
|
||
|
|
... [ 1.0,1.0]])
|
||
|
|
>>> whitened = whiten(features)
|
||
|
|
>>> book = np.array((whitened[0],whitened[2]))
|
||
|
|
>>> kmeans(whitened,book)
|
||
|
|
(array([[ 2.3110306 , 2.86287398], # random
|
||
|
|
[ 0.93218041, 1.24398691]]), 0.85684700941625547)
|
||
|
|
|
||
|
|
>>> codes = 3
|
||
|
|
>>> kmeans(whitened,codes)
|
||
|
|
(array([[ 2.3110306 , 2.86287398], # random
|
||
|
|
[ 1.32544402, 0.65607529],
|
||
|
|
[ 0.40782893, 2.02786907]]), 0.5196582527686241)
|
||
|
|
|
||
|
|
>>> # Create 50 datapoints in two clusters a and b
|
||
|
|
>>> pts = 50
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> a = rng.multivariate_normal([0, 0], [[4, 1], [1, 4]], size=pts)
|
||
|
|
>>> b = rng.multivariate_normal([30, 10],
|
||
|
|
... [[10, 2], [2, 1]],
|
||
|
|
... size=pts)
|
||
|
|
>>> features = np.concatenate((a, b))
|
||
|
|
>>> # Whiten data
|
||
|
|
>>> whitened = whiten(features)
|
||
|
|
>>> # Find 2 clusters in the data
|
||
|
|
>>> codebook, distortion = kmeans(whitened, 2)
|
||
|
|
>>> # Plot whitened data and cluster centers in red
|
||
|
|
>>> plt.scatter(whitened[:, 0], whitened[:, 1])
|
||
|
|
>>> plt.scatter(codebook[:, 0], codebook[:, 1], c='r')
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
obs = _asarray_validated(obs, check_finite=check_finite)
|
||
|
|
if iter < 1:
|
||
|
|
raise ValueError("iter must be at least 1, got %s" % iter)
|
||
|
|
|
||
|
|
# Determine whether a count (scalar) or an initial guess (array) was passed.
|
||
|
|
if not np.isscalar(k_or_guess):
|
||
|
|
guess = _asarray_validated(k_or_guess, check_finite=check_finite)
|
||
|
|
if guess.size < 1:
|
||
|
|
raise ValueError("Asked for 0 clusters. Initial book was %s" %
|
||
|
|
guess)
|
||
|
|
return _kmeans(obs, guess, thresh=thresh)
|
||
|
|
|
||
|
|
# k_or_guess is a scalar, now verify that it's an integer
|
||
|
|
k = int(k_or_guess)
|
||
|
|
if k != k_or_guess:
|
||
|
|
raise ValueError("If k_or_guess is a scalar, it must be an integer.")
|
||
|
|
if k < 1:
|
||
|
|
raise ValueError("Asked for %d clusters." % k)
|
||
|
|
|
||
|
|
rng = check_random_state(seed)
|
||
|
|
|
||
|
|
# initialize best distance value to a large value
|
||
|
|
best_dist = np.inf
|
||
|
|
for i in range(iter):
|
||
|
|
# the initial code book is randomly selected from observations
|
||
|
|
guess = _kpoints(obs, k, rng)
|
||
|
|
book, dist = _kmeans(obs, guess, thresh=thresh)
|
||
|
|
if dist < best_dist:
|
||
|
|
best_book = book
|
||
|
|
best_dist = dist
|
||
|
|
return best_book, best_dist
|
||
|
|
|
||
|
|
|
||
|
|
def _kpoints(data, k, rng):
|
||
|
|
"""Pick k points at random in data (one row = one observation).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : ndarray
|
||
|
|
Expect a rank 1 or 2 array. Rank 1 are assumed to describe one
|
||
|
|
dimensional data, rank 2 multidimensional data, in which case one
|
||
|
|
row is one observation.
|
||
|
|
k : int
|
||
|
|
Number of samples to generate.
|
||
|
|
rng : `numpy.random.Generator` or `numpy.random.RandomState`
|
||
|
|
Random number generator.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : ndarray
|
||
|
|
A 'k' by 'N' containing the initial centroids
|
||
|
|
|
||
|
|
"""
|
||
|
|
idx = rng.choice(data.shape[0], size=k, replace=False)
|
||
|
|
return data[idx]
|
||
|
|
|
||
|
|
|
||
|
|
def _krandinit(data, k, rng):
|
||
|
|
"""Returns k samples of a random variable whose parameters depend on data.
|
||
|
|
|
||
|
|
More precisely, it returns k observations sampled from a Gaussian random
|
||
|
|
variable whose mean and covariances are the ones estimated from the data.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : ndarray
|
||
|
|
Expect a rank 1 or 2 array. Rank 1 is assumed to describe 1-D
|
||
|
|
data, rank 2 multidimensional data, in which case one
|
||
|
|
row is one observation.
|
||
|
|
k : int
|
||
|
|
Number of samples to generate.
|
||
|
|
rng : `numpy.random.Generator` or `numpy.random.RandomState`
|
||
|
|
Random number generator.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : ndarray
|
||
|
|
A 'k' by 'N' containing the initial centroids
|
||
|
|
|
||
|
|
"""
|
||
|
|
mu = data.mean(axis=0)
|
||
|
|
|
||
|
|
if data.ndim == 1:
|
||
|
|
cov = np.cov(data)
|
||
|
|
x = rng.standard_normal(size=k)
|
||
|
|
x *= np.sqrt(cov)
|
||
|
|
elif data.shape[1] > data.shape[0]:
|
||
|
|
# initialize when the covariance matrix is rank deficient
|
||
|
|
_, s, vh = np.linalg.svd(data - mu, full_matrices=False)
|
||
|
|
x = rng.standard_normal(size=(k, s.size))
|
||
|
|
sVh = s[:, None] * vh / np.sqrt(data.shape[0] - 1)
|
||
|
|
x = x.dot(sVh)
|
||
|
|
else:
|
||
|
|
cov = np.atleast_2d(np.cov(data, rowvar=False))
|
||
|
|
|
||
|
|
# k rows, d cols (one row = one obs)
|
||
|
|
# Generate k sample of a random variable ~ Gaussian(mu, cov)
|
||
|
|
x = rng.standard_normal(size=(k, mu.size))
|
||
|
|
x = x.dot(np.linalg.cholesky(cov).T)
|
||
|
|
|
||
|
|
x += mu
|
||
|
|
return x
|
||
|
|
|
||
|
|
|
||
|
|
def _kpp(data, k, rng):
|
||
|
|
""" Picks k points in the data based on the kmeans++ method.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : ndarray
|
||
|
|
Expect a rank 1 or 2 array. Rank 1 is assumed to describe 1-D
|
||
|
|
data, rank 2 multidimensional data, in which case one
|
||
|
|
row is one observation.
|
||
|
|
k : int
|
||
|
|
Number of samples to generate.
|
||
|
|
rng : `numpy.random.Generator` or `numpy.random.RandomState`
|
||
|
|
Random number generator.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
init : ndarray
|
||
|
|
A 'k' by 'N' containing the initial centroids.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] D. Arthur and S. Vassilvitskii, "k-means++: the advantages of
|
||
|
|
careful seeding", Proceedings of the Eighteenth Annual ACM-SIAM Symposium
|
||
|
|
on Discrete Algorithms, 2007.
|
||
|
|
"""
|
||
|
|
|
||
|
|
dims = data.shape[1] if len(data.shape) > 1 else 1
|
||
|
|
init = np.ndarray((k, dims))
|
||
|
|
|
||
|
|
for i in range(k):
|
||
|
|
if i == 0:
|
||
|
|
init[i, :] = data[rng_integers(rng, data.shape[0])]
|
||
|
|
|
||
|
|
else:
|
||
|
|
D2 = cdist(init[:i,:], data, metric='sqeuclidean').min(axis=0)
|
||
|
|
probs = D2/D2.sum()
|
||
|
|
cumprobs = probs.cumsum()
|
||
|
|
r = rng.uniform()
|
||
|
|
init[i, :] = data[np.searchsorted(cumprobs, r)]
|
||
|
|
|
||
|
|
return init
|
||
|
|
|
||
|
|
|
||
|
|
_valid_init_meth = {'random': _krandinit, 'points': _kpoints, '++': _kpp}
|
||
|
|
|
||
|
|
|
||
|
|
def _missing_warn():
|
||
|
|
"""Print a warning when called."""
|
||
|
|
warnings.warn("One of the clusters is empty. "
|
||
|
|
"Re-run kmeans with a different initialization.")
|
||
|
|
|
||
|
|
|
||
|
|
def _missing_raise():
|
||
|
|
"""Raise a ClusterError when called."""
|
||
|
|
raise ClusterError("One of the clusters is empty. "
|
||
|
|
"Re-run kmeans with a different initialization.")
|
||
|
|
|
||
|
|
|
||
|
|
_valid_miss_meth = {'warn': _missing_warn, 'raise': _missing_raise}
|
||
|
|
|
||
|
|
|
||
|
|
def kmeans2(data, k, iter=10, thresh=1e-5, minit='random',
|
||
|
|
missing='warn', check_finite=True, *, seed=None):
|
||
|
|
"""
|
||
|
|
Classify a set of observations into k clusters using the k-means algorithm.
|
||
|
|
|
||
|
|
The algorithm attempts to minimize the Euclidean distance between
|
||
|
|
observations and centroids. Several initialization methods are
|
||
|
|
included.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : ndarray
|
||
|
|
A 'M' by 'N' array of 'M' observations in 'N' dimensions or a length
|
||
|
|
'M' array of 'M' 1-D observations.
|
||
|
|
k : int or ndarray
|
||
|
|
The number of clusters to form as well as the number of
|
||
|
|
centroids to generate. If `minit` initialization string is
|
||
|
|
'matrix', or if a ndarray is given instead, it is
|
||
|
|
interpreted as initial cluster to use instead.
|
||
|
|
iter : int, optional
|
||
|
|
Number of iterations of the k-means algorithm to run. Note
|
||
|
|
that this differs in meaning from the iters parameter to
|
||
|
|
the kmeans function.
|
||
|
|
thresh : float, optional
|
||
|
|
(not used yet)
|
||
|
|
minit : str, optional
|
||
|
|
Method for initialization. Available methods are 'random',
|
||
|
|
'points', '++' and 'matrix':
|
||
|
|
|
||
|
|
'random': generate k centroids from a Gaussian with mean and
|
||
|
|
variance estimated from the data.
|
||
|
|
|
||
|
|
'points': choose k observations (rows) at random from data for
|
||
|
|
the initial centroids.
|
||
|
|
|
||
|
|
'++': choose k observations accordingly to the kmeans++ method
|
||
|
|
(careful seeding)
|
||
|
|
|
||
|
|
'matrix': interpret the k parameter as a k by M (or length k
|
||
|
|
array for 1-D data) array of initial centroids.
|
||
|
|
missing : str, optional
|
||
|
|
Method to deal with empty clusters. Available methods are
|
||
|
|
'warn' and 'raise':
|
||
|
|
|
||
|
|
'warn': give a warning and continue.
|
||
|
|
|
||
|
|
'raise': raise an ClusterError and terminate the algorithm.
|
||
|
|
check_finite : bool, optional
|
||
|
|
Whether to check that the input matrices contain only finite numbers.
|
||
|
|
Disabling may give a performance gain, but may result in problems
|
||
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
||
|
|
Default: True
|
||
|
|
seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
|
||
|
|
Seed for initializing the pseudo-random number generator.
|
||
|
|
If `seed` is None (or `numpy.random`), the `numpy.random.RandomState`
|
||
|
|
singleton is used.
|
||
|
|
If `seed` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `seed`.
|
||
|
|
If `seed` is already a ``Generator`` or ``RandomState`` instance then
|
||
|
|
that instance is used.
|
||
|
|
The default is None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
centroid : ndarray
|
||
|
|
A 'k' by 'N' array of centroids found at the last iteration of
|
||
|
|
k-means.
|
||
|
|
label : ndarray
|
||
|
|
label[i] is the code or index of the centroid the
|
||
|
|
ith observation is closest to.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
kmeans
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] D. Arthur and S. Vassilvitskii, "k-means++: the advantages of
|
||
|
|
careful seeding", Proceedings of the Eighteenth Annual ACM-SIAM Symposium
|
||
|
|
on Discrete Algorithms, 2007.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.cluster.vq import kmeans2
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> import numpy as np
|
||
|
|
|
||
|
|
Create z, an array with shape (100, 2) containing a mixture of samples
|
||
|
|
from three multivariate normal distributions.
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> a = rng.multivariate_normal([0, 6], [[2, 1], [1, 1.5]], size=45)
|
||
|
|
>>> b = rng.multivariate_normal([2, 0], [[1, -1], [-1, 3]], size=30)
|
||
|
|
>>> c = rng.multivariate_normal([6, 4], [[5, 0], [0, 1.2]], size=25)
|
||
|
|
>>> z = np.concatenate((a, b, c))
|
||
|
|
>>> rng.shuffle(z)
|
||
|
|
|
||
|
|
Compute three clusters.
|
||
|
|
|
||
|
|
>>> centroid, label = kmeans2(z, 3, minit='points')
|
||
|
|
>>> centroid
|
||
|
|
array([[ 2.22274463, -0.61666946], # may vary
|
||
|
|
[ 0.54069047, 5.86541444],
|
||
|
|
[ 6.73846769, 4.01991898]])
|
||
|
|
|
||
|
|
How many points are in each cluster?
|
||
|
|
|
||
|
|
>>> counts = np.bincount(label)
|
||
|
|
>>> counts
|
||
|
|
array([29, 51, 20]) # may vary
|
||
|
|
|
||
|
|
Plot the clusters.
|
||
|
|
|
||
|
|
>>> w0 = z[label == 0]
|
||
|
|
>>> w1 = z[label == 1]
|
||
|
|
>>> w2 = z[label == 2]
|
||
|
|
>>> plt.plot(w0[:, 0], w0[:, 1], 'o', alpha=0.5, label='cluster 0')
|
||
|
|
>>> plt.plot(w1[:, 0], w1[:, 1], 'd', alpha=0.5, label='cluster 1')
|
||
|
|
>>> plt.plot(w2[:, 0], w2[:, 1], 's', alpha=0.5, label='cluster 2')
|
||
|
|
>>> plt.plot(centroid[:, 0], centroid[:, 1], 'k*', label='centroids')
|
||
|
|
>>> plt.axis('equal')
|
||
|
|
>>> plt.legend(shadow=True)
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
"""
|
||
|
|
if int(iter) < 1:
|
||
|
|
raise ValueError("Invalid iter (%s), "
|
||
|
|
"must be a positive integer." % iter)
|
||
|
|
try:
|
||
|
|
miss_meth = _valid_miss_meth[missing]
|
||
|
|
except KeyError as e:
|
||
|
|
raise ValueError("Unknown missing method %r" % (missing,)) from e
|
||
|
|
|
||
|
|
data = _asarray_validated(data, check_finite=check_finite)
|
||
|
|
if data.ndim == 1:
|
||
|
|
d = 1
|
||
|
|
elif data.ndim == 2:
|
||
|
|
d = data.shape[1]
|
||
|
|
else:
|
||
|
|
raise ValueError("Input of rank > 2 is not supported.")
|
||
|
|
|
||
|
|
if data.size < 1:
|
||
|
|
raise ValueError("Empty input is not supported.")
|
||
|
|
|
||
|
|
# If k is not a single value, it should be compatible with data's shape
|
||
|
|
if minit == 'matrix' or not np.isscalar(k):
|
||
|
|
code_book = np.array(k, copy=True)
|
||
|
|
if data.ndim != code_book.ndim:
|
||
|
|
raise ValueError("k array doesn't match data rank")
|
||
|
|
nc = len(code_book)
|
||
|
|
if data.ndim > 1 and code_book.shape[1] != d:
|
||
|
|
raise ValueError("k array doesn't match data dimension")
|
||
|
|
else:
|
||
|
|
nc = int(k)
|
||
|
|
|
||
|
|
if nc < 1:
|
||
|
|
raise ValueError("Cannot ask kmeans2 for %d clusters"
|
||
|
|
" (k was %s)" % (nc, k))
|
||
|
|
elif nc != k:
|
||
|
|
warnings.warn("k was not an integer, was converted.")
|
||
|
|
|
||
|
|
try:
|
||
|
|
init_meth = _valid_init_meth[minit]
|
||
|
|
except KeyError as e:
|
||
|
|
raise ValueError("Unknown init method %r" % (minit,)) from e
|
||
|
|
else:
|
||
|
|
rng = check_random_state(seed)
|
||
|
|
code_book = init_meth(data, k, rng)
|
||
|
|
|
||
|
|
for i in range(iter):
|
||
|
|
# Compute the nearest neighbor for each obs using the current code book
|
||
|
|
label = vq(data, code_book)[0]
|
||
|
|
# Update the code book by computing centroids
|
||
|
|
new_code_book, has_members = _vq.update_cluster_means(data, label, nc)
|
||
|
|
if not has_members.all():
|
||
|
|
miss_meth()
|
||
|
|
# Set the empty clusters to their previous positions
|
||
|
|
new_code_book[~has_members] = code_book[~has_members]
|
||
|
|
code_book = new_code_book
|
||
|
|
|
||
|
|
return code_book, label
|