431 lines
16 KiB
Python
431 lines
16 KiB
Python
"""
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Multi-dimensional Scaling (MDS)
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"""
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# author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
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# License: BSD
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import numpy as np
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import warnings
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from ..base import BaseEstimator
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from ..metrics import euclidean_distances
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from ..utils import check_random_state, check_array, check_symmetric
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from ..externals.joblib import Parallel
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from ..externals.joblib import delayed
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from ..isotonic import IsotonicRegression
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def _smacof_single(dissimilarities, metric=True, n_components=2, init=None,
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max_iter=300, verbose=0, eps=1e-3, random_state=None):
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"""Computes multidimensional scaling using SMACOF algorithm
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Parameters
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----------
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dissimilarities : ndarray, shape (n_samples, n_samples)
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Pairwise dissimilarities between the points. Must be symmetric.
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metric : boolean, optional, default: True
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Compute metric or nonmetric SMACOF algorithm.
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n_components : int, optional, default: 2
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Number of dimensions in which to immerse the dissimilarities. If an
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``init`` array is provided, this option is overridden and the shape of
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``init`` is used to determine the dimensionality of the embedding
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space.
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init : ndarray, shape (n_samples, n_components), optional, default: None
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Starting configuration of the embedding to initialize the algorithm. By
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default, the algorithm is initialized with a randomly chosen array.
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max_iter : int, optional, default: 300
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Maximum number of iterations of the SMACOF algorithm for a single run.
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verbose : int, optional, default: 0
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Level of verbosity.
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eps : float, optional, default: 1e-3
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Relative tolerance with respect to stress at which to declare
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convergence.
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random_state : int, RandomState instance or None, optional, default: None
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The generator used to initialize the centers. If int, random_state is
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the seed used by the random number generator; If RandomState instance,
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random_state is the random number generator; If None, the random number
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generator is the RandomState instance used by `np.random`.
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Returns
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-------
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X : ndarray, shape (n_samples, n_components)
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Coordinates of the points in a ``n_components``-space.
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stress : float
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The final value of the stress (sum of squared distance of the
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disparities and the distances for all constrained points).
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n_iter : int
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The number of iterations corresponding to the best stress.
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"""
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dissimilarities = check_symmetric(dissimilarities, raise_exception=True)
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n_samples = dissimilarities.shape[0]
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random_state = check_random_state(random_state)
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sim_flat = ((1 - np.tri(n_samples)) * dissimilarities).ravel()
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sim_flat_w = sim_flat[sim_flat != 0]
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if init is None:
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# Randomly choose initial configuration
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X = random_state.rand(n_samples * n_components)
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X = X.reshape((n_samples, n_components))
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else:
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# overrides the parameter p
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n_components = init.shape[1]
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if n_samples != init.shape[0]:
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raise ValueError("init matrix should be of shape (%d, %d)" %
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(n_samples, n_components))
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X = init
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old_stress = None
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ir = IsotonicRegression()
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for it in range(max_iter):
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# Compute distance and monotonic regression
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dis = euclidean_distances(X)
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if metric:
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disparities = dissimilarities
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else:
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dis_flat = dis.ravel()
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# dissimilarities with 0 are considered as missing values
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dis_flat_w = dis_flat[sim_flat != 0]
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# Compute the disparities using a monotonic regression
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disparities_flat = ir.fit_transform(sim_flat_w, dis_flat_w)
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disparities = dis_flat.copy()
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disparities[sim_flat != 0] = disparities_flat
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disparities = disparities.reshape((n_samples, n_samples))
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disparities *= np.sqrt((n_samples * (n_samples - 1) / 2) /
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(disparities ** 2).sum())
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# Compute stress
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stress = ((dis.ravel() - disparities.ravel()) ** 2).sum() / 2
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# Update X using the Guttman transform
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dis[dis == 0] = 1e-5
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ratio = disparities / dis
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B = - ratio
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B[np.arange(len(B)), np.arange(len(B))] += ratio.sum(axis=1)
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X = 1. / n_samples * np.dot(B, X)
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dis = np.sqrt((X ** 2).sum(axis=1)).sum()
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if verbose >= 2:
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print('it: %d, stress %s' % (it, stress))
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if old_stress is not None:
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if(old_stress - stress / dis) < eps:
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if verbose:
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print('breaking at iteration %d with stress %s' % (it,
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stress))
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break
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old_stress = stress / dis
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return X, stress, it + 1
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def smacof(dissimilarities, metric=True, n_components=2, init=None, n_init=8,
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n_jobs=1, max_iter=300, verbose=0, eps=1e-3, random_state=None,
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return_n_iter=False):
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"""Computes multidimensional scaling using the SMACOF algorithm.
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The SMACOF (Scaling by MAjorizing a COmplicated Function) algorithm is a
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multidimensional scaling algorithm which minimizes an objective function
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(the *stress*) using a majorization technique. Stress majorization, also
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known as the Guttman Transform, guarantees a monotone convergence of
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stress, and is more powerful than traditional techniques such as gradient
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descent.
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The SMACOF algorithm for metric MDS can summarized by the following steps:
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1. Set an initial start configuration, randomly or not.
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2. Compute the stress
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3. Compute the Guttman Transform
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4. Iterate 2 and 3 until convergence.
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The nonmetric algorithm adds a monotonic regression step before computing
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the stress.
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Parameters
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----------
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dissimilarities : ndarray, shape (n_samples, n_samples)
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Pairwise dissimilarities between the points. Must be symmetric.
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metric : boolean, optional, default: True
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Compute metric or nonmetric SMACOF algorithm.
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n_components : int, optional, default: 2
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Number of dimensions in which to immerse the dissimilarities. If an
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``init`` array is provided, this option is overridden and the shape of
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``init`` is used to determine the dimensionality of the embedding
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space.
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init : ndarray, shape (n_samples, n_components), optional, default: None
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Starting configuration of the embedding to initialize the algorithm. By
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default, the algorithm is initialized with a randomly chosen array.
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n_init : int, optional, default: 8
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Number of times the SMACOF algorithm will be run with different
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initializations. The final results will be the best output of the runs,
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determined by the run with the smallest final stress. If ``init`` is
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provided, this option is overridden and a single run is performed.
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n_jobs : int, optional, default: 1
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The number of jobs to use for the computation. If multiple
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initializations are used (``n_init``), each run of the algorithm is
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computed in parallel.
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If -1 all CPUs are used. If 1 is given, no parallel computing code is
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used at all, which is useful for debugging. For ``n_jobs`` below -1,
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(``n_cpus + 1 + n_jobs``) are used. Thus for ``n_jobs = -2``, all CPUs
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but one are used.
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max_iter : int, optional, default: 300
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Maximum number of iterations of the SMACOF algorithm for a single run.
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verbose : int, optional, default: 0
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Level of verbosity.
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eps : float, optional, default: 1e-3
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Relative tolerance with respect to stress at which to declare
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convergence.
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random_state : int, RandomState instance or None, optional, default: None
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The generator used to initialize the centers. If int, random_state is
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the seed used by the random number generator; If RandomState instance,
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random_state is the random number generator; If None, the random number
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generator is the RandomState instance used by `np.random`.
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return_n_iter : bool, optional, default: False
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Whether or not to return the number of iterations.
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Returns
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-------
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X : ndarray, shape (n_samples, n_components)
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Coordinates of the points in a ``n_components``-space.
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stress : float
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The final value of the stress (sum of squared distance of the
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disparities and the distances for all constrained points).
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n_iter : int
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The number of iterations corresponding to the best stress. Returned
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only if ``return_n_iter`` is set to ``True``.
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Notes
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-----
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"Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
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Groenen P. Springer Series in Statistics (1997)
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"Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
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Psychometrika, 29 (1964)
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"Multidimensional scaling by optimizing goodness of fit to a nonmetric
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hypothesis" Kruskal, J. Psychometrika, 29, (1964)
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"""
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dissimilarities = check_array(dissimilarities)
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random_state = check_random_state(random_state)
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if hasattr(init, '__array__'):
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init = np.asarray(init).copy()
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if not n_init == 1:
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warnings.warn(
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'Explicit initial positions passed: '
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'performing only one init of the MDS instead of %d'
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% n_init)
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n_init = 1
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best_pos, best_stress = None, None
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if n_jobs == 1:
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for it in range(n_init):
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pos, stress, n_iter_ = _smacof_single(
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dissimilarities, metric=metric,
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n_components=n_components, init=init,
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max_iter=max_iter, verbose=verbose,
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eps=eps, random_state=random_state)
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if best_stress is None or stress < best_stress:
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best_stress = stress
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best_pos = pos.copy()
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best_iter = n_iter_
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else:
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seeds = random_state.randint(np.iinfo(np.int32).max, size=n_init)
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results = Parallel(n_jobs=n_jobs, verbose=max(verbose - 1, 0))(
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delayed(_smacof_single)(
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dissimilarities, metric=metric, n_components=n_components,
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init=init, max_iter=max_iter, verbose=verbose, eps=eps,
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random_state=seed)
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for seed in seeds)
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positions, stress, n_iters = zip(*results)
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best = np.argmin(stress)
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best_stress = stress[best]
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best_pos = positions[best]
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best_iter = n_iters[best]
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if return_n_iter:
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return best_pos, best_stress, best_iter
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else:
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return best_pos, best_stress
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class MDS(BaseEstimator):
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"""Multidimensional scaling
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Read more in the :ref:`User Guide <multidimensional_scaling>`.
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Parameters
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----------
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n_components : int, optional, default: 2
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Number of dimensions in which to immerse the dissimilarities.
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metric : boolean, optional, default: True
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If ``True``, perform metric MDS; otherwise, perform nonmetric MDS.
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n_init : int, optional, default: 4
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Number of times the SMACOF algorithm will be run with different
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initializations. The final results will be the best output of the runs,
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determined by the run with the smallest final stress.
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max_iter : int, optional, default: 300
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Maximum number of iterations of the SMACOF algorithm for a single run.
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verbose : int, optional, default: 0
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Level of verbosity.
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eps : float, optional, default: 1e-3
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Relative tolerance with respect to stress at which to declare
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convergence.
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n_jobs : int, optional, default: 1
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The number of jobs to use for the computation. If multiple
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initializations are used (``n_init``), each run of the algorithm is
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computed in parallel.
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If -1 all CPUs are used. If 1 is given, no parallel computing code is
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used at all, which is useful for debugging. For ``n_jobs`` below -1,
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(``n_cpus + 1 + n_jobs``) are used. Thus for ``n_jobs = -2``, all CPUs
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but one are used.
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random_state : int, RandomState instance or None, optional, default: None
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The generator used to initialize the centers. If int, random_state is
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the seed used by the random number generator; If RandomState instance,
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random_state is the random number generator; If None, the random number
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generator is the RandomState instance used by `np.random`.
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dissimilarity : 'euclidean' | 'precomputed', optional, default: 'euclidean'
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Dissimilarity measure to use:
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- 'euclidean':
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Pairwise Euclidean distances between points in the dataset.
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- 'precomputed':
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Pre-computed dissimilarities are passed directly to ``fit`` and
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``fit_transform``.
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Attributes
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----------
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embedding_ : array-like, shape (n_components, n_samples)
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Stores the position of the dataset in the embedding space.
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stress_ : float
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The final value of the stress (sum of squared distance of the
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disparities and the distances for all constrained points).
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References
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----------
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"Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
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Groenen P. Springer Series in Statistics (1997)
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"Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
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Psychometrika, 29 (1964)
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"Multidimensional scaling by optimizing goodness of fit to a nonmetric
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hypothesis" Kruskal, J. Psychometrika, 29, (1964)
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"""
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def __init__(self, n_components=2, metric=True, n_init=4,
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max_iter=300, verbose=0, eps=1e-3, n_jobs=1,
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random_state=None, dissimilarity="euclidean"):
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self.n_components = n_components
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self.dissimilarity = dissimilarity
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self.metric = metric
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self.n_init = n_init
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self.max_iter = max_iter
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self.eps = eps
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self.verbose = verbose
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self.n_jobs = n_jobs
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self.random_state = random_state
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@property
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def _pairwise(self):
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return self.kernel == "precomputed"
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def fit(self, X, y=None, init=None):
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"""
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Computes the position of the points in the embedding space
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Parameters
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----------
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X : array, shape (n_samples, n_features) or (n_samples, n_samples)
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Input data. If ``dissimilarity=='precomputed'``, the input should
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be the dissimilarity matrix.
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y: Ignored.
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init : ndarray, shape (n_samples,), optional, default: None
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Starting configuration of the embedding to initialize the SMACOF
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algorithm. By default, the algorithm is initialized with a randomly
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chosen array.
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"""
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self.fit_transform(X, init=init)
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return self
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def fit_transform(self, X, y=None, init=None):
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"""
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Fit the data from X, and returns the embedded coordinates
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Parameters
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----------
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X : array, shape (n_samples, n_features) or (n_samples, n_samples)
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Input data. If ``dissimilarity=='precomputed'``, the input should
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be the dissimilarity matrix.
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y: Ignored.
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init : ndarray, shape (n_samples,), optional, default: None
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Starting configuration of the embedding to initialize the SMACOF
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algorithm. By default, the algorithm is initialized with a randomly
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chosen array.
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"""
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X = check_array(X)
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if X.shape[0] == X.shape[1] and self.dissimilarity != "precomputed":
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warnings.warn("The MDS API has changed. ``fit`` now constructs an"
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" dissimilarity matrix from data. To use a custom "
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"dissimilarity matrix, set "
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"``dissimilarity='precomputed'``.")
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if self.dissimilarity == "precomputed":
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self.dissimilarity_matrix_ = X
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elif self.dissimilarity == "euclidean":
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self.dissimilarity_matrix_ = euclidean_distances(X)
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else:
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raise ValueError("Proximity must be 'precomputed' or 'euclidean'."
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" Got %s instead" % str(self.dissimilarity))
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self.embedding_, self.stress_, self.n_iter_ = smacof(
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self.dissimilarity_matrix_, metric=self.metric,
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n_components=self.n_components, init=init, n_init=self.n_init,
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n_jobs=self.n_jobs, max_iter=self.max_iter, verbose=self.verbose,
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eps=self.eps, random_state=self.random_state,
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return_n_iter=True)
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return self.embedding_
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