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656
venv/lib/python3.11/site-packages/sklearn/manifold/_mds.py
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venv/lib/python3.11/site-packages/sklearn/manifold/_mds.py
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"""
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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 warnings
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from numbers import Integral, Real
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import numpy as np
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from joblib import effective_n_jobs
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from ..base import BaseEstimator, _fit_context
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from ..isotonic import IsotonicRegression
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from ..metrics import euclidean_distances
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from ..utils import check_array, check_random_state, check_symmetric
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from ..utils._param_validation import Interval, StrOptions, validate_params
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from ..utils.parallel import Parallel, delayed
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def _smacof_single(
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dissimilarities,
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metric=True,
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n_components=2,
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init=None,
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max_iter=300,
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verbose=0,
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eps=1e-3,
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random_state=None,
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normalized_stress=False,
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):
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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 of shape (n_samples, n_samples)
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Pairwise dissimilarities between the points. Must be symmetric.
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metric : bool, default=True
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Compute metric or nonmetric SMACOF algorithm.
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When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as
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missing values.
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n_components : int, 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 of shape (n_samples, n_components), 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, default=300
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Maximum number of iterations of the SMACOF algorithm for a single run.
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verbose : int, default=0
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Level of verbosity.
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eps : float, default=1e-3
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Relative tolerance with respect to stress at which to declare
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convergence. The value of `eps` should be tuned separately depending
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on whether or not `normalized_stress` is being used.
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random_state : int, RandomState instance or None, default=None
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Determines the random number generator used to initialize the centers.
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Pass an int for reproducible results across multiple function calls.
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See :term:`Glossary <random_state>`.
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normalized_stress : bool, default=False
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Whether use and return normed stress value (Stress-1) instead of raw
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stress calculated by default. Only supported in non-metric MDS. The
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caller must ensure that if `normalized_stress=True` then `metric=False`
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.. versionadded:: 1.2
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Returns
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-------
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X : ndarray of 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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If `normalized_stress=True`, and `metric=False` returns Stress-1.
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A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good,
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0.1 fair, and 0.2 poor [1]_.
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n_iter : int
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The number of iterations corresponding to the best stress.
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References
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----------
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.. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
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Psychometrika, 29 (1964)
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.. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric
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hypothesis" Kruskal, J. Psychometrika, 29, (1964)
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.. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
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Groenen P. Springer Series in Statistics (1997)
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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.uniform(size=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(
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"init matrix should be of shape (%d, %d)" % (n_samples, n_components)
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)
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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(
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(n_samples * (n_samples - 1) / 2) / (disparities**2).sum()
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)
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# Compute stress
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stress = ((dis.ravel() - disparities.ravel()) ** 2).sum() / 2
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if normalized_stress:
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stress = np.sqrt(stress / ((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 = 1.0 / 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, 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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@validate_params(
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{
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"dissimilarities": ["array-like"],
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"metric": ["boolean"],
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"n_components": [Interval(Integral, 1, None, closed="left")],
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"init": ["array-like", None],
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"n_init": [Interval(Integral, 1, None, closed="left")],
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"n_jobs": [Integral, None],
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"max_iter": [Interval(Integral, 1, None, closed="left")],
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"verbose": ["verbose"],
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"eps": [Interval(Real, 0, None, closed="left")],
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"random_state": ["random_state"],
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"return_n_iter": ["boolean"],
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"normalized_stress": ["boolean", StrOptions({"auto"})],
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},
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prefer_skip_nested_validation=True,
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)
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def smacof(
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dissimilarities,
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*,
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metric=True,
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n_components=2,
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init=None,
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n_init=8,
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n_jobs=None,
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max_iter=300,
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verbose=0,
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eps=1e-3,
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random_state=None,
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return_n_iter=False,
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normalized_stress="auto",
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):
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"""Compute 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 be summarized by the following
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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 : array-like of shape (n_samples, n_samples)
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Pairwise dissimilarities between the points. Must be symmetric.
|
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|
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metric : bool, default=True
|
||||
Compute metric or nonmetric SMACOF algorithm.
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When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as
|
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missing values.
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|
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n_components : int, 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
|
||||
``init`` is used to determine the dimensionality of the embedding
|
||||
space.
|
||||
|
||||
init : array-like of shape (n_samples, n_components), default=None
|
||||
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, default=8
|
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Number of times the SMACOF algorithm will be run with different
|
||||
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
|
||||
provided, this option is overridden and a single run is performed.
|
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|
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n_jobs : int, default=None
|
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The number of jobs to use for the computation. If multiple
|
||||
initializations are used (``n_init``), each run of the algorithm is
|
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computed in parallel.
|
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|
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``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
|
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``-1`` means using all processors. See :term:`Glossary <n_jobs>`
|
||||
for more details.
|
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|
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max_iter : int, default=300
|
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Maximum number of iterations of the SMACOF algorithm for a single run.
|
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|
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verbose : int, default=0
|
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Level of verbosity.
|
||||
|
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eps : float, default=1e-3
|
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Relative tolerance with respect to stress at which to declare
|
||||
convergence. The value of `eps` should be tuned separately depending
|
||||
on whether or not `normalized_stress` is being used.
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|
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random_state : int, RandomState instance or None, default=None
|
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Determines the random number generator used to initialize the centers.
|
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Pass an int for reproducible results across multiple function calls.
|
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See :term:`Glossary <random_state>`.
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return_n_iter : bool, default=False
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Whether or not to return the number of iterations.
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normalized_stress : bool or "auto" default="auto"
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Whether use and return normed stress value (Stress-1) instead of raw
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stress calculated by default. Only supported in non-metric MDS.
|
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|
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.. versionadded:: 1.2
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.. versionchanged:: 1.4
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The default value changed from `False` to `"auto"` in version 1.4.
|
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|
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Returns
|
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-------
|
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X : ndarray of 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
|
||||
disparities and the distances for all constrained points).
|
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If `normalized_stress=True`, and `metric=False` returns Stress-1.
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A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good,
|
||||
0.1 fair, and 0.2 poor [1]_.
|
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|
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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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|
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References
|
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----------
|
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.. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
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Psychometrika, 29 (1964)
|
||||
|
||||
.. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric
|
||||
hypothesis" Kruskal, J. Psychometrika, 29, (1964)
|
||||
|
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.. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
|
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Groenen P. Springer Series in Statistics (1997)
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Examples
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--------
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>>> import numpy as np
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>>> from sklearn.manifold import smacof
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>>> from sklearn.metrics import euclidean_distances
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>>> X = np.array([[0, 1, 2], [1, 0, 3],[2, 3, 0]])
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>>> dissimilarities = euclidean_distances(X)
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>>> mds_result, stress = smacof(dissimilarities, n_components=2, random_state=42)
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>>> mds_result
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array([[ 0.05... -1.07... ],
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[ 1.74..., -0.75...],
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[-1.79..., 1.83...]])
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>>> stress
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np.float64(0.0012...)
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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 normalized_stress == "auto":
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normalized_stress = not metric
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|
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if normalized_stress and metric:
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raise ValueError(
|
||||
"Normalized stress is not supported for metric MDS. Either set"
|
||||
" `normalized_stress=False` or use `metric=False`."
|
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)
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if hasattr(init, "__array__"):
|
||||
init = np.asarray(init).copy()
|
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if not n_init == 1:
|
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warnings.warn(
|
||||
"Explicit initial positions passed: "
|
||||
"performing only one init of the MDS instead of %d" % n_init
|
||||
)
|
||||
n_init = 1
|
||||
|
||||
best_pos, best_stress = None, None
|
||||
|
||||
if effective_n_jobs(n_jobs) == 1:
|
||||
for it in range(n_init):
|
||||
pos, stress, n_iter_ = _smacof_single(
|
||||
dissimilarities,
|
||||
metric=metric,
|
||||
n_components=n_components,
|
||||
init=init,
|
||||
max_iter=max_iter,
|
||||
verbose=verbose,
|
||||
eps=eps,
|
||||
random_state=random_state,
|
||||
normalized_stress=normalized_stress,
|
||||
)
|
||||
if best_stress is None or stress < best_stress:
|
||||
best_stress = stress
|
||||
best_pos = pos.copy()
|
||||
best_iter = n_iter_
|
||||
else:
|
||||
seeds = random_state.randint(np.iinfo(np.int32).max, size=n_init)
|
||||
results = Parallel(n_jobs=n_jobs, verbose=max(verbose - 1, 0))(
|
||||
delayed(_smacof_single)(
|
||||
dissimilarities,
|
||||
metric=metric,
|
||||
n_components=n_components,
|
||||
init=init,
|
||||
max_iter=max_iter,
|
||||
verbose=verbose,
|
||||
eps=eps,
|
||||
random_state=seed,
|
||||
normalized_stress=normalized_stress,
|
||||
)
|
||||
for seed in seeds
|
||||
)
|
||||
positions, stress, n_iters = zip(*results)
|
||||
best = np.argmin(stress)
|
||||
best_stress = stress[best]
|
||||
best_pos = positions[best]
|
||||
best_iter = n_iters[best]
|
||||
|
||||
if return_n_iter:
|
||||
return best_pos, best_stress, best_iter
|
||||
else:
|
||||
return best_pos, best_stress
|
||||
|
||||
|
||||
class MDS(BaseEstimator):
|
||||
"""Multidimensional scaling.
|
||||
|
||||
Read more in the :ref:`User Guide <multidimensional_scaling>`.
|
||||
|
||||
Parameters
|
||||
----------
|
||||
n_components : int, default=2
|
||||
Number of dimensions in which to immerse the dissimilarities.
|
||||
|
||||
metric : bool, default=True
|
||||
If ``True``, perform metric MDS; otherwise, perform nonmetric MDS.
|
||||
When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as
|
||||
missing values.
|
||||
|
||||
n_init : int, default=4
|
||||
Number of times the SMACOF algorithm will be run with different
|
||||
initializations. The final results will be the best output of the runs,
|
||||
determined by the run with the smallest final stress.
|
||||
|
||||
max_iter : int, default=300
|
||||
Maximum number of iterations of the SMACOF algorithm for a single run.
|
||||
|
||||
verbose : int, default=0
|
||||
Level of verbosity.
|
||||
|
||||
eps : float, default=1e-3
|
||||
Relative tolerance with respect to stress at which to declare
|
||||
convergence. The value of `eps` should be tuned separately depending
|
||||
on whether or not `normalized_stress` is being used.
|
||||
|
||||
n_jobs : int, default=None
|
||||
The number of jobs to use for the computation. If multiple
|
||||
initializations are used (``n_init``), each run of the algorithm is
|
||||
computed in parallel.
|
||||
|
||||
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
|
||||
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
|
||||
for more details.
|
||||
|
||||
random_state : int, RandomState instance or None, default=None
|
||||
Determines the random number generator used to initialize the centers.
|
||||
Pass an int for reproducible results across multiple function calls.
|
||||
See :term:`Glossary <random_state>`.
|
||||
|
||||
dissimilarity : {'euclidean', 'precomputed'}, default='euclidean'
|
||||
Dissimilarity measure to use:
|
||||
|
||||
- 'euclidean':
|
||||
Pairwise Euclidean distances between points in the dataset.
|
||||
|
||||
- 'precomputed':
|
||||
Pre-computed dissimilarities are passed directly to ``fit`` and
|
||||
``fit_transform``.
|
||||
|
||||
normalized_stress : bool or "auto" default="auto"
|
||||
Whether use and return normed stress value (Stress-1) instead of raw
|
||||
stress calculated by default. Only supported in non-metric MDS.
|
||||
|
||||
.. versionadded:: 1.2
|
||||
|
||||
.. versionchanged:: 1.4
|
||||
The default value changed from `False` to `"auto"` in version 1.4.
|
||||
|
||||
Attributes
|
||||
----------
|
||||
embedding_ : ndarray of shape (n_samples, n_components)
|
||||
Stores the position of the dataset in the embedding space.
|
||||
|
||||
stress_ : float
|
||||
The final value of the stress (sum of squared distance of the
|
||||
disparities and the distances for all constrained points).
|
||||
If `normalized_stress=True`, and `metric=False` returns Stress-1.
|
||||
A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good,
|
||||
0.1 fair, and 0.2 poor [1]_.
|
||||
|
||||
dissimilarity_matrix_ : ndarray of shape (n_samples, n_samples)
|
||||
Pairwise dissimilarities between the points. Symmetric matrix that:
|
||||
|
||||
- either uses a custom dissimilarity matrix by setting `dissimilarity`
|
||||
to 'precomputed';
|
||||
- or constructs a dissimilarity matrix from data using
|
||||
Euclidean distances.
|
||||
|
||||
n_features_in_ : int
|
||||
Number of features seen during :term:`fit`.
|
||||
|
||||
.. versionadded:: 0.24
|
||||
|
||||
feature_names_in_ : ndarray of shape (`n_features_in_`,)
|
||||
Names of features seen during :term:`fit`. Defined only when `X`
|
||||
has feature names that are all strings.
|
||||
|
||||
.. versionadded:: 1.0
|
||||
|
||||
n_iter_ : int
|
||||
The number of iterations corresponding to the best stress.
|
||||
|
||||
See Also
|
||||
--------
|
||||
sklearn.decomposition.PCA : Principal component analysis that is a linear
|
||||
dimensionality reduction method.
|
||||
sklearn.decomposition.KernelPCA : Non-linear dimensionality reduction using
|
||||
kernels and PCA.
|
||||
TSNE : T-distributed Stochastic Neighbor Embedding.
|
||||
Isomap : Manifold learning based on Isometric Mapping.
|
||||
LocallyLinearEmbedding : Manifold learning using Locally Linear Embedding.
|
||||
SpectralEmbedding : Spectral embedding for non-linear dimensionality.
|
||||
|
||||
References
|
||||
----------
|
||||
.. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
|
||||
Psychometrika, 29 (1964)
|
||||
|
||||
.. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric
|
||||
hypothesis" Kruskal, J. Psychometrika, 29, (1964)
|
||||
|
||||
.. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
|
||||
Groenen P. Springer Series in Statistics (1997)
|
||||
|
||||
Examples
|
||||
--------
|
||||
>>> from sklearn.datasets import load_digits
|
||||
>>> from sklearn.manifold import MDS
|
||||
>>> X, _ = load_digits(return_X_y=True)
|
||||
>>> X.shape
|
||||
(1797, 64)
|
||||
>>> embedding = MDS(n_components=2, normalized_stress='auto')
|
||||
>>> X_transformed = embedding.fit_transform(X[:100])
|
||||
>>> X_transformed.shape
|
||||
(100, 2)
|
||||
|
||||
For a more detailed example of usage, see
|
||||
:ref:`sphx_glr_auto_examples_manifold_plot_mds.py`.
|
||||
|
||||
For a comparison of manifold learning techniques, see
|
||||
:ref:`sphx_glr_auto_examples_manifold_plot_compare_methods.py`.
|
||||
"""
|
||||
|
||||
_parameter_constraints: dict = {
|
||||
"n_components": [Interval(Integral, 1, None, closed="left")],
|
||||
"metric": ["boolean"],
|
||||
"n_init": [Interval(Integral, 1, None, closed="left")],
|
||||
"max_iter": [Interval(Integral, 1, None, closed="left")],
|
||||
"verbose": ["verbose"],
|
||||
"eps": [Interval(Real, 0.0, None, closed="left")],
|
||||
"n_jobs": [None, Integral],
|
||||
"random_state": ["random_state"],
|
||||
"dissimilarity": [StrOptions({"euclidean", "precomputed"})],
|
||||
"normalized_stress": ["boolean", StrOptions({"auto"})],
|
||||
}
|
||||
|
||||
def __init__(
|
||||
self,
|
||||
n_components=2,
|
||||
*,
|
||||
metric=True,
|
||||
n_init=4,
|
||||
max_iter=300,
|
||||
verbose=0,
|
||||
eps=1e-3,
|
||||
n_jobs=None,
|
||||
random_state=None,
|
||||
dissimilarity="euclidean",
|
||||
normalized_stress="auto",
|
||||
):
|
||||
self.n_components = n_components
|
||||
self.dissimilarity = dissimilarity
|
||||
self.metric = metric
|
||||
self.n_init = n_init
|
||||
self.max_iter = max_iter
|
||||
self.eps = eps
|
||||
self.verbose = verbose
|
||||
self.n_jobs = n_jobs
|
||||
self.random_state = random_state
|
||||
self.normalized_stress = normalized_stress
|
||||
|
||||
def _more_tags(self):
|
||||
return {"pairwise": self.dissimilarity == "precomputed"}
|
||||
|
||||
def fit(self, X, y=None, init=None):
|
||||
"""
|
||||
Compute the position of the points in the embedding space.
|
||||
|
||||
Parameters
|
||||
----------
|
||||
X : array-like of shape (n_samples, n_features) or \
|
||||
(n_samples, n_samples)
|
||||
Input data. If ``dissimilarity=='precomputed'``, the input should
|
||||
be the dissimilarity matrix.
|
||||
|
||||
y : Ignored
|
||||
Not used, present for API consistency by convention.
|
||||
|
||||
init : ndarray of shape (n_samples, n_components), default=None
|
||||
Starting configuration of the embedding to initialize the SMACOF
|
||||
algorithm. By default, the algorithm is initialized with a randomly
|
||||
chosen array.
|
||||
|
||||
Returns
|
||||
-------
|
||||
self : object
|
||||
Fitted estimator.
|
||||
"""
|
||||
self.fit_transform(X, init=init)
|
||||
return self
|
||||
|
||||
@_fit_context(prefer_skip_nested_validation=True)
|
||||
def fit_transform(self, X, y=None, init=None):
|
||||
"""
|
||||
Fit the data from `X`, and returns the embedded coordinates.
|
||||
|
||||
Parameters
|
||||
----------
|
||||
X : array-like of shape (n_samples, n_features) or \
|
||||
(n_samples, n_samples)
|
||||
Input data. If ``dissimilarity=='precomputed'``, the input should
|
||||
be the dissimilarity matrix.
|
||||
|
||||
y : Ignored
|
||||
Not used, present for API consistency by convention.
|
||||
|
||||
init : ndarray of shape (n_samples, n_components), default=None
|
||||
Starting configuration of the embedding to initialize the SMACOF
|
||||
algorithm. By default, the algorithm is initialized with a randomly
|
||||
chosen array.
|
||||
|
||||
Returns
|
||||
-------
|
||||
X_new : ndarray of shape (n_samples, n_components)
|
||||
X transformed in the new space.
|
||||
"""
|
||||
X = self._validate_data(X)
|
||||
if X.shape[0] == X.shape[1] and self.dissimilarity != "precomputed":
|
||||
warnings.warn(
|
||||
"The MDS API has changed. ``fit`` now constructs an"
|
||||
" dissimilarity matrix from data. To use a custom "
|
||||
"dissimilarity matrix, set "
|
||||
"``dissimilarity='precomputed'``."
|
||||
)
|
||||
|
||||
if self.dissimilarity == "precomputed":
|
||||
self.dissimilarity_matrix_ = X
|
||||
elif self.dissimilarity == "euclidean":
|
||||
self.dissimilarity_matrix_ = euclidean_distances(X)
|
||||
|
||||
self.embedding_, self.stress_, self.n_iter_ = smacof(
|
||||
self.dissimilarity_matrix_,
|
||||
metric=self.metric,
|
||||
n_components=self.n_components,
|
||||
init=init,
|
||||
n_init=self.n_init,
|
||||
n_jobs=self.n_jobs,
|
||||
max_iter=self.max_iter,
|
||||
verbose=self.verbose,
|
||||
eps=self.eps,
|
||||
random_state=self.random_state,
|
||||
return_n_iter=True,
|
||||
normalized_stress=self.normalized_stress,
|
||||
)
|
||||
|
||||
return self.embedding_
|
||||
Reference in New Issue
Block a user