_mds.py

"""
Multi-dimensional Scaling (MDS).
"""

# author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
# License: BSD

import numpy as np
from joblib import Parallel, effective_n_jobs

import warnings

from ..base import BaseEstimator
from ..metrics import euclidean_distances
from ..utils import check_random_state, check_array, check_symmetric
from ..isotonic import IsotonicRegression
from ..utils.fixes import delayed


def _smacof_single(
    dissimilarities,
    metric=True,
    n_components=2,
    init=None,
    max_iter=300,
    verbose=0,
    eps=1e-3,
    random_state=None,
):
    """Computes multidimensional scaling using SMACOF algorithm.

    Parameters
    ----------
    dissimilarities : ndarray of shape (n_samples, n_samples)
        Pairwise dissimilarities between the points. Must be symmetric.

    metric : bool, default=True
        Compute metric or nonmetric SMACOF algorithm.

    n_components : int, default=2
        Number of dimensions in which to immerse the dissimilarities. If an
        ``init`` array is provided, this option is overridden and the shape of
        ``init`` is used to determine the dimensionality of the embedding
        space.

    init : ndarray of shape (n_samples, n_components), default=None
        Starting configuration of the embedding to initialize the algorithm. By
        default, the algorithm is initialized with a randomly chosen array.

    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.

    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>`.

    Returns
    -------
    X : ndarray of shape (n_samples, n_components)
        Coordinates of the points in a ``n_components``-space.

    stress : float
        The final value of the stress (sum of squared distance of the
        disparities and the distances for all constrained points).

    n_iter : int
        The number of iterations corresponding to the best stress.
    """
    dissimilarities = check_symmetric(dissimilarities, raise_exception=True)

    n_samples = dissimilarities.shape[0]
    random_state = check_random_state(random_state)

    sim_flat = ((1 - np.tri(n_samples)) * dissimilarities).ravel()
    sim_flat_w = sim_flat[sim_flat != 0]
    if init is None:
        # Randomly choose initial configuration
        X = random_state.uniform(size=n_samples * n_components)
        X = X.reshape((n_samples, n_components))
    else:
        # overrides the parameter p
        n_components = init.shape[1]
        if n_samples != init.shape[0]:
            raise ValueError(
                "init matrix should be of shape (%d, %d)" % (n_samples, n_components)
            )
        X = init

    old_stress = None
    ir = IsotonicRegression()
    for it in range(max_iter):
        # Compute distance and monotonic regression
        dis = euclidean_distances(X)

        if metric:
            disparities = dissimilarities
        else:
            dis_flat = dis.ravel()
            # dissimilarities with 0 are considered as missing values
            dis_flat_w = dis_flat[sim_flat != 0]

            # Compute the disparities using a monotonic regression
            disparities_flat = ir.fit_transform(sim_flat_w, dis_flat_w)
            disparities = dis_flat.copy()
            disparities[sim_flat != 0] = disparities_flat
            disparities = disparities.reshape((n_samples, n_samples))
            disparities *= np.sqrt(
                (n_samples * (n_samples - 1) / 2) / (disparities**2).sum()
            )

        # Compute stress
        stress = ((dis.ravel() - disparities.ravel()) ** 2).sum() / 2

        # Update X using the Guttman transform
        dis[dis == 0] = 1e-5
        ratio = disparities / dis
        B = -ratio
        B[np.arange(len(B)), np.arange(len(B))] += ratio.sum(axis=1)
        X = 1.0 / n_samples * np.dot(B, X)

        dis = np.sqrt((X**2).sum(axis=1)).sum()
        if verbose >= 2:
            print("it: %d, stress %s" % (it, stress))
        if old_stress is not None:
            if (old_stress - stress / dis) < eps:
                if verbose:
                    print("breaking at iteration %d with stress %s" % (it, stress))
                break
        old_stress = stress / dis

    return X, stress, it + 1


def smacof(
    dissimilarities,
    *,
    metric=True,
    n_components=2,
    init=None,
    n_init=8,
    n_jobs=None,
    max_iter=300,
    verbose=0,
    eps=1e-3,
    random_state=None,
    return_n_iter=False,
):
    """Compute multidimensional scaling using the SMACOF algorithm.

    The SMACOF (Scaling by MAjorizing a COmplicated Function) algorithm is a
    multidimensional scaling algorithm which minimizes an objective function
    (the *stress*) using a majorization technique. Stress majorization, also
    known as the Guttman Transform, guarantees a monotone convergence of
    stress, and is more powerful than traditional techniques such as gradient
    descent.

    The SMACOF algorithm for metric MDS can be summarized by the following
    steps:

    1. Set an initial start configuration, randomly or not.
    2. Compute the stress
    3. Compute the Guttman Transform
    4. Iterate 2 and 3 until convergence.

    The nonmetric algorithm adds a monotonic regression step before computing
    the stress.

    Parameters
    ----------
    dissimilarities : ndarray of shape (n_samples, n_samples)
        Pairwise dissimilarities between the points. Must be symmetric.

    metric : bool, default=True
        Compute metric or nonmetric SMACOF algorithm.

    n_components : int, default=2
        Number of dimensions in which to immerse the dissimilarities. If an
        ``init`` array is provided, this option is overridden and the shape of
        ``init`` is used to determine the dimensionality of the embedding
        space.

    init : ndarray of shape (n_samples, n_components), default=None
        Starting configuration of the embedding to initialize the algorithm. By
        default, the algorithm is initialized with a randomly chosen array.

    n_init : int, default=8
        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. If ``init`` is
        provided, this option is overridden and a single run is performed.

    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.

    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.

    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>`.

    return_n_iter : bool, default=False
        Whether or not to return the number of iterations.

    Returns
    -------
    X : ndarray of shape (n_samples, n_components)
        Coordinates of the points in a ``n_components``-space.

    stress : float
        The final value of the stress (sum of squared distance of the
        disparities and the distances for all constrained points).

    n_iter : int
        The number of iterations corresponding to the best stress. Returned
        only if ``return_n_iter`` is set to ``True``.

    Notes
    -----
    "Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
    Groenen P. Springer Series in Statistics (1997)

    "Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
    Psychometrika, 29 (1964)

    "Multidimensional scaling by optimizing goodness of fit to a nonmetric
    hypothesis" Kruskal, J. Psychometrika, 29, (1964)
    """

    dissimilarities = check_array(dissimilarities)
    random_state = check_random_state(random_state)

    if hasattr(init, "__array__"):
        init = np.asarray(init).copy()
        if not n_init == 1:
            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,
            )
            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,
            )
            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.

    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.

    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``.

    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).

    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
    ----------
    "Modern Multidimensional Scaling - Theory and Applications" Borg, I.;
    Groenen P. Springer Series in Statistics (1997)

    "Nonmetric multidimensional scaling: a numerical method" Kruskal, J.
    Psychometrika, 29 (1964)

    "Multidimensional scaling by optimizing goodness of fit to a nonmetric
    hypothesis" Kruskal, J. Psychometrika, 29, (1964)

    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)
    >>> X_transformed = embedding.fit_transform(X[:100])
    >>> X_transformed.shape
    (100, 2)
    """

    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",
    ):
        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

    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,), 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

    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,), 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)
        else:
            raise ValueError(
                "Proximity must be 'precomputed' or 'euclidean'. Got %s instead"
                % str(self.dissimilarity)
            )

        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,
        )

        return self.embedding_