"""
Multi-dimensional Scaling (MDS)
"""
# author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
# License: BSD
import numpy as np
import warnings
from ..base import BaseEstimator
from ..metrics import euclidean_distances
from ..utils import check_random_state, check_array, check_symmetric
from ..externals.joblib import Parallel
from ..externals.joblib import delayed
from ..isotonic import IsotonicRegression
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, shape (n_samples, n_samples)
Pairwise dissimilarities between the points. Must be symmetric.
metric : boolean, optional, default: True
Compute metric or nonmetric SMACOF algorithm.
n_components : int, optional, 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, shape (n_samples, n_components), optional, 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, optional, default: 300
Maximum number of iterations of the SMACOF algorithm for a single run.
verbose : int, optional, default: 0
Level of verbosity.
eps : float, optional, default: 1e-3
Relative tolerance with respect to stress at which to declare
convergence.
random_state : int, RandomState instance or None, optional, default: None
The generator used to initialize the centers. If int, random_state is
the seed used by the random number generator; If RandomState instance,
random_state is the random number generator; If None, the random number
generator is the RandomState instance used by `np.random`.
Returns
-------
X : ndarray, 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.rand(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. / 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=1, max_iter=300, verbose=0, eps=1e-3, random_state=None,
return_n_iter=False):
"""Computes 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 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, shape (n_samples, n_samples)
Pairwise dissimilarities between the points. Must be symmetric.
metric : boolean, optional, default: True
Compute metric or nonmetric SMACOF algorithm.
n_components : int, optional, 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, shape (n_samples, n_components), optional, 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, optional, 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, optional, default: 1
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.
If -1 all CPUs are used. If 1 is given, no parallel computing code is
used at all, which is useful for debugging. For ``n_jobs`` below -1,
(``n_cpus + 1 + n_jobs``) are used. Thus for ``n_jobs = -2``, all CPUs
but one are used.
max_iter : int, optional, default: 300
Maximum number of iterations of the SMACOF algorithm for a single run.
verbose : int, optional, default: 0
Level of verbosity.
eps : float, optional, default: 1e-3
Relative tolerance with respect to stress at which to declare
convergence.
random_state : int, RandomState instance or None, optional, default: None
The generator used to initialize the centers. If int, random_state is
the seed used by the random number generator; If RandomState instance,
random_state is the random number generator; If None, the random number
generator is the RandomState instance used by `np.random`.
return_n_iter : bool, optional, default: False
Whether or not to return the number of iterations.
Returns
-------
X : ndarray, 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 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, optional, default: 2
Number of dimensions in which to immerse the dissimilarities.
metric : boolean, optional, default: True
If ``True``, perform metric MDS; otherwise, perform nonmetric MDS.
n_init : int, optional, 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, optional, default: 300
Maximum number of iterations of the SMACOF algorithm for a single run.
verbose : int, optional, default: 0
Level of verbosity.
eps : float, optional, default: 1e-3
Relative tolerance with respect to stress at which to declare
convergence.
n_jobs : int, optional, default: 1
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.
If -1 all CPUs are used. If 1 is given, no parallel computing code is
used at all, which is useful for debugging. For ``n_jobs`` below -1,
(``n_cpus + 1 + n_jobs``) are used. Thus for ``n_jobs = -2``, all CPUs
but one are used.
random_state : int, RandomState instance or None, optional, default: None
The generator used to initialize the centers. If int, random_state is
the seed used by the random number generator; If RandomState instance,
random_state is the random number generator; If None, the random number
generator is the RandomState instance used by `np.random`.
dissimilarity : 'euclidean' | 'precomputed', optional, 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_ : array-like, shape (n_components, n_samples)
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).
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)
"""
def __init__(self, n_components=2, metric=True, n_init=4,
max_iter=300, verbose=0, eps=1e-3, n_jobs=1,
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
@property
def _pairwise(self):
return self.kernel == "precomputed"
[docs] def fit(self, X, y=None, init=None):
"""
Computes the position of the points in the embedding space
Parameters
----------
X : array, shape (n_samples, n_features) or (n_samples, n_samples)
Input data. If ``dissimilarity=='precomputed'``, the input should
be the dissimilarity matrix.
init : ndarray, shape (n_samples,), optional, default: None
Starting configuration of the embedding to initialize the SMACOF
algorithm. By default, the algorithm is initialized with a randomly
chosen array.
"""
self.fit_transform(X, init=init)
return self
