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Multi-dimensional Scaling (MDS)
"""
# author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
# Licence: BSD
import numpy as np
import warnings
from ..base import BaseEstimator
from ..metrics import euclidean_distances
from ..utils import check_random_state, check_arrays
from ..externals.joblib import Parallel
from ..externals.joblib import delayed
from ..isotonic import IsotonicRegression
def _smacof_single(similarities, 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
----------
similarities: symmetric ndarray, shape [n * n]
similarities between the points
metric: boolean, optional, default: True
compute metric or nonmetric SMACOF algorithm
n_components: int, optional, default: 2
number of dimension in which to immerse the similarities
overwritten if initial array is provided.
init: {None or ndarray}, optional
if None, randomly chooses the initial configuration
if ndarray, initialize the SMACOF algorithm with this 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-6
relative tolerance w.r.t stress to declare converge
random_state: integer or numpy.RandomState, optional
The generator used to initialize the centers. If an integer is
given, it fixes the seed. Defaults to the global numpy random
number generator.
Returns
-------
X: ndarray (n_samples, n_components), float
coordinates of the n_samples 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_samples = similarities.shape[0]
random_state = check_random_state(random_state)
if similarities.shape[0] != similarities.shape[1]:
raise ValueError("similarities must be a square array (shape=%d)" %
n_samples)
res = 100 * np.finfo(np.float).resolution
if np.any((similarities - similarities.T) > res):
raise ValueError("similarities must be symmetric")
sim_flat = ((1 - np.tri(n_samples)) * similarities).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 = similarities
else:
dis_flat = dis.ravel()
# similarities 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
def smacof(similarities, metric=True, n_components=2, init=None, n_init=8,
n_jobs=1, max_iter=300, verbose=0, eps=1e-3, random_state=None):
"""
Computes multidimensional scaling using SMACOF (Scaling by Majorizing a
Complicated Function) algorithm
The SMACOF algorithm is a multidimensional scaling algorithm: it minimizes
a objective function, the *stress*, using a majorization technique. The
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 steps before computing
the stress.
Parameters
----------
similarities : symmetric ndarray, shape (n_samples, n_samples)
similarities between the points
metric : boolean, optional, default: True
compute metric or nonmetric SMACOF algorithm
n_components : int, optional, default: 2
number of dimension in which to immerse the similarities
overridden if initial array is provided.
init : {None or ndarray of shape (n_samples, n_components)}, optional
if None, randomly chooses the initial configuration
if ndarray, initialize the SMACOF algorithm with this array
n_init : int, optional, default: 8
Number of time the smacof algorithm will be run with different
initialisation. The final results will be the best output of the
n_init consecutive runs in terms of stress.
n_jobs : int, optional, default: 1
The number of jobs to use for the computation. This works by breaking
down the pairwise matrix into n_jobs even slices and computing them 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-6
relative tolerance w.r.t stress to declare converge
random_state : integer or numpy.RandomState, optional
The generator used to initialize the centers. If an integer is
given, it fixes the seed. Defaults to the global numpy random
number generator.
Returns
-------
X : ndarray (n_samples,n_components)
Coordinates of the n_samples 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)
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)
"""
similarities, = check_arrays(similarities, sparse_format='dense')
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 = _smacof_single(similarities, 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()
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)(
similarities, 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 = zip(*results)
best = np.argmin(stress)
best_stress = stress[best]
best_pos = positions[best]
return best_pos, best_stress
class MDS(BaseEstimator):
"""Multidimensional scaling
Parameters
----------
metric : boolean, optional, default: True
compute metric or nonmetric SMACOF (Scaling by Majorizing a
Complicated Function) algorithm
n_components : int, optional, default: 2
number of dimension in which to immerse the similarities
overridden if initial array is provided.
n_init : int, optional, default: 4
Number of time the smacof algorithm will be run with different
initialisation. The final results will be the best output of the
n_init consecutive runs in terms of 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-6
relative tolerance w.r.t stress to declare converge
n_jobs : int, optional, default: 1
The number of jobs to use for the computation. This works by breaking
down the pairwise matrix into n_jobs even slices and computing them 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 : integer or numpy.RandomState, optional
The generator used to initialize the centers. If an integer is
given, it fixes the seed. Defaults to the global numpy random
number generator.
dissimilarity : string
Which dissimilarity measure to use.
Supported are 'euclidean' and 'precomputed'.
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"
def fit(self, X, init=None, y=None):
"""
Computes the position of the points in the embedding space
Parameters
----------
X : array, shape=[n_samples, n_features]
Input data.
init : {None or ndarray, shape (n_samples,)}, optional
If None, randomly chooses the initial configuration
if ndarray, initialize the SMACOF algorithm with this array.
"""
self.fit_transform(X, init=init)
return self
def fit_transform(self, X, init=None, y=None):
"""
Fit the data from X, and returns the embedded coordinates
Parameters
----------
X : array, shape=[n_samples, n_features]
Input data.
init : {None or ndarray, shape (n_samples,)}, optional
If None, randomly chooses the initial configuration
if ndarray, initialize the SMACOF algorithm with this array.
"""
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 is "precomputed":
self.dissimilarity_matrix_ = X
elif self.dissimilarity is "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_ = 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 self.embedding_
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