# Source code for rsatoolbox.util.vis_utils

```#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""Collection of helper methods for vis module

+ Weighted_MDS:       an MDS class that incorporates weighting
+ weight_to_matrices: batch squareform() to weight matrices

@author: baihan

Notice:

The functions of MDS in this module are modified from
the Python package scikit-learn, originally written by
Nelle Varoquaux <nelle.varoquaux@gmail.com> under BSD
We modified the MDS function to include an additional
functionality of having an important matrix as an input.
"""

import warnings
import numpy as np
from joblib import Parallel, delayed, effective_n_jobs
from sklearn.base import BaseEstimator
from sklearn.metrics import euclidean_distances
from sklearn.utils import check_random_state, check_array, check_symmetric
from sklearn.isotonic import IsotonicRegression
from scipy.spatial.distance import squareform
from rsatoolbox.util.rdm_utils import _get_n_from_reduced_vectors

[docs]def weight_to_matrices(x):
"""converts a *stack* of weights in vector or matrix form into matrix form

Args:
**x** (np.ndarray): stack of weight matrices or weight vectors

Returns:
tuple: **v** (np.ndarray): 3D, matrix form of the stack of weight matrices
"""
if x.ndim == 2:
v = x
n_rdm = x.shape[0]
n_cond = _get_n_from_reduced_vectors(x)
m = np.ndarray((n_rdm, n_cond, n_cond))
for idx in np.arange(n_rdm):
m[idx, :, :] = squareform(v[idx, :])
elif x.ndim == 3:
m = x
return m

def _smacof_single(dissimilarities, metric=True, n_components=2, init=None,
max_iter=300, verbose=0, eps=1e-3, random_state=None,
weight=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>`.

weight : ndarray of shape (n_samples, n_samples), default=None
symmetric weighting matrix of similarities.
In default, all weights are 1.

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.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
if weight is None:
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)
else:
ratio = weight * disparities / dis
B = - ratio
B[np.arange(len(B)), np.arange(len(B))] += ratio.sum(axis=1)
V = np.zeros((n_samples, n_samples))
for nn in range(n_samples):
for mm in range(nn, n_samples):
v = np.zeros((n_samples, 1))
v[nn], v[mm] = 1, -1
V += weight[nn, mm] * np.dot(v, v.T)
X = np.dot(np.linalg.pinv(V), 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

[docs]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, weight=None):
"""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
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 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.

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.

weight : ndarray of shape (n_samples, n_samples), default=None
symmetric weighting matrix of similarities.
In default, all weights are 1.

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,
weight=weight)
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, weight=weight)
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

[docs]class Weighted_MDS(BaseEstimator):
"""Multidimensional scaling with weighting options.

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.

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_iter_ : int
The number of iterations corresponding to the best stress.

Examples
--------
>>> from sklearn.manifold import MDS
>>> X.shape
(1797, 64)
>>> embedding = MDS(n_components=2)
>>> X_transformed = embedding.fit_transform(X[:100])
>>> X_transformed.shape
(100, 2)

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=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.dissimilarity_matrix_ = None
self.embedding_ = None
self.stress_ = None
self.n_iter_ = None
# not in use, declared for consistency with sklearn:
self.normalized_stress = normalized_stress

@property
def _pairwise(self):
return self.dissimilarity == "precomputed"

[docs]    def fit(self, X, y=None, init=None, weight=None):
"""
Computes 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

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.

weight : ndarray of shape (n_samples, n_samples), default=None
symmetric weighting matrix of similarities.
In default, all weights are 1.
"""
self.fit_transform(X, init=init, weight=weight)
return self

[docs]    def fit_transform(self, X, y=None, init=None, weight=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

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.
weight : ndarray of shape (n_samples, n_samples), default=None
symmetric weighting matrix of similarities.
In default, all weights are 1.
"""
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, weight=weight)

return self.embedding_
```