""" Jointly perform multi-dimensional Scaling (MDS) on two datasets """ # original author: Nelle Varoquaux # modified by: Lila Rieber # License: BSD import numpy as np import sys import warnings from sklearn.base import BaseEstimator from sklearn.metrics import euclidean_distances from sklearn.utils import check_random_state, check_array, check_symmetric from sklearn.externals.joblib import Parallel from sklearn.externals.joblib import delayed from sklearn.isotonic import IsotonicRegression def squared_dist(x1, x2): """Computes squared Euclidean distance between coordinate x1 and coordinate x2""" return sum([(i1 - i2)**2 for i1, i2 in zip(x1, x2)]) def ssd(X1, X2): """Computes sum of squared distances between coordinates X1 and coordinates X2""" return sum([squared_dist(x1, x2) for x1, x2 in zip(X1, X2)]) def moore_penrose(V): """Computes Moore-Penrose inverse of matrix V""" n = len(V) return np.linalg.inv(V + np.ones((n,n))) - n**-2 * np.ones((n,n)) def initialize(dissimilarities, random_state, init, n_samples, n_components): 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: 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 return X, sim_flat, sim_flat_w def nonmetric_disparities(dis, sim_flat, n_samples): 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()) return disparities def guttman(X1, X2, disparities, inv_V, V2, dis): # avoid division by 0 dis[dis == 0] = 1e-5 # B: error between distance matrix and embedding ratio = disparities / dis B = - ratio B[np.arange(len(B)), np.arange(len(B))] += ratio.sum(axis=1) return np.dot(inv_V, (np.dot(B, X1) + np.dot(V2, X2))) def _smacof_single(dissimilarities1, dissimilarities2, p, weights1=None, weights2=None, metric=True, n_components=2, init1=None, init2=None, max_iter=300, verbose=0, eps=1e-3, random_state1=None, random_state2=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 : 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, 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. """ dissimilarities1 = check_symmetric(dissimilarities1, raise_exception=True) dissimilarities2 = check_symmetric(dissimilarities2, raise_exception=True) if dissimilarities1.shape != dissimilarities2.shape: print("Error. Distance matrices have different shapes.") sys.exit("Error. Distance matrices have different shapes.") n_samples = dissimilarities1.shape[0] X1, sim_flat1, sim_flat_w1 = initialize(dissimilarities1, random_state1, init1, n_samples, n_components) X2, sim_flat2, sim_flat_w2 = initialize(dissimilarities2, random_state2, init2, n_samples, n_components) #Default: equal weights if weights1 is None: weights1 = np.ones((n_samples, n_samples)) if weights2 is None: weights2 = np.ones(n_samples) # Disparity-specific weights (V in Borg) V1 = np.zeros((n_samples,n_samples)) for i in range(n_samples): diagonal = 0 for j in range(n_samples): V1[i,j] = -weights1[i,j] diagonal += weights1[i,j] V1[i,i] = diagonal # Locus-specific weights V2 = np.zeros((n_samples,n_samples)) for i, weight in enumerate(weights2): V2[i,i] = weight * p * n_samples inv_V = moore_penrose(V1+V2) old_stress = None ir = IsotonicRegression() for it in range(max_iter): # Compute distance and monotonic regression dis1 = euclidean_distances(X1) dis2 = euclidean_distances(X2) if metric: disparities1 = dissimilarities1 disparities2 = dissimilarities2 else: disparities1 = nonmetric_disparities1(dis1, sim_flat1, n_samples) disparities2 = nonmetric_disparities2(dis2, sim_flat2, n_samples) # Compute stress stress = ((dis1.ravel() - disparities1.ravel()) ** 2).sum() + ((dis2.ravel() - disparities2.ravel()) ** 2).sum() + n_samples * p * ssd(X1, X2) #multiply by n_samples to make ssd term comparable in magnitude to embedding error terms # Update X1 using the Guttman transform X1 = guttman(X1, X2, disparities1, inv_V, V2, dis1) # Update X2 using the Guttman transform X2 = guttman(X2, X1, disparities2, inv_V, V2, dis2) # Test stress dis1 = np.sqrt((X1 ** 2).sum(axis=1)).sum() dis2 = np.sqrt((X2 ** 2).sum(axis=1)).sum() dis = np.mean((dis1, dis2)) if verbose >= 2: print('it: %d, stress %s' % (it, stress)) if old_stress is not None: if np.abs(old_stress - stress / dis) < eps: if verbose: print('breaking at iteration %d with stress %s' % (it, stress)) break old_stress = stress / dis return X1, X2, stress, it + 1 def smacof(dissimilarities1, dissimilarities2, p, weights1, weights2, metric=True, n_components=2, init1=None, init2=None, n_init=8, n_jobs=1, max_iter=300, verbose=0, eps=1e-3, random_state1=None, random_state2=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 : integer or numpy.RandomState, optional, default: None The generator used to initialize the centers. If an integer is given, it fixes the seed. Defaults to the global numpy random number generator. 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) """ if p < 0: sys.exit('Error. Penalty must be non-negative.') dissimilarities1 = check_array(dissimilarities1) dissimilarities2 = check_array(dissimilarities2) random_state1 = check_random_state(random_state1) random_state2 = check_random_state(random_state2) if hasattr(init1, '__array__'): init1 = np.asarray(init1).copy() if not n_init == 1: warnings.warn( 'Explicit initial positions passed: ' 'performing only one init of the MDS instead of {}'.format(n_init)) n_init = 1 if hasattr(init2, '__array__'): init2 = np.asarray(init2).copy() if not n_init == 1: warnings.warn( 'Explicit initial positions passed: ' 'performing only one init of the MDS instead of {}'.format(n_init)) n_init = 1 best_pos1, best_pos2, best_stress = None, None, None if n_jobs == 1: for it in range(n_init): pos1, pos2, stress, n_iter_ = _smacof_single( dissimilarities1, dissimilarities2, p, metric=metric, n_components=n_components, init1=init1, init2=init2, max_iter=max_iter, verbose=verbose, eps=eps, random_state1=random_state1, random_state2=random_state2) if best_stress is None or stress < best_stress: best_stress = stress best_pos1 = pos1.copy() best_pos2 = pos2.copy() best_iter = n_iter_ else: seeds1 = random_state1.randint(np.iinfo(np.int32).max, size=n_init) seeds2 = random_state2.randint(np.iinfo(np.int32).max, size=n_init) results = Parallel(n_jobs=n_jobs, verbose=max(verbose - 1, 0))( delayed(_smacof_single)( dissimilarities1, dissimilarities2, p, weights1=weights1, weights2=weights2, metric=metric, n_components=n_components, init1=init1, init2=init2, max_iter=max_iter, verbose=verbose, eps=eps, random_state1=seed1, random_state2=seed2) for seed1, seed2 in zip(seeds1, seeds2)) positions1, positions2, stress, n_iters = zip(*results) best = np.argmin(stress) best_stress = stress[best] best_pos1 = positions1[best] best_pos2 = positions2[best] best_iter = n_iters[best] if return_n_iter: return best_pos1, best_pos2, best_stress, best_iter else: return best_pos1, best_pos2, best_stress class Joint_MDS(BaseEstimator): """Multidimensional scaling Read more in the :ref:`User Guide `. 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 : integer or numpy.RandomState, optional, default: None 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 : '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, weights1=None, weights2=None, p=0, metric=True, n_init=4, max_iter=300, verbose=0, eps=1e-3, n_jobs=1, random_state1=None, random_state2=None, dissimilarity="euclidean"): self.n_components = n_components self.weights1 = weights1 self.weights2 = weights2 self.p = p 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_state1 = random_state1 self.random_state2 = random_state2 @property def _pairwise(self): return self.kernel == "precomputed" def fit(self, X1, X2, weights1=None, weights2=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(X1, X2, weights1=weights1, weights2=weights2, init=init) return self def fit_transform(self, X1, X2, weights1=None, weights2=None, init1=None, init2=None): """ Fit the data from X, and returns the embedded coordinates 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. """ X1 = check_array(X1) if X1.shape[0] == X1.shape[1] and self.dissimilarity != "precomputed": warnings.warn("The MDS API has changed. ``fit`` now constructs a" " dissimilarity matrix from data. To use a custom " "dissimilarity matrix, set " "``dissimilarity='precomputed'``.") if self.dissimilarity == "precomputed": self.dissimilarity_matrix1_ = X1 elif self.dissimilarity == "euclidean": self.dissimilarity_matrix1_ = euclidean_distances(X1) else: raise ValueError("Proximity must be 'precomputed' or 'euclidean'." " Got %s instead" % str(self.dissimilarity)) X2 = check_array(X2) if X2.shape[0] == X2.shape[1] and self.dissimilarity != "precomputed": warnings.warn("The MDS API has changed. ``fit`` now constructs a" " dissimilarity matrix from data. To use a custom " "dissimilarity matrix, set " "``dissimilarity='precomputed'``.") if self.dissimilarity == "precomputed": self.dissimilarity_matrix2_ = X2 elif self.dissimilarity == "euclidean": self.dissimilarity_matrix2_ = euclidean_distances(X2) else: raise ValueError("Proximity must be 'precomputed' or 'euclidean'." " Got %s instead" % str(self.dissimilarity)) self.embedding1_, self.embedding2_, self.stress_, self.n_iter_ = smacof( self.dissimilarity_matrix1_, self.dissimilarity_matrix2_, p=self.p, weights1=self.weights1, weights2=self.weights2, metric=self.metric, n_components=self.n_components, init1=init1, init2=init2, n_init=self.n_init, n_jobs=self.n_jobs, max_iter=self.max_iter, verbose=self.verbose, eps=self.eps, random_state1=self.random_state1, random_state2=self.random_state2, return_n_iter=True) return self.embedding1_, self.embedding2_