"""Spectral biclustering algorithms. Authors : Kemal Eren License: BSD 3 clause """ from abc import ABCMeta, abstractmethod import numpy as np from scipy.linalg import norm from scipy.sparse import dia_matrix, issparse from scipy.sparse.linalg import eigsh, svds from . import KMeans, MiniBatchKMeans from ..base import BaseEstimator, BiclusterMixin from ..externals import six from ..utils import check_random_state from ..utils.extmath import (make_nonnegative, randomized_svd, safe_sparse_dot) from ..utils.validation import assert_all_finite, check_array __all__ = ['SpectralCoclustering', 'SpectralBiclustering'] def _scale_normalize(X): """Normalize ``X`` by scaling rows and columns independently. Returns the normalized matrix and the row and column scaling factors. """ X = make_nonnegative(X) row_diag = np.asarray(1.0 / np.sqrt(X.sum(axis=1))).squeeze() col_diag = np.asarray(1.0 / np.sqrt(X.sum(axis=0))).squeeze() row_diag = np.where(np.isnan(row_diag), 0, row_diag) col_diag = np.where(np.isnan(col_diag), 0, col_diag) if issparse(X): n_rows, n_cols = X.shape r = dia_matrix((row_diag, [0]), shape=(n_rows, n_rows)) c = dia_matrix((col_diag, [0]), shape=(n_cols, n_cols)) an = r * X * c else: an = row_diag[:, np.newaxis] * X * col_diag return an, row_diag, col_diag def _bistochastic_normalize(X, max_iter=1000, tol=1e-5): """Normalize rows and columns of ``X`` simultaneously so that all rows sum to one constant and all columns sum to a different constant. """ # According to paper, this can also be done more efficiently with # deviation reduction and balancing algorithms. X = make_nonnegative(X) X_scaled = X dist = None for _ in range(max_iter): X_new, _, _ = _scale_normalize(X_scaled) if issparse(X): dist = norm(X_scaled.data - X.data) else: dist = norm(X_scaled - X_new) X_scaled = X_new if dist is not None and dist < tol: break return X_scaled def _log_normalize(X): """Normalize ``X`` according to Kluger's log-interactions scheme.""" X = make_nonnegative(X, min_value=1) if issparse(X): raise ValueError("Cannot compute log of a sparse matrix," " because log(x) diverges to -infinity as x" " goes to 0.") L = np.log(X) row_avg = L.mean(axis=1)[:, np.newaxis] col_avg = L.mean(axis=0) avg = L.mean() return L - row_avg - col_avg + avg class BaseSpectral(six.with_metaclass(ABCMeta, BaseEstimator, BiclusterMixin)): """Base class for spectral biclustering.""" @abstractmethod def __init__(self, n_clusters=3, svd_method="randomized", n_svd_vecs=None, mini_batch=False, init="k-means++", n_init=10, n_jobs=1, random_state=None): self.n_clusters = n_clusters self.svd_method = svd_method self.n_svd_vecs = n_svd_vecs self.mini_batch = mini_batch self.init = init self.n_init = n_init self.n_jobs = n_jobs self.random_state = random_state def _check_parameters(self): legal_svd_methods = ('randomized', 'arpack') if self.svd_method not in legal_svd_methods: raise ValueError("Unknown SVD method: '{0}'. svd_method must be" " one of {1}.".format(self.svd_method, legal_svd_methods)) def fit(self, X, y=None): """Creates a biclustering for X. Parameters ---------- X : array-like, shape (n_samples, n_features) y : Ignored """ X = check_array(X, accept_sparse='csr', dtype=np.float64) self._check_parameters() self._fit(X) return self def _svd(self, array, n_components, n_discard): """Returns first `n_components` left and right singular vectors u and v, discarding the first `n_discard`. """ if self.svd_method == 'randomized': kwargs = {} if self.n_svd_vecs is not None: kwargs['n_oversamples'] = self.n_svd_vecs u, _, vt = randomized_svd(array, n_components, random_state=self.random_state, **kwargs) elif self.svd_method == 'arpack': u, _, vt = svds(array, k=n_components, ncv=self.n_svd_vecs) if np.any(np.isnan(vt)): # some eigenvalues of A * A.T are negative, causing # sqrt() to be np.nan. This causes some vectors in vt # to be np.nan. A = safe_sparse_dot(array.T, array) random_state = check_random_state(self.random_state) # initialize with [-1,1] as in ARPACK v0 = random_state.uniform(-1, 1, A.shape[0]) _, v = eigsh(A, ncv=self.n_svd_vecs, v0=v0) vt = v.T if np.any(np.isnan(u)): A = safe_sparse_dot(array, array.T) random_state = check_random_state(self.random_state) # initialize with [-1,1] as in ARPACK v0 = random_state.uniform(-1, 1, A.shape[0]) _, u = eigsh(A, ncv=self.n_svd_vecs, v0=v0) assert_all_finite(u) assert_all_finite(vt) u = u[:, n_discard:] vt = vt[n_discard:] return u, vt.T def _k_means(self, data, n_clusters): if self.mini_batch: model = MiniBatchKMeans(n_clusters, init=self.init, n_init=self.n_init, random_state=self.random_state) else: model = KMeans(n_clusters, init=self.init, n_init=self.n_init, n_jobs=self.n_jobs, random_state=self.random_state) model.fit(data) centroid = model.cluster_centers_ labels = model.labels_ return centroid, labels class SpectralCoclustering(BaseSpectral): """Spectral Co-Clustering algorithm (Dhillon, 2001). Clusters rows and columns of an array `X` to solve the relaxed normalized cut of the bipartite graph created from `X` as follows: the edge between row vertex `i` and column vertex `j` has weight `X[i, j]`. The resulting bicluster structure is block-diagonal, since each row and each column belongs to exactly one bicluster. Supports sparse matrices, as long as they are nonnegative. Read more in the :ref:`User Guide `. Parameters ---------- n_clusters : integer, optional, default: 3 The number of biclusters to find. svd_method : string, optional, default: 'randomized' Selects the algorithm for finding singular vectors. May be 'randomized' or 'arpack'. If 'randomized', use :func:`sklearn.utils.extmath.randomized_svd`, which may be faster for large matrices. If 'arpack', use :func:`scipy.sparse.linalg.svds`, which is more accurate, but possibly slower in some cases. n_svd_vecs : int, optional, default: None Number of vectors to use in calculating the SVD. Corresponds to `ncv` when `svd_method=arpack` and `n_oversamples` when `svd_method` is 'randomized`. mini_batch : bool, optional, default: False Whether to use mini-batch k-means, which is faster but may get different results. init : {'k-means++', 'random' or an ndarray} Method for initialization of k-means algorithm; defaults to 'k-means++'. n_init : int, optional, default: 10 Number of random initializations that are tried with the k-means algorithm. If mini-batch k-means is used, the best initialization is chosen and the algorithm runs once. Otherwise, the algorithm is run for each initialization and the best solution chosen. 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 : int, RandomState instance or None, optional, default: None 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`. Attributes ---------- rows_ : array-like, shape (n_row_clusters, n_rows) Results of the clustering. `rows[i, r]` is True if cluster `i` contains row `r`. Available only after calling ``fit``. columns_ : array-like, shape (n_column_clusters, n_columns) Results of the clustering, like `rows`. row_labels_ : array-like, shape (n_rows,) The bicluster label of each row. column_labels_ : array-like, shape (n_cols,) The bicluster label of each column. References ---------- * Dhillon, Inderjit S, 2001. `Co-clustering documents and words using bipartite spectral graph partitioning `__. """ def __init__(self, n_clusters=3, svd_method='randomized', n_svd_vecs=None, mini_batch=False, init='k-means++', n_init=10, n_jobs=1, random_state=None): super(SpectralCoclustering, self).__init__(n_clusters, svd_method, n_svd_vecs, mini_batch, init, n_init, n_jobs, random_state) def _fit(self, X): normalized_data, row_diag, col_diag = _scale_normalize(X) n_sv = 1 + int(np.ceil(np.log2(self.n_clusters))) u, v = self._svd(normalized_data, n_sv, n_discard=1) z = np.vstack((row_diag[:, np.newaxis] * u, col_diag[:, np.newaxis] * v)) _, labels = self._k_means(z, self.n_clusters) n_rows = X.shape[0] self.row_labels_ = labels[:n_rows] self.column_labels_ = labels[n_rows:] self.rows_ = np.vstack(self.row_labels_ == c for c in range(self.n_clusters)) self.columns_ = np.vstack(self.column_labels_ == c for c in range(self.n_clusters)) class SpectralBiclustering(BaseSpectral): """Spectral biclustering (Kluger, 2003). Partitions rows and columns under the assumption that the data has an underlying checkerboard structure. For instance, if there are two row partitions and three column partitions, each row will belong to three biclusters, and each column will belong to two biclusters. The outer product of the corresponding row and column label vectors gives this checkerboard structure. Read more in the :ref:`User Guide `. Parameters ---------- n_clusters : integer or tuple (n_row_clusters, n_column_clusters) The number of row and column clusters in the checkerboard structure. method : string, optional, default: 'bistochastic' Method of normalizing and converting singular vectors into biclusters. May be one of 'scale', 'bistochastic', or 'log'. The authors recommend using 'log'. If the data is sparse, however, log normalization will not work, which is why the default is 'bistochastic'. CAUTION: if `method='log'`, the data must not be sparse. n_components : integer, optional, default: 6 Number of singular vectors to check. n_best : integer, optional, default: 3 Number of best singular vectors to which to project the data for clustering. svd_method : string, optional, default: 'randomized' Selects the algorithm for finding singular vectors. May be 'randomized' or 'arpack'. If 'randomized', uses `sklearn.utils.extmath.randomized_svd`, which may be faster for large matrices. If 'arpack', uses `scipy.sparse.linalg.svds`, which is more accurate, but possibly slower in some cases. n_svd_vecs : int, optional, default: None Number of vectors to use in calculating the SVD. Corresponds to `ncv` when `svd_method=arpack` and `n_oversamples` when `svd_method` is 'randomized`. mini_batch : bool, optional, default: False Whether to use mini-batch k-means, which is faster but may get different results. init : {'k-means++', 'random' or an ndarray} Method for initialization of k-means algorithm; defaults to 'k-means++'. n_init : int, optional, default: 10 Number of random initializations that are tried with the k-means algorithm. If mini-batch k-means is used, the best initialization is chosen and the algorithm runs once. Otherwise, the algorithm is run for each initialization and the best solution chosen. 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 : int, RandomState instance or None, optional, default: None 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`. Attributes ---------- rows_ : array-like, shape (n_row_clusters, n_rows) Results of the clustering. `rows[i, r]` is True if cluster `i` contains row `r`. Available only after calling ``fit``. columns_ : array-like, shape (n_column_clusters, n_columns) Results of the clustering, like `rows`. row_labels_ : array-like, shape (n_rows,) Row partition labels. column_labels_ : array-like, shape (n_cols,) Column partition labels. References ---------- * Kluger, Yuval, et. al., 2003. `Spectral biclustering of microarray data: coclustering genes and conditions `__. """ def __init__(self, n_clusters=3, method='bistochastic', n_components=6, n_best=3, svd_method='randomized', n_svd_vecs=None, mini_batch=False, init='k-means++', n_init=10, n_jobs=1, random_state=None): super(SpectralBiclustering, self).__init__(n_clusters, svd_method, n_svd_vecs, mini_batch, init, n_init, n_jobs, random_state) self.method = method self.n_components = n_components self.n_best = n_best def _check_parameters(self): super(SpectralBiclustering, self)._check_parameters() legal_methods = ('bistochastic', 'scale', 'log') if self.method not in legal_methods: raise ValueError("Unknown method: '{0}'. method must be" " one of {1}.".format(self.method, legal_methods)) try: int(self.n_clusters) except TypeError: try: r, c = self.n_clusters int(r) int(c) except (ValueError, TypeError): raise ValueError("Incorrect parameter n_clusters has value:" " {}. It should either be a single integer" " or an iterable with two integers:" " (n_row_clusters, n_column_clusters)") if self.n_components < 1: raise ValueError("Parameter n_components must be greater than 0," " but its value is {}".format(self.n_components)) if self.n_best < 1: raise ValueError("Parameter n_best must be greater than 0," " but its value is {}".format(self.n_best)) if self.n_best > self.n_components: raise ValueError("n_best cannot be larger than" " n_components, but {} > {}" "".format(self.n_best, self.n_components)) def _fit(self, X): n_sv = self.n_components if self.method == 'bistochastic': normalized_data = _bistochastic_normalize(X) n_sv += 1 elif self.method == 'scale': normalized_data, _, _ = _scale_normalize(X) n_sv += 1 elif self.method == 'log': normalized_data = _log_normalize(X) n_discard = 0 if self.method == 'log' else 1 u, v = self._svd(normalized_data, n_sv, n_discard) ut = u.T vt = v.T try: n_row_clusters, n_col_clusters = self.n_clusters except TypeError: n_row_clusters = n_col_clusters = self.n_clusters best_ut = self._fit_best_piecewise(ut, self.n_best, n_row_clusters) best_vt = self._fit_best_piecewise(vt, self.n_best, n_col_clusters) self.row_labels_ = self._project_and_cluster(X, best_vt.T, n_row_clusters) self.column_labels_ = self._project_and_cluster(X.T, best_ut.T, n_col_clusters) self.rows_ = np.vstack(self.row_labels_ == label for label in range(n_row_clusters) for _ in range(n_col_clusters)) self.columns_ = np.vstack(self.column_labels_ == label for _ in range(n_row_clusters) for label in range(n_col_clusters)) def _fit_best_piecewise(self, vectors, n_best, n_clusters): """Find the ``n_best`` vectors that are best approximated by piecewise constant vectors. The piecewise vectors are found by k-means; the best is chosen according to Euclidean distance. """ def make_piecewise(v): centroid, labels = self._k_means(v.reshape(-1, 1), n_clusters) return centroid[labels].ravel() piecewise_vectors = np.apply_along_axis(make_piecewise, axis=1, arr=vectors) dists = np.apply_along_axis(norm, axis=1, arr=(vectors - piecewise_vectors)) result = vectors[np.argsort(dists)[:n_best]] return result def _project_and_cluster(self, data, vectors, n_clusters): """Project ``data`` to ``vectors`` and cluster the result.""" projected = safe_sparse_dot(data, vectors) _, labels = self._k_means(projected, n_clusters) return labels