scikit-learn/sklearn/utils/extmath.py

"""
Extended math utilities.
"""
# Authors: G. Varoquaux, A. Gramfort, A. Passos, O. Grisel
# License: BSD

import warnings
import numpy as np
from scipy import linalg

from . import check_random_state
from .fixes import qr_economic


def norm(v):
    v = np.asarray(v)
    __nrm2, = linalg.get_blas_funcs(['nrm2'], [v])
    return __nrm2(v)


def _fast_logdet(A):
    """Compute log(det(A)) for A symmetric

    Equivalent to : np.log(np.linalg.det(A)) but more robust.
    It returns -Inf if det(A) is non positive or is not defined.
    """
    # XXX: Should be implemented as in numpy, using ATLAS
    # http://projects.scipy.org/numpy/browser/ \
    #        trunk/numpy/linalg/linalg.py#L1559
    ld = np.sum(np.log(np.diag(A)))
    a = np.exp(ld / A.shape[0])
    d = np.linalg.det(A / a)
    ld += np.log(d)
    if not np.isfinite(ld):
        return -np.inf
    return ld


def _fast_logdet_numpy(A):
    """Compute log(det(A)) for A symmetric

    Equivalent to : np.log(nl.det(A)) but more robust.
    It returns -Inf if det(A) is non positive or is not defined.
    """
    sign, ld = np.linalg.slogdet(A)
    if not sign > 0:
        return -np.inf
    return ld


# Numpy >= 1.5 provides a fast logdet
if hasattr(np.linalg, 'slogdet'):
    fast_logdet = _fast_logdet_numpy
else:
    fast_logdet = _fast_logdet


def density(w, **kwargs):
    """Compute density of a sparse vector

    Return a value between 0 and 1
    """
    if hasattr(w, "toarray"):
        d = float(w.nnz) / (w.shape[0] * w.shape[1])
    else:
        d = 0 if w is None else float((w != 0).sum()) / w.size
    return d


def safe_sparse_dot(a, b, dense_output=False):
    """Dot product that handle the sparse matrix case correctly"""
    from scipy import sparse
    if sparse.issparse(a) or sparse.issparse(b):
        ret = a * b
        if dense_output and hasattr(ret, "toarray"):
            ret = ret.toarray()
        return ret
    else:
        return np.dot(a, b)


def randomized_range_finder(A, size, n_iter, random_state=None,
        n_iterations=None):
    """Computes an orthonormal matrix whose range approximates the range of A.

    Parameters
    ----------
    A: 2D array
        The input data matrix
    size: integer
        Size of the return array
    n_iter: integer
        Number of power iterations used to stabilize the result
    random_state: RandomState or an int seed (0 by default)
        A random number generator instance

    Returns
    -------
    Q: 2D array
        A (size x size) projection matrix, the range of which
        approximates well the range of the input matrix A.

    Notes
    -----

    Follows Algorithm 4.3 of
    Finding structure with randomness: Stochastic algorithms for constructing
    approximate matrix decompositions
    Halko, et al., 2009 (arXiv:909) http://arxiv.org/pdf/0909.4061
    """
    if n_iterations is not None:
        warnings.warn("n_iterations was renamed to n_iter for consistency "
                "and will be removed in 0.16.", DeprecationWarning)
        n_iter = n_iterations
    random_state = check_random_state(random_state)

    # generating random gaussian vectors r with shape: (A.shape[1], size)
    R = random_state.normal(size=(A.shape[1], size))

    # sampling the range of A using by linear projection of r
    Y = safe_sparse_dot(A, R)
    del R

    # perform power iterations with Y to further 'imprint' the top
    # singular vectors of A in Y
    for i in xrange(n_iter):
        Y = safe_sparse_dot(A, safe_sparse_dot(A.T, Y))

    # extracting an orthonormal basis of the A range samples
    Q, R = qr_economic(Y)
    return Q


def randomized_svd(M, n_components, n_oversamples=10, n_iter=0,
                   transpose='auto', random_state=0, n_iterations=None):
    """Computes a truncated randomized SVD

    Parameters
    ----------
    M: ndarray or sparse matrix
        Matrix to decompose

    n_components: int
        Number of singular values and vectors to extract.

    n_oversamples: int (default is 10)
        Additional number of random vectors to sample the range of M so as
        to ensure proper conditioning. The total number of random vectors
        used to find the range of M is n_components + n_oversamples.

    n_iter: int (default is 0)
        Number of power iterations (can be used to deal with very noisy
        problems).

    transpose: True, False or 'auto' (default)
        Whether the algorithm should be applied to M.T instead of M. The
        result should approximately be the same. The 'auto' mode will
        trigger the transposition if M.shape[1] > M.shape[0] since this
        implementation of randomized SVD tend to be a little faster in that
        case).

    random_state: RandomState or an int seed (0 by default)
        A random number generator instance to make behavior

    Notes
    -----
    This algorithm finds a (usually very good) approximate truncated
    singular value decomposition using randomization to speed up the
    computations. It is particularly fast on large matrices on which
    you wish to extract only a small number of components.

    References
    ----------
    * Finding structure with randomness: Stochastic algorithms for constructing
      approximate matrix decompositions
      Halko, et al., 2009 http://arxiv.org/abs/arXiv:0909.4061

    * A randomized algorithm for the decomposition of matrices
      Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert
    """
    if n_iterations is not None:
        warnings.warn("n_iterations was renamed to n_iter for consistency "
                "and will be removed in 0.16.", DeprecationWarning)
        n_iter = n_iterations

    random_state = check_random_state(random_state)
    n_random = n_components + n_oversamples
    n_samples, n_features = M.shape

    if transpose == 'auto' and n_samples > n_features:
        transpose = True
    if transpose:
        # this implementation is a bit faster with smaller shape[1]
        M = M.T

    Q = randomized_range_finder(M, n_random, n_iter, random_state)

    # project M to the (k + p) dimensional space using the basis vectors
    B = safe_sparse_dot(Q.T, M)

    # compute the SVD on the thin matrix: (k + p) wide
    Uhat, s, V = linalg.svd(B, full_matrices=False)
    del B
    U = np.dot(Q, Uhat)

    if transpose:
        # transpose back the results according to the input convention
        return V[:n_components, :].T, s[:n_components], U[:, :n_components].T
    else:
        return U[:, :n_components], s[:n_components], V[:n_components, :]


def logsumexp(arr, axis=0):
    """Computes the sum of arr assuming arr is in the log domain.

    Returns log(sum(exp(arr))) while minimizing the possibility of
    over/underflow.

    Examples
    --------

    >>> import numpy as np
    >>> from sklearn.utils.extmath import logsumexp
    >>> a = np.arange(10)
    >>> np.log(np.sum(np.exp(a)))
    9.4586297444267107
    >>> logsumexp(a)
    9.4586297444267107
    """
    arr = np.rollaxis(arr, axis)
    # Use the max to normalize, as with the log this is what accumulates
    # the less errors
    vmax = arr.max(axis=0)
    out = np.log(np.sum(np.exp(arr - vmax), axis=0))
    out += vmax
    return out


def weighted_mode(a, w, axis=0):
    """Returns an array of the weighted modal (most common) value in a

    If there is more than one such value, only the first is returned.
    The bin-count for the modal bins is also returned.

    This is an extension of the algorithm in scipy.stats.mode.

    Parameters
    ----------
    a : array_like
        n-dimensional array of which to find mode(s).
    w : array_like
        n-dimensional array of weights for each value
    axis : int, optional
        Axis along which to operate. Default is 0, i.e. the first axis.

    Returns
    -------
    vals : ndarray
        Array of modal values.
    score : ndarray
        Array of weighted counts for each mode.

    Examples
    --------
    >>> from sklearn.utils.extmath import weighted_mode
    >>> x = [4, 1, 4, 2, 4, 2]
    >>> weights = [1, 1, 1, 1, 1, 1]
    >>> weighted_mode(x, weights)
    (array([ 4.]), array([ 3.]))

    The value 4 appears three times: with uniform weights, the result is
    simply the mode of the distribution.

    >>> weights = [1, 3, 0.5, 1.5, 1, 2] # deweight the 4's
    >>> weighted_mode(x, weights)
    (array([ 2.]), array([ 3.5]))

    The value 2 has the highest score: it appears twice with weights of
    1.5 and 2: the sum of these is 3.

    See Also
    --------
    scipy.stats.mode
    """
    if axis is None:
        a = np.ravel(a)
        w = np.ravel(w)
        axis = 0
    else:
        a = np.asarray(a)
        w = np.asarray(w)
        axis = axis

    if a.shape != w.shape:
        w = np.zeros(a.shape, dtype=w.dtype) + w

    scores = np.unique(np.ravel(a))       # get ALL unique values
    testshape = list(a.shape)
    testshape[axis] = 1
    oldmostfreq = np.zeros(testshape)
    oldcounts = np.zeros(testshape)
    for score in scores:
        template = np.zeros(a.shape)
        ind = (a == score)
        template[ind] = w[ind]
        counts = np.expand_dims(np.sum(template, axis), axis)
        mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
        oldcounts = np.maximum(counts, oldcounts)
        oldmostfreq = mostfrequent
    return mostfrequent, oldcounts


def pinvh(a, cond=None, rcond=None, lower=True):
    """Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.

    Calculate a generalized inverse of a symmetric matrix using its
    eigenvalue decomposition and including all 'large' eigenvalues.

    Parameters
    ----------
    a : array, shape (N, N)
        Real symmetric or complex hermetian matrix to be pseudo-inverted
    cond, rcond : float or None
        Cutoff for 'small' eigenvalues.
        Singular values smaller than rcond * largest_eigenvalue are considered
        zero.

        If None or -1, suitable machine precision is used.
    lower : boolean
        Whether the pertinent array data is taken from the lower or upper
        triangle of a. (Default: lower)

    Returns
    -------
    B : array, shape (N, N)

    Raises
    ------
    LinAlgError
        If eigenvalue does not converge

    Examples
    --------
    >>> from numpy import *
    >>> a = random.randn(9, 6)
    >>> a = np.dot(a, a.T)
    >>> B = pinvh(a)
    >>> allclose(a, dot(a, dot(B, a)))
    True
    >>> allclose(B, dot(B, dot(a, B)))
    True

    """
    a = np.asarray_chkfinite(a)
    s, u = linalg.eigh(a, lower=lower)

    if rcond is not None:
        cond = rcond
    if cond in [None, -1]:
        t = u.dtype.char.lower()
        factor = {'f': 1E3, 'd': 1E6}
        cond = factor[t] * np.finfo(t).eps

    # unlike svd case, eigh can lead to negative eigenvalues
    above_cutoff = (abs(s) > cond * np.max(abs(s)))
    psigma_diag = np.zeros_like(s)
    psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]

    return np.dot(u * psigma_diag, np.conjugate(u).T)