shrunk_covariance_.py

"""
Covariance estimators using shrinkage.

Shrinkage corresponds to regularising `cov` using a convex combination:
shrunk_cov = (1-shrinkage)*cov + shrinkage*structured_estimate.

"""

# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#         Gael Varoquaux <gael.varoquaux@normalesup.org>
#         Virgile Fritsch <virgile.fritsch@inria.fr>
#
# License: BSD 3 clause

# avoid division truncation
import warnings
import numpy as np

from .empirical_covariance_ import empirical_covariance, EmpiricalCovariance
from ..utils import check_array


# ShrunkCovariance estimator

def shrunk_covariance(emp_cov, shrinkage=0.1):
    """Calculates a covariance matrix shrunk on the diagonal

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    Parameters
    ----------
    emp_cov : array-like, shape (n_features, n_features)
        Covariance matrix to be shrunk

    shrinkage : float, 0 <= shrinkage <= 1
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Returns
    -------
    shrunk_cov : array-like
        Shrunk covariance.

    Notes
    -----
    The regularized (shrunk) covariance is given by:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features

    """
    emp_cov = check_array(emp_cov)
    n_features = emp_cov.shape[0]

    mu = np.trace(emp_cov) / n_features
    shrunk_cov = (1. - shrinkage) * emp_cov
    shrunk_cov.flat[::n_features + 1] += shrinkage * mu

    return shrunk_cov


class ShrunkCovariance(EmpiricalCovariance):
    """Covariance estimator with shrinkage

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    Parameters
    ----------
    store_precision : boolean, default True
        Specify if the estimated precision is stored

    assume_centered : boolean, default False
        If True, data will not be centered before computation.
        Useful when working with data whose mean is almost, but not exactly
        zero.
        If False, data will be centered before computation.

    shrinkage : float, 0 <= shrinkage <= 1, default 0.1
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Attributes
    ----------
    location_ : array-like, shape (n_features,)
        Estimated location, i.e. the estimated mean.

    covariance_ : array-like, shape (n_features, n_features)
        Estimated covariance matrix

    precision_ : array-like, shape (n_features, n_features)
        Estimated pseudo inverse matrix.
        (stored only if store_precision is True)

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.covariance import ShrunkCovariance
    >>> from sklearn.datasets import make_gaussian_quantiles
    >>> real_cov = np.array([[.8, .3],
    ...                      [.3, .4]])
    >>> rng = np.random.RandomState(0)
    >>> X = rng.multivariate_normal(mean=[0, 0],
    ...                                   cov=real_cov,
    ...                                   size=500)
    >>> cov = ShrunkCovariance().fit(X)
    >>> cov.covariance_
    array([[0.7387..., 0.2536...],
           [0.2536..., 0.4110...]])
    >>> cov.location_
    array([0.0622..., 0.0193...])

    Notes
    -----
    The regularized covariance is given by:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features

    """
    def __init__(self, store_precision=True, assume_centered=False,
                 shrinkage=0.1):
        super().__init__(store_precision=store_precision,
                         assume_centered=assume_centered)
        self.shrinkage = shrinkage

    def fit(self, X, y=None):
        """ Fits the shrunk covariance model
        according to the given training data and parameters.

        Parameters
        ----------
        X : array-like, shape = [n_samples, n_features]
            Training data, where n_samples is the number of samples
            and n_features is the number of features.

        y
            not used, present for API consistence purpose.

        Returns
        -------
        self : object

        """
        X = check_array(X)
        # Not calling the parent object to fit, to avoid a potential
        # matrix inversion when setting the precision
        if self.assume_centered:
            self.location_ = np.zeros(X.shape[1])
        else:
            self.location_ = X.mean(0)
        covariance = empirical_covariance(
            X, assume_centered=self.assume_centered)
        covariance = shrunk_covariance(covariance, self.shrinkage)
        self._set_covariance(covariance)

        return self


# Ledoit-Wolf estimator

def ledoit_wolf_shrinkage(X, assume_centered=False, block_size=1000):
    """Estimates the shrunk Ledoit-Wolf covariance matrix.

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)
        Data from which to compute the Ledoit-Wolf shrunk covariance shrinkage.

    assume_centered : bool
        If True, data will not be centered before computation.
        Useful to work with data whose mean is significantly equal to
        zero but is not exactly zero.
        If False, data will be centered before computation.

    block_size : int
        Size of the blocks into which the covariance matrix will be split.

    Returns
    -------
    shrinkage : float
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Notes
    -----
    The regularized (shrunk) covariance is:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features

    """
    X = np.asarray(X)
    # for only one feature, the result is the same whatever the shrinkage
    if len(X.shape) == 2 and X.shape[1] == 1:
        return 0.
    if X.ndim == 1:
        X = np.reshape(X, (1, -1))

    if X.shape[0] == 1:
        warnings.warn("Only one sample available. "
                      "You may want to reshape your data array")
    n_samples, n_features = X.shape

    # optionally center data
    if not assume_centered:
        X = X - X.mean(0)

    # A non-blocked version of the computation is present in the tests
    # in tests/test_covariance.py

    # number of blocks to split the covariance matrix into
    n_splits = int(n_features / block_size)
    X2 = X ** 2
    emp_cov_trace = np.sum(X2, axis=0) / n_samples
    mu = np.sum(emp_cov_trace) / n_features
    beta_ = 0.  # sum of the coefficients of <X2.T, X2>
    delta_ = 0.  # sum of the *squared* coefficients of <X.T, X>
    # starting block computation
    for i in range(n_splits):
        for j in range(n_splits):
            rows = slice(block_size * i, block_size * (i + 1))
            cols = slice(block_size * j, block_size * (j + 1))
            beta_ += np.sum(np.dot(X2.T[rows], X2[:, cols]))
            delta_ += np.sum(np.dot(X.T[rows], X[:, cols]) ** 2)
        rows = slice(block_size * i, block_size * (i + 1))
        beta_ += np.sum(np.dot(X2.T[rows], X2[:, block_size * n_splits:]))
        delta_ += np.sum(
            np.dot(X.T[rows], X[:, block_size * n_splits:]) ** 2)
    for j in range(n_splits):
        cols = slice(block_size * j, block_size * (j + 1))
        beta_ += np.sum(np.dot(X2.T[block_size * n_splits:], X2[:, cols]))
        delta_ += np.sum(
            np.dot(X.T[block_size * n_splits:], X[:, cols]) ** 2)
    delta_ += np.sum(np.dot(X.T[block_size * n_splits:],
                            X[:, block_size * n_splits:]) ** 2)
    delta_ /= n_samples ** 2
    beta_ += np.sum(np.dot(X2.T[block_size * n_splits:],
                           X2[:, block_size * n_splits:]))
    # use delta_ to compute beta
    beta = 1. / (n_features * n_samples) * (beta_ / n_samples - delta_)
    # delta is the sum of the squared coefficients of (<X.T,X> - mu*Id) / p
    delta = delta_ - 2. * mu * emp_cov_trace.sum() + n_features * mu ** 2
    delta /= n_features
    # get final beta as the min between beta and delta
    # We do this to prevent shrinking more than "1", which whould invert
    # the value of covariances
    beta = min(beta, delta)
    # finally get shrinkage
    shrinkage = 0 if beta == 0 else beta / delta
    return shrinkage


def ledoit_wolf(X, assume_centered=False, block_size=1000):
    """Estimates the shrunk Ledoit-Wolf covariance matrix.

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)
        Data from which to compute the covariance estimate

    assume_centered : boolean, default=False
        If True, data will not be centered before computation.
        Useful to work with data whose mean is significantly equal to
        zero but is not exactly zero.
        If False, data will be centered before computation.

    block_size : int, default=1000
        Size of the blocks into which the covariance matrix will be split.
        This is purely a memory optimization and does not affect results.

    Returns
    -------
    shrunk_cov : array-like, shape (n_features, n_features)
        Shrunk covariance.

    shrinkage : float
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Notes
    -----
    The regularized (shrunk) covariance is:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features

    """
    X = np.asarray(X)
    # for only one feature, the result is the same whatever the shrinkage
    if len(X.shape) == 2 and X.shape[1] == 1:
        if not assume_centered:
            X = X - X.mean()
        return np.atleast_2d((X ** 2).mean()), 0.
    if X.ndim == 1:
        X = np.reshape(X, (1, -1))
        warnings.warn("Only one sample available. "
                      "You may want to reshape your data array")
        n_features = X.size
    else:
        _, n_features = X.shape

    # get Ledoit-Wolf shrinkage
    shrinkage = ledoit_wolf_shrinkage(
        X, assume_centered=assume_centered, block_size=block_size)
    emp_cov = empirical_covariance(X, assume_centered=assume_centered)
    mu = np.sum(np.trace(emp_cov)) / n_features
    shrunk_cov = (1. - shrinkage) * emp_cov
    shrunk_cov.flat[::n_features + 1] += shrinkage * mu

    return shrunk_cov, shrinkage


class LedoitWolf(EmpiricalCovariance):
    """LedoitWolf Estimator

    Ledoit-Wolf is a particular form of shrinkage, where the shrinkage
    coefficient is computed using O. Ledoit and M. Wolf's formula as
    described in "A Well-Conditioned Estimator for Large-Dimensional
    Covariance Matrices", Ledoit and Wolf, Journal of Multivariate
    Analysis, Volume 88, Issue 2, February 2004, pages 365-411.

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    Parameters
    ----------
    store_precision : bool, default=True
        Specify if the estimated precision is stored.

    assume_centered : bool, default=False
        If True, data will not be centered before computation.
        Useful when working with data whose mean is almost, but not exactly
        zero.
        If False (default), data will be centered before computation.

    block_size : int, default=1000
        Size of the blocks into which the covariance matrix will be split
        during its Ledoit-Wolf estimation. This is purely a memory
        optimization and does not affect results.

    Attributes
    ----------
    location_ : array-like, shape (n_features,)
        Estimated location, i.e. the estimated mean.

    covariance_ : array-like, shape (n_features, n_features)
        Estimated covariance matrix

    precision_ : array-like, shape (n_features, n_features)
        Estimated pseudo inverse matrix.
        (stored only if store_precision is True)

    shrinkage_ : float, 0 <= shrinkage <= 1
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.covariance import LedoitWolf
    >>> real_cov = np.array([[.4, .2],
    ...                      [.2, .8]])
    >>> np.random.seed(0)
    >>> X = np.random.multivariate_normal(mean=[0, 0],
    ...                                   cov=real_cov,
    ...                                   size=50)
    >>> cov = LedoitWolf().fit(X)
    >>> cov.covariance_
    array([[0.4406..., 0.1616...],
           [0.1616..., 0.8022...]])
    >>> cov.location_
    array([ 0.0595... , -0.0075...])

    Notes
    -----
    The regularised covariance is:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features
    and shrinkage is given by the Ledoit and Wolf formula (see References)

    References
    ----------
    "A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices",
    Ledoit and Wolf, Journal of Multivariate Analysis, Volume 88, Issue 2,
    February 2004, pages 365-411.

    """
    def __init__(self, store_precision=True, assume_centered=False,
                 block_size=1000):
        super().__init__(store_precision=store_precision,
                         assume_centered=assume_centered)
        self.block_size = block_size

    def fit(self, X, y=None):
        """ Fits the Ledoit-Wolf shrunk covariance model
        according to the given training data and parameters.

        Parameters
        ----------
        X : array-like, shape = [n_samples, n_features]
            Training data, where n_samples is the number of samples
            and n_features is the number of features.
        y
            not used, present for API consistence purpose.

        Returns
        -------
        self : object

        """
        # Not calling the parent object to fit, to avoid computing the
        # covariance matrix (and potentially the precision)
        X = check_array(X)
        if self.assume_centered:
            self.location_ = np.zeros(X.shape[1])
        else:
            self.location_ = X.mean(0)
        covariance, shrinkage = ledoit_wolf(X - self.location_,
                                            assume_centered=True,
                                            block_size=self.block_size)
        self.shrinkage_ = shrinkage
        self._set_covariance(covariance)

        return self


# OAS estimator

def oas(X, assume_centered=False):
    """Estimate covariance with the Oracle Approximating Shrinkage algorithm.

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)
        Data from which to compute the covariance estimate.

    assume_centered : boolean
      If True, data will not be centered before computation.
      Useful to work with data whose mean is significantly equal to
      zero but is not exactly zero.
      If False, data will be centered before computation.

    Returns
    -------
    shrunk_cov : array-like, shape (n_features, n_features)
        Shrunk covariance.

    shrinkage : float
        Coefficient in the convex combination used for the computation
        of the shrunk estimate.

    Notes
    -----
    The regularised (shrunk) covariance is:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features

    The formula we used to implement the OAS is slightly modified compared
    to the one given in the article. See :class:`OAS` for more details.

    """
    X = np.asarray(X)
    # for only one feature, the result is the same whatever the shrinkage
    if len(X.shape) == 2 and X.shape[1] == 1:
        if not assume_centered:
            X = X - X.mean()
        return np.atleast_2d((X ** 2).mean()), 0.
    if X.ndim == 1:
        X = np.reshape(X, (1, -1))
        warnings.warn("Only one sample available. "
                      "You may want to reshape your data array")
        n_samples = 1
        n_features = X.size
    else:
        n_samples, n_features = X.shape

    emp_cov = empirical_covariance(X, assume_centered=assume_centered)
    mu = np.trace(emp_cov) / n_features

    # formula from Chen et al.'s **implementation**
    alpha = np.mean(emp_cov ** 2)
    num = alpha + mu ** 2
    den = (n_samples + 1.) * (alpha - (mu ** 2) / n_features)

    shrinkage = 1. if den == 0 else min(num / den, 1.)
    shrunk_cov = (1. - shrinkage) * emp_cov
    shrunk_cov.flat[::n_features + 1] += shrinkage * mu

    return shrunk_cov, shrinkage


class OAS(EmpiricalCovariance):
    """Oracle Approximating Shrinkage Estimator

    Read more in the :ref:`User Guide <shrunk_covariance>`.

    OAS is a particular form of shrinkage described in
    "Shrinkage Algorithms for MMSE Covariance Estimation"
    Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.

    The formula used here does not correspond to the one given in the
    article. In the original article, formula (23) states that 2/p is
    multiplied by Trace(cov*cov) in both the numerator and denominator, but
    this operation is omitted because for a large p, the value of 2/p is
    so small that it doesn't affect the value of the estimator.

    Parameters
    ----------
    store_precision : bool, default=True
        Specify if the estimated precision is stored.

    assume_centered : bool, default=False
        If True, data will not be centered before computation.
        Useful when working with data whose mean is almost, but not exactly
        zero.
        If False (default), data will be centered before computation.

    Attributes
    ----------
    covariance_ : array-like, shape (n_features, n_features)
        Estimated covariance matrix.

    precision_ : array-like, shape (n_features, n_features)
        Estimated pseudo inverse matrix.
        (stored only if store_precision is True)

    shrinkage_ : float, 0 <= shrinkage <= 1
      coefficient in the convex combination used for the computation
      of the shrunk estimate.

    Notes
    -----
    The regularised covariance is:

    (1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

    where mu = trace(cov) / n_features
    and shrinkage is given by the OAS formula (see References)

    References
    ----------
    "Shrinkage Algorithms for MMSE Covariance Estimation"
    Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.

    """

    def fit(self, X, y=None):
        """ Fits the Oracle Approximating Shrinkage covariance model
        according to the given training data and parameters.

        Parameters
        ----------
        X : array-like, shape = [n_samples, n_features]
            Training data, where n_samples is the number of samples
            and n_features is the number of features.
        y
            not used, present for API consistence purpose.

        Returns
        -------
        self : object

        """
        X = check_array(X)
        # Not calling the parent object to fit, to avoid computing the
        # covariance matrix (and potentially the precision)
        if self.assume_centered:
            self.location_ = np.zeros(X.shape[1])
        else:
            self.location_ = X.mean(0)

        covariance, shrinkage = oas(X - self.location_, assume_centered=True)
        self.shrinkage_ = shrinkage
        self._set_covariance(covariance)

        return self