venv/lib/python3.11/site-packages/scipy/stats/_covariance.py

from functools import cached_property

import numpy as np
from scipy import linalg
from scipy.stats import _multivariate


__all__ = ["Covariance"]


class Covariance:
    """
    Representation of a covariance matrix

    Calculations involving covariance matrices (e.g. data whitening,
    multivariate normal function evaluation) are often performed more
    efficiently using a decomposition of the covariance matrix instead of the
    covariance matrix itself. This class allows the user to construct an
    object representing a covariance matrix using any of several
    decompositions and perform calculations using a common interface.

    .. note::

        The `Covariance` class cannot be instantiated directly. Instead, use
        one of the factory methods (e.g. `Covariance.from_diagonal`).

    Examples
    --------
    The `Covariance` class is used by calling one of its
    factory methods to create a `Covariance` object, then pass that
    representation of the `Covariance` matrix as a shape parameter of a
    multivariate distribution.

    For instance, the multivariate normal distribution can accept an array
    representing a covariance matrix:

    >>> from scipy import stats
    >>> import numpy as np
    >>> d = [1, 2, 3]
    >>> A = np.diag(d)  # a diagonal covariance matrix
    >>> x = [4, -2, 5]  # a point of interest
    >>> dist = stats.multivariate_normal(mean=[0, 0, 0], cov=A)
    >>> dist.pdf(x)
    4.9595685102808205e-08

    but the calculations are performed in a very generic way that does not
    take advantage of any special properties of the covariance matrix. Because
    our covariance matrix is diagonal, we can use ``Covariance.from_diagonal``
    to create an object representing the covariance matrix, and
    `multivariate_normal` can use this to compute the probability density
    function more efficiently.

    >>> cov = stats.Covariance.from_diagonal(d)
    >>> dist = stats.multivariate_normal(mean=[0, 0, 0], cov=cov)
    >>> dist.pdf(x)
    4.9595685102808205e-08

    """
    def __init__(self):
        message = ("The `Covariance` class cannot be instantiated directly. "
                   "Please use one of the factory methods "
                   "(e.g. `Covariance.from_diagonal`).")
        raise NotImplementedError(message)

    @staticmethod
    def from_diagonal(diagonal):
        r"""
        Return a representation of a covariance matrix from its diagonal.

        Parameters
        ----------
        diagonal : array_like
            The diagonal elements of a diagonal matrix.

        Notes
        -----
        Let the diagonal elements of a diagonal covariance matrix :math:`D` be
        stored in the vector :math:`d`.

        When all elements of :math:`d` are strictly positive, whitening of a
        data point :math:`x` is performed by computing
        :math:`x \cdot d^{-1/2}`, where the inverse square root can be taken
        element-wise.
        :math:`\log\det{D}` is calculated as :math:`-2 \sum(\log{d})`,
        where the :math:`\log` operation is performed element-wise.

        This `Covariance` class supports singular covariance matrices. When
        computing ``_log_pdet``, non-positive elements of :math:`d` are
        ignored. Whitening is not well defined when the point to be whitened
        does not lie in the span of the columns of the covariance matrix. The
        convention taken here is to treat the inverse square root of
        non-positive elements of :math:`d` as zeros.

        Examples
        --------
        Prepare a symmetric positive definite covariance matrix ``A`` and a
        data point ``x``.

        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng()
        >>> n = 5
        >>> A = np.diag(rng.random(n))
        >>> x = rng.random(size=n)

        Extract the diagonal from ``A`` and create the `Covariance` object.

        >>> d = np.diag(A)
        >>> cov = stats.Covariance.from_diagonal(d)

        Compare the functionality of the `Covariance` object against a
        reference implementations.

        >>> res = cov.whiten(x)
        >>> ref = np.diag(d**-0.5) @ x
        >>> np.allclose(res, ref)
        True
        >>> res = cov.log_pdet
        >>> ref = np.linalg.slogdet(A)[-1]
        >>> np.allclose(res, ref)
        True

        """
        return CovViaDiagonal(diagonal)

    @staticmethod
    def from_precision(precision, covariance=None):
        r"""
        Return a representation of a covariance from its precision matrix.

        Parameters
        ----------
        precision : array_like
            The precision matrix; that is, the inverse of a square, symmetric,
            positive definite covariance matrix.
        covariance : array_like, optional
            The square, symmetric, positive definite covariance matrix. If not
            provided, this may need to be calculated (e.g. to evaluate the
            cumulative distribution function of
            `scipy.stats.multivariate_normal`) by inverting `precision`.

        Notes
        -----
        Let the covariance matrix be :math:`A`, its precision matrix be
        :math:`P = A^{-1}`, and :math:`L` be the lower Cholesky factor such
        that :math:`L L^T = P`.
        Whitening of a data point :math:`x` is performed by computing
        :math:`x^T L`. :math:`\log\det{A}` is calculated as
        :math:`-2tr(\log{L})`, where the :math:`\log` operation is performed
        element-wise.

        This `Covariance` class does not support singular covariance matrices
        because the precision matrix does not exist for a singular covariance
        matrix.

        Examples
        --------
        Prepare a symmetric positive definite precision matrix ``P`` and a
        data point ``x``. (If the precision matrix is not already available,
        consider the other factory methods of the `Covariance` class.)

        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng()
        >>> n = 5
        >>> P = rng.random(size=(n, n))
        >>> P = P @ P.T  # a precision matrix must be positive definite
        >>> x = rng.random(size=n)

        Create the `Covariance` object.

        >>> cov = stats.Covariance.from_precision(P)

        Compare the functionality of the `Covariance` object against
        reference implementations.

        >>> res = cov.whiten(x)
        >>> ref = x @ np.linalg.cholesky(P)
        >>> np.allclose(res, ref)
        True
        >>> res = cov.log_pdet
        >>> ref = -np.linalg.slogdet(P)[-1]
        >>> np.allclose(res, ref)
        True

        """
        return CovViaPrecision(precision, covariance)

    @staticmethod
    def from_cholesky(cholesky):
        r"""
        Representation of a covariance provided via the (lower) Cholesky factor

        Parameters
        ----------
        cholesky : array_like
            The lower triangular Cholesky factor of the covariance matrix.

        Notes
        -----
        Let the covariance matrix be :math:`A` and :math:`L` be the lower
        Cholesky factor such that :math:`L L^T = A`.
        Whitening of a data point :math:`x` is performed by computing
        :math:`L^{-1} x`. :math:`\log\det{A}` is calculated as
        :math:`2tr(\log{L})`, where the :math:`\log` operation is performed
        element-wise.

        This `Covariance` class does not support singular covariance matrices
        because the Cholesky decomposition does not exist for a singular
        covariance matrix.

        Examples
        --------
        Prepare a symmetric positive definite covariance matrix ``A`` and a
        data point ``x``.

        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng()
        >>> n = 5
        >>> A = rng.random(size=(n, n))
        >>> A = A @ A.T  # make the covariance symmetric positive definite
        >>> x = rng.random(size=n)

        Perform the Cholesky decomposition of ``A`` and create the
        `Covariance` object.

        >>> L = np.linalg.cholesky(A)
        >>> cov = stats.Covariance.from_cholesky(L)

        Compare the functionality of the `Covariance` object against
        reference implementation.

        >>> from scipy.linalg import solve_triangular
        >>> res = cov.whiten(x)
        >>> ref = solve_triangular(L, x, lower=True)
        >>> np.allclose(res, ref)
        True
        >>> res = cov.log_pdet
        >>> ref = np.linalg.slogdet(A)[-1]
        >>> np.allclose(res, ref)
        True

        """
        return CovViaCholesky(cholesky)

    @staticmethod
    def from_eigendecomposition(eigendecomposition):
        r"""
        Representation of a covariance provided via eigendecomposition

        Parameters
        ----------
        eigendecomposition : sequence
            A sequence (nominally a tuple) containing the eigenvalue and
            eigenvector arrays as computed by `scipy.linalg.eigh` or
            `numpy.linalg.eigh`.

        Notes
        -----
        Let the covariance matrix be :math:`A`, let :math:`V` be matrix of
        eigenvectors, and let :math:`W` be the diagonal matrix of eigenvalues
        such that `V W V^T = A`.

        When all of the eigenvalues are strictly positive, whitening of a
        data point :math:`x` is performed by computing
        :math:`x^T (V W^{-1/2})`, where the inverse square root can be taken
        element-wise.
        :math:`\log\det{A}` is calculated as  :math:`tr(\log{W})`,
        where the :math:`\log` operation is performed element-wise.

        This `Covariance` class supports singular covariance matrices. When
        computing ``_log_pdet``, non-positive eigenvalues are ignored.
        Whitening is not well defined when the point to be whitened
        does not lie in the span of the columns of the covariance matrix. The
        convention taken here is to treat the inverse square root of
        non-positive eigenvalues as zeros.

        Examples
        --------
        Prepare a symmetric positive definite covariance matrix ``A`` and a
        data point ``x``.

        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng()
        >>> n = 5
        >>> A = rng.random(size=(n, n))
        >>> A = A @ A.T  # make the covariance symmetric positive definite
        >>> x = rng.random(size=n)

        Perform the eigendecomposition of ``A`` and create the `Covariance`
        object.

        >>> w, v = np.linalg.eigh(A)
        >>> cov = stats.Covariance.from_eigendecomposition((w, v))

        Compare the functionality of the `Covariance` object against
        reference implementations.

        >>> res = cov.whiten(x)
        >>> ref = x @ (v @ np.diag(w**-0.5))
        >>> np.allclose(res, ref)
        True
        >>> res = cov.log_pdet
        >>> ref = np.linalg.slogdet(A)[-1]
        >>> np.allclose(res, ref)
        True

        """
        return CovViaEigendecomposition(eigendecomposition)

    def whiten(self, x):
        """
        Perform a whitening transformation on data.

        "Whitening" ("white" as in "white noise", in which each frequency has
        equal magnitude) transforms a set of random variables into a new set of
        random variables with unit-diagonal covariance. When a whitening
        transform is applied to a sample of points distributed according to
        a multivariate normal distribution with zero mean, the covariance of
        the transformed sample is approximately the identity matrix.

        Parameters
        ----------
        x : array_like
            An array of points. The last dimension must correspond with the
            dimensionality of the space, i.e., the number of columns in the
            covariance matrix.

        Returns
        -------
        x_ : array_like
            The transformed array of points.

        References
        ----------
        .. [1] "Whitening Transformation". Wikipedia.
               https://en.wikipedia.org/wiki/Whitening_transformation
        .. [2] Novak, Lukas, and Miroslav Vorechovsky. "Generalization of
               coloring linear transformation". Transactions of VSB 18.2
               (2018): 31-35. :doi:`10.31490/tces-2018-0013`

        Examples
        --------
        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng()
        >>> n = 3
        >>> A = rng.random(size=(n, n))
        >>> cov_array = A @ A.T  # make matrix symmetric positive definite
        >>> precision = np.linalg.inv(cov_array)
        >>> cov_object = stats.Covariance.from_precision(precision)
        >>> x = rng.multivariate_normal(np.zeros(n), cov_array, size=(10000))
        >>> x_ = cov_object.whiten(x)
        >>> np.cov(x_, rowvar=False)  # near-identity covariance
        array([[0.97862122, 0.00893147, 0.02430451],
               [0.00893147, 0.96719062, 0.02201312],
               [0.02430451, 0.02201312, 0.99206881]])

        """
        return self._whiten(np.asarray(x))

    def colorize(self, x):
        """
        Perform a colorizing transformation on data.

        "Colorizing" ("color" as in "colored noise", in which different
        frequencies may have different magnitudes) transforms a set of
        uncorrelated random variables into a new set of random variables with
        the desired covariance. When a coloring transform is applied to a
        sample of points distributed according to a multivariate normal
        distribution with identity covariance and zero mean, the covariance of
        the transformed sample is approximately the covariance matrix used
        in the coloring transform.

        Parameters
        ----------
        x : array_like
            An array of points. The last dimension must correspond with the
            dimensionality of the space, i.e., the number of columns in the
            covariance matrix.

        Returns
        -------
        x_ : array_like
            The transformed array of points.

        References
        ----------
        .. [1] "Whitening Transformation". Wikipedia.
               https://en.wikipedia.org/wiki/Whitening_transformation
        .. [2] Novak, Lukas, and Miroslav Vorechovsky. "Generalization of
               coloring linear transformation". Transactions of VSB 18.2
               (2018): 31-35. :doi:`10.31490/tces-2018-0013`

        Examples
        --------
        >>> import numpy as np
        >>> from scipy import stats
        >>> rng = np.random.default_rng(1638083107694713882823079058616272161)
        >>> n = 3
        >>> A = rng.random(size=(n, n))
        >>> cov_array = A @ A.T  # make matrix symmetric positive definite
        >>> cholesky = np.linalg.cholesky(cov_array)
        >>> cov_object = stats.Covariance.from_cholesky(cholesky)
        >>> x = rng.multivariate_normal(np.zeros(n), np.eye(n), size=(10000))
        >>> x_ = cov_object.colorize(x)
        >>> cov_data = np.cov(x_, rowvar=False)
        >>> np.allclose(cov_data, cov_array, rtol=3e-2)
        True
        """
        return self._colorize(np.asarray(x))

    @property
    def log_pdet(self):
        """
        Log of the pseudo-determinant of the covariance matrix
        """
        return np.array(self._log_pdet, dtype=float)[()]

    @property
    def rank(self):
        """
        Rank of the covariance matrix
        """
        return np.array(self._rank, dtype=int)[()]

    @property
    def covariance(self):
        """
        Explicit representation of the covariance matrix
        """
        return self._covariance

    @property
    def shape(self):
        """
        Shape of the covariance array
        """
        return self._shape

    def _validate_matrix(self, A, name):
        A = np.atleast_2d(A)
        m, n = A.shape[-2:]
        if m != n or A.ndim != 2 or not (np.issubdtype(A.dtype, np.integer) or
                                         np.issubdtype(A.dtype, np.floating)):
            message = (f"The input `{name}` must be a square, "
                       "two-dimensional array of real numbers.")
            raise ValueError(message)
        return A

    def _validate_vector(self, A, name):
        A = np.atleast_1d(A)
        if A.ndim != 1 or not (np.issubdtype(A.dtype, np.integer) or
                               np.issubdtype(A.dtype, np.floating)):
            message = (f"The input `{name}` must be a one-dimensional array "
                       "of real numbers.")
            raise ValueError(message)
        return A


class CovViaPrecision(Covariance):

    def __init__(self, precision, covariance=None):
        precision = self._validate_matrix(precision, 'precision')
        if covariance is not None:
            covariance = self._validate_matrix(covariance, 'covariance')
            message = "`precision.shape` must equal `covariance.shape`."
            if precision.shape != covariance.shape:
                raise ValueError(message)

        self._chol_P = np.linalg.cholesky(precision)
        self._log_pdet = -2*np.log(np.diag(self._chol_P)).sum(axis=-1)
        self._rank = precision.shape[-1]  # must be full rank if invertible
        self._precision = precision
        self._cov_matrix = covariance
        self._shape = precision.shape
        self._allow_singular = False

    def _whiten(self, x):
        return x @ self._chol_P

    @cached_property
    def _covariance(self):
        n = self._shape[-1]
        return (linalg.cho_solve((self._chol_P, True), np.eye(n))
                if self._cov_matrix is None else self._cov_matrix)

    def _colorize(self, x):
        return linalg.solve_triangular(self._chol_P.T, x.T, lower=False).T


def _dot_diag(x, d):
    # If d were a full diagonal matrix, x @ d would always do what we want.
    # Special treatment is needed for n-dimensional `d` in which each row
    # includes only the diagonal elements of a covariance matrix.
    return x * d if x.ndim < 2 else x * np.expand_dims(d, -2)


class CovViaDiagonal(Covariance):

    def __init__(self, diagonal):
        diagonal = self._validate_vector(diagonal, 'diagonal')

        i_zero = diagonal <= 0
        positive_diagonal = np.array(diagonal, dtype=np.float64)

        positive_diagonal[i_zero] = 1  # ones don't affect determinant
        self._log_pdet = np.sum(np.log(positive_diagonal), axis=-1)

        psuedo_reciprocals = 1 / np.sqrt(positive_diagonal)
        psuedo_reciprocals[i_zero] = 0

        self._sqrt_diagonal = np.sqrt(diagonal)
        self._LP = psuedo_reciprocals
        self._rank = positive_diagonal.shape[-1] - i_zero.sum(axis=-1)
        self._covariance = np.apply_along_axis(np.diag, -1, diagonal)
        self._i_zero = i_zero
        self._shape = self._covariance.shape
        self._allow_singular = True

    def _whiten(self, x):
        return _dot_diag(x, self._LP)

    def _colorize(self, x):
        return _dot_diag(x, self._sqrt_diagonal)

    def _support_mask(self, x):
        """
        Check whether x lies in the support of the distribution.
        """
        return ~np.any(_dot_diag(x, self._i_zero), axis=-1)


class CovViaCholesky(Covariance):

    def __init__(self, cholesky):
        L = self._validate_matrix(cholesky, 'cholesky')

        self._factor = L
        self._log_pdet = 2*np.log(np.diag(self._factor)).sum(axis=-1)
        self._rank = L.shape[-1]  # must be full rank for cholesky
        self._shape = L.shape
        self._allow_singular = False

    @cached_property
    def _covariance(self):
        return self._factor @ self._factor.T

    def _whiten(self, x):
        res = linalg.solve_triangular(self._factor, x.T, lower=True).T
        return res

    def _colorize(self, x):
        return x @ self._factor.T


class CovViaEigendecomposition(Covariance):

    def __init__(self, eigendecomposition):
        eigenvalues, eigenvectors = eigendecomposition
        eigenvalues = self._validate_vector(eigenvalues, 'eigenvalues')
        eigenvectors = self._validate_matrix(eigenvectors, 'eigenvectors')
        message = ("The shapes of `eigenvalues` and `eigenvectors` "
                   "must be compatible.")
        try:
            eigenvalues = np.expand_dims(eigenvalues, -2)
            eigenvectors, eigenvalues = np.broadcast_arrays(eigenvectors,
                                                            eigenvalues)
            eigenvalues = eigenvalues[..., 0, :]
        except ValueError:
            raise ValueError(message)

        i_zero = eigenvalues <= 0
        positive_eigenvalues = np.array(eigenvalues, dtype=np.float64)

        positive_eigenvalues[i_zero] = 1  # ones don't affect determinant
        self._log_pdet = np.sum(np.log(positive_eigenvalues), axis=-1)

        psuedo_reciprocals = 1 / np.sqrt(positive_eigenvalues)
        psuedo_reciprocals[i_zero] = 0

        self._LP = eigenvectors * psuedo_reciprocals
        self._LA = eigenvectors * np.sqrt(eigenvalues)
        self._rank = positive_eigenvalues.shape[-1] - i_zero.sum(axis=-1)
        self._w = eigenvalues
        self._v = eigenvectors
        self._shape = eigenvectors.shape
        self._null_basis = eigenvectors * i_zero
        # This is only used for `_support_mask`, not to decide whether
        # the covariance is singular or not.
        self._eps = _multivariate._eigvalsh_to_eps(eigenvalues) * 10**3
        self._allow_singular = True

    def _whiten(self, x):
        return x @ self._LP

    def _colorize(self, x):
        return x @ self._LA.T

    @cached_property
    def _covariance(self):
        return (self._v * self._w) @ self._v.T

    def _support_mask(self, x):
        """
        Check whether x lies in the support of the distribution.
        """
        residual = np.linalg.norm(x @ self._null_basis, axis=-1)
        in_support = residual < self._eps
        return in_support


class CovViaPSD(Covariance):
    """
    Representation of a covariance provided via an instance of _PSD
    """

    def __init__(self, psd):
        self._LP = psd.U
        self._log_pdet = psd.log_pdet
        self._rank = psd.rank
        self._covariance = psd._M
        self._shape = psd._M.shape
        self._psd = psd
        self._allow_singular = False  # by default

    def _whiten(self, x):
        return x @ self._LP

    def _support_mask(self, x):
        return self._psd._support_mask(x)