634 lines
22 KiB
Python
634 lines
22 KiB
Python
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from functools import cached_property
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import numpy as np
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from scipy import linalg
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from scipy.stats import _multivariate
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__all__ = ["Covariance"]
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class Covariance:
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"""
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Representation of a covariance matrix
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Calculations involving covariance matrices (e.g. data whitening,
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multivariate normal function evaluation) are often performed more
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efficiently using a decomposition of the covariance matrix instead of the
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covariance matrix itself. This class allows the user to construct an
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object representing a covariance matrix using any of several
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decompositions and perform calculations using a common interface.
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.. note::
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The `Covariance` class cannot be instantiated directly. Instead, use
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one of the factory methods (e.g. `Covariance.from_diagonal`).
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Examples
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--------
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The `Covariance` class is is used by calling one of its
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factory methods to create a `Covariance` object, then pass that
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representation of the `Covariance` matrix as a shape parameter of a
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multivariate distribution.
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For instance, the multivariate normal distribution can accept an array
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representing a covariance matrix:
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>>> from scipy import stats
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>>> import numpy as np
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>>> d = [1, 2, 3]
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>>> A = np.diag(d) # a diagonal covariance matrix
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>>> x = [4, -2, 5] # a point of interest
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>>> dist = stats.multivariate_normal(mean=[0, 0, 0], cov=A)
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>>> dist.pdf(x)
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4.9595685102808205e-08
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but the calculations are performed in a very generic way that does not
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take advantage of any special properties of the covariance matrix. Because
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our covariance matrix is diagonal, we can use ``Covariance.from_diagonal``
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to create an object representing the covariance matrix, and
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`multivariate_normal` can use this to compute the probability density
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function more efficiently.
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>>> cov = stats.Covariance.from_diagonal(d)
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>>> dist = stats.multivariate_normal(mean=[0, 0, 0], cov=cov)
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>>> dist.pdf(x)
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4.9595685102808205e-08
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"""
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def __init__(self):
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message = ("The `Covariance` class cannot be instantiated directly. "
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"Please use one of the factory methods "
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"(e.g. `Covariance.from_diagonal`).")
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raise NotImplementedError(message)
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@staticmethod
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def from_diagonal(diagonal):
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r"""
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Return a representation of a covariance matrix from its diagonal.
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Parameters
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----------
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diagonal : array_like
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The diagonal elements of a diagonal matrix.
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Notes
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-----
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Let the diagonal elements of a diagonal covariance matrix :math:`D` be
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stored in the vector :math:`d`.
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When all elements of :math:`d` are strictly positive, whitening of a
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data point :math:`x` is performed by computing
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:math:`x \cdot d^{-1/2}`, where the inverse square root can be taken
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element-wise.
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:math:`\log\det{D}` is calculated as :math:`-2 \sum(\log{d})`,
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where the :math:`\log` operation is performed element-wise.
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This `Covariance` class supports singular covariance matrices. When
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computing ``_log_pdet``, non-positive elements of :math:`d` are
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ignored. Whitening is not well defined when the point to be whitened
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does not lie in the span of the columns of the covariance matrix. The
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convention taken here is to treat the inverse square root of
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non-positive elements of :math:`d` as zeros.
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Examples
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--------
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Prepare a symmetric positive definite covariance matrix ``A`` and a
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data point ``x``.
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>>> import numpy as np
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>>> from scipy import stats
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>>> rng = np.random.default_rng()
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>>> n = 5
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>>> A = np.diag(rng.random(n))
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>>> x = rng.random(size=n)
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Extract the diagonal from ``A`` and create the `Covariance` object.
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>>> d = np.diag(A)
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>>> cov = stats.Covariance.from_diagonal(d)
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Compare the functionality of the `Covariance` object against a
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reference implementations.
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>>> res = cov.whiten(x)
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>>> ref = np.diag(d**-0.5) @ x
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>>> np.allclose(res, ref)
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True
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>>> res = cov.log_pdet
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>>> ref = np.linalg.slogdet(A)[-1]
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>>> np.allclose(res, ref)
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True
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"""
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return CovViaDiagonal(diagonal)
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@staticmethod
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def from_precision(precision, covariance=None):
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r"""
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Return a representation of a covariance from its precision matrix.
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Parameters
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----------
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precision : array_like
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The precision matrix; that is, the inverse of a square, symmetric,
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positive definite covariance matrix.
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covariance : array_like, optional
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The square, symmetric, positive definite covariance matrix. If not
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provided, this may need to be calculated (e.g. to evaluate the
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cumulative distribution function of
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`scipy.stats.multivariate_normal`) by inverting `precision`.
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Notes
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-----
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Let the covariance matrix be :math:`A`, its precision matrix be
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:math:`P = A^{-1}`, and :math:`L` be the lower Cholesky factor such
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that :math:`L L^T = P`.
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Whitening of a data point :math:`x` is performed by computing
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:math:`x^T L`. :math:`\log\det{A}` is calculated as
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:math:`-2tr(\log{L})`, where the :math:`\log` operation is performed
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element-wise.
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This `Covariance` class does not support singular covariance matrices
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because the precision matrix does not exist for a singular covariance
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matrix.
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Examples
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--------
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Prepare a symmetric positive definite precision matrix ``P`` and a
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data point ``x``. (If the precision matrix is not already available,
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consider the other factory methods of the `Covariance` class.)
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>>> import numpy as np
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>>> from scipy import stats
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>>> rng = np.random.default_rng()
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>>> n = 5
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>>> P = rng.random(size=(n, n))
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>>> P = P @ P.T # a precision matrix must be positive definite
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>>> x = rng.random(size=n)
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Create the `Covariance` object.
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>>> cov = stats.Covariance.from_precision(P)
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Compare the functionality of the `Covariance` object against
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reference implementations.
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>>> res = cov.whiten(x)
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>>> ref = x @ np.linalg.cholesky(P)
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>>> np.allclose(res, ref)
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True
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>>> res = cov.log_pdet
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>>> ref = -np.linalg.slogdet(P)[-1]
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>>> np.allclose(res, ref)
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True
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"""
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return CovViaPrecision(precision, covariance)
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@staticmethod
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def from_cholesky(cholesky):
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r"""
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Representation of a covariance provided via the (lower) Cholesky factor
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Parameters
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----------
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cholesky : array_like
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The lower triangular Cholesky factor of the covariance matrix.
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Notes
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-----
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Let the covariance matrix be :math:`A` and :math:`L` be the lower
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Cholesky factor such that :math:`L L^T = A`.
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Whitening of a data point :math:`x` is performed by computing
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:math:`L^{-1} x`. :math:`\log\det{A}` is calculated as
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:math:`2tr(\log{L})`, where the :math:`\log` operation is performed
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element-wise.
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This `Covariance` class does not support singular covariance matrices
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because the Cholesky decomposition does not exist for a singular
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covariance matrix.
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Examples
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--------
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Prepare a symmetric positive definite covariance matrix ``A`` and a
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data point ``x``.
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>>> import numpy as np
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>>> from scipy import stats
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>>> rng = np.random.default_rng()
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>>> n = 5
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>>> A = rng.random(size=(n, n))
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>>> A = A @ A.T # make the covariance symmetric positive definite
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>>> x = rng.random(size=n)
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Perform the Cholesky decomposition of ``A`` and create the
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`Covariance` object.
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>>> L = np.linalg.cholesky(A)
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>>> cov = stats.Covariance.from_cholesky(L)
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Compare the functionality of the `Covariance` object against
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reference implementation.
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>>> from scipy.linalg import solve_triangular
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>>> res = cov.whiten(x)
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>>> ref = solve_triangular(L, x, lower=True)
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>>> np.allclose(res, ref)
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True
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>>> res = cov.log_pdet
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>>> ref = np.linalg.slogdet(A)[-1]
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>>> np.allclose(res, ref)
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True
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"""
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return CovViaCholesky(cholesky)
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@staticmethod
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def from_eigendecomposition(eigendecomposition):
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r"""
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Representation of a covariance provided via eigendecomposition
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Parameters
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----------
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eigendecomposition : sequence
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A sequence (nominally a tuple) containing the eigenvalue and
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eigenvector arrays as computed by `scipy.linalg.eigh` or
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`numpy.linalg.eigh`.
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Notes
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-----
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Let the covariance matrix be :math:`A`, let :math:`V` be matrix of
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eigenvectors, and let :math:`W` be the diagonal matrix of eigenvalues
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such that `V W V^T = A`.
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When all of the eigenvalues are strictly positive, whitening of a
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data point :math:`x` is performed by computing
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:math:`x^T (V W^{-1/2})`, where the inverse square root can be taken
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element-wise.
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:math:`\log\det{A}` is calculated as :math:`tr(\log{W})`,
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where the :math:`\log` operation is performed element-wise.
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This `Covariance` class supports singular covariance matrices. When
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computing ``_log_pdet``, non-positive eigenvalues are ignored.
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Whitening is not well defined when the point to be whitened
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does not lie in the span of the columns of the covariance matrix. The
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convention taken here is to treat the inverse square root of
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non-positive eigenvalues as zeros.
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Examples
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--------
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Prepare a symmetric positive definite covariance matrix ``A`` and a
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data point ``x``.
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>>> import numpy as np
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>>> from scipy import stats
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>>> rng = np.random.default_rng()
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>>> n = 5
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>>> A = rng.random(size=(n, n))
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>>> A = A @ A.T # make the covariance symmetric positive definite
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>>> x = rng.random(size=n)
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Perform the eigendecomposition of ``A`` and create the `Covariance`
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object.
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>>> w, v = np.linalg.eigh(A)
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>>> cov = stats.Covariance.from_eigendecomposition((w, v))
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Compare the functionality of the `Covariance` object against
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reference implementations.
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>>> res = cov.whiten(x)
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>>> ref = x @ (v @ np.diag(w**-0.5))
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>>> np.allclose(res, ref)
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True
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>>> res = cov.log_pdet
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>>> ref = np.linalg.slogdet(A)[-1]
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>>> np.allclose(res, ref)
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True
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"""
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return CovViaEigendecomposition(eigendecomposition)
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def whiten(self, x):
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"""
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Perform a whitening transformation on data.
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"Whitening" ("white" as in "white noise", in which each frequency has
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equal magnitude) transforms a set of random variables into a new set of
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random variables with unit-diagonal covariance. When a whitening
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transform is applied to a sample of points distributed according to
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a multivariate normal distribution with zero mean, the covariance of
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the transformed sample is approximately the identity matrix.
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Parameters
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----------
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x : array_like
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An array of points. The last dimension must correspond with the
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dimensionality of the space, i.e., the number of columns in the
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covariance matrix.
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Returns
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-------
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x_ : array_like
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The transformed array of points.
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References
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----------
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.. [1] "Whitening Transformation". Wikipedia.
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https://en.wikipedia.org/wiki/Whitening_transformation
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.. [2] Novak, Lukas, and Miroslav Vorechovsky. "Generalization of
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coloring linear transformation". Transactions of VSB 18.2
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(2018): 31-35. :doi:`10.31490/tces-2018-0013`
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Examples
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--------
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>>> import numpy as np
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>>> from scipy import stats
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>>> rng = np.random.default_rng()
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>>> n = 3
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>>> A = rng.random(size=(n, n))
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>>> cov_array = A @ A.T # make matrix symmetric positive definite
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>>> precision = np.linalg.inv(cov_array)
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>>> cov_object = stats.Covariance.from_precision(precision)
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>>> x = rng.multivariate_normal(np.zeros(n), cov_array, size=(10000))
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>>> x_ = cov_object.whiten(x)
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>>> np.cov(x_, rowvar=False) # near-identity covariance
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array([[0.97862122, 0.00893147, 0.02430451],
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[0.00893147, 0.96719062, 0.02201312],
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[0.02430451, 0.02201312, 0.99206881]])
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"""
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return self._whiten(np.asarray(x))
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def colorize(self, x):
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"""
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Perform a colorizing transformation on data.
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"Colorizing" ("color" as in "colored noise", in which different
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frequencies may have different magnitudes) transforms a set of
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uncorrelated random variables into a new set of random variables with
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the desired covariance. When a coloring transform is applied to a
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sample of points distributed according to a multivariate normal
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distribution with identity covariance and zero mean, the covariance of
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the transformed sample is approximately the covariance matrix used
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in the coloring transform.
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Parameters
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----------
|
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|
|
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.
|
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|
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Returns
|
||
|
|
-------
|
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|
|
x_ : array_like
|
||
|
|
The transformed array of points.
|
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|
|
|
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|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] "Whitening Transformation". Wikipedia.
|
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|
|
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`
|
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|
|
|
||
|
|
Examples
|
||
|
|
--------
|
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|
|
>>> import numpy as np
|
||
|
|
>>> from scipy import stats
|
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|
|
>>> rng = np.random.default_rng(1638083107694713882823079058616272161)
|
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|
|
>>> 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)
|