def update(self, x, weights=None):
'''Batch-update the properties for the current distribution.
This assumes that all samples are being provided at once.
Parameters
----------
x : numpy.ndarray
a DxN array containing the dataset used to update the
distribution's parameters
weights : numpy.ndarray
a 1xN array containing the relative weighting of each component for
this distribution; if not provided then assumed to all be '1'
'''
self._samples = x.shape[1]
if weights is None:
weights = np.ones((1, self._samples))
# Compute new means.
Nk = np.sum(weights)
self.mu = np.sum(weights * x, axis=1)[:, np.newaxis] / Nk
# Compute new covariances.
Sigma = linalg.weighted_scatter_matrix(weights, x, self.mu) / Nk
self.L = cholesky.from_matrix(Sigma)
# Compute the new model weight.
self.weight = Nk / self._samples
# Update any variables that don't change during a log-likelihood
# estimate.
detL = 2.0 * np.sum(np.log(np.diag(self.L)))
self._const = -(np.log(2 * np.pi) * x.shape[0] + detL) / 2.0
def __init__(self, alpha, d, num_models, k=1, mu=None, Sigma=None):
'''
Parameters
----------
alpha : float
concentration parameter
d : float
dimensionality of the data
num_models : int
number of total models
k : float
initial scaling factor
mu : numpy.ndarray
initial prior on the distribution mean; if unknown then it is set
to zero
Sigma : numpy.ndarray
initial prior on the covariance; if unknown then it is set to
identity
'''
super().__init__()
# Internal variables
self._num_models = num_models
self._alpha0 = alpha
self._beta0 = k
self._nu0 = d + 1
if mu is None:
self._mu0 = np.zeros((d, 1))
else:
if mu.shape[0] != d:
raise ValueError(
'Mean prior should have a dimensionality of %d.' % d)
self._mu0 = mu.copy()
# Note: this stores the *covariance*, not the precision even though
# that's how the Wishart distribution is defined. The math in the
# update equations is actually a little bit cleaner if the Cholesky
# decomposition of the covariance matrix is used (basically it ends up
# being nearly the same as the Gaussian version).
if Sigma is None:
self._Sigma0 = np.eye(d)
self._L0 = np.eye(d)
else:
if Sigma.shape[0] != d and Sigma.shape[1] != d:
raise ValueError('Covariance prior should be a %dx%d matrix.' %
(d, d))
self._Sigma0 = Sigma.copy()
self._L0 = cholesky.from_matrix(self._Sigma0)
# Public attributes
self.alpha = self._alpha0
self.alpha_sum = 0
self.beta = self._beta0
self.nu = self._nu0
self.mu = self._mu0.copy()
self.L = self._L0.copy()
def test_decomp():
'''Cholesky decomposition is called correctly.'''
np.random.seed(1234)
# Generate a positive-definite matrix.
x = np.random.uniform(size=(5, 100))
S = x @ x.T
Lwrapper = cholesky.from_matrix(S)
Lscipy = linalg.cholesky(S, lower=True)
assert np.linalg.norm(Lwrapper - Lscipy) == pytest.approx(0)
def test_reconst():
'''Matrix is recovered from a Cholesky decomposition.'''
np.random.seed(1234)
# Generate a positive-definite matrix.
x = np.random.uniform(size=(5, 100))
S = x @ x.T
L = cholesky.from_matrix(S)
T = cholesky.to_matrix(L)
print(S)
print(L)
print(T)
assert np.linalg.norm(S - T) == pytest.approx(0)
def update(self, x, weights=None):
'''Batch-update the properties for the current distribution.
This assumes that all samples are being provided at once.
Parameters
----------
x : numpy.ndarray
a DxN array containing the dataset used to update the
distribution's parameters
weights : numpy.ndarray
a 1xN array containing the relative weighting of each component for
this distribution; if not provided then assumed to all be '1'
'''
self._samples = x.shape[1]
# print('Samples: %d' % self._samples)
if weights is None:
weights = np.ones((1, self._samples)) / self._num_models
# NOTE All equations are from the cited resource.
# "Pattern Recognition and Machine Learning". Bishop. 2006
# Gaussian Update
Nk = np.sum(weights) + np.finfo(float).eps # (10.51)
x_mean = np.sum(weights * x, axis=1)[:, np.newaxis] / Nk # (10.52)
Sk = linalg.weighted_scatter_matrix(weights, x, x_mean) / Nk # (10.53)
# Dirichlet Update
self.alpha = self._alpha0 + Nk # (10.58)
# Wishart Update
self.nu = self._nu0 + Nk # (10.63)
self.beta = self._beta0 + Nk # (10.60)
self.mu = (self._beta0 * self._mu0 +
Nk * x_mean) / self.beta # (10.61)
delta = x_mean - self._mu0
Winv = (self._Sigma0 + Nk * Sk + (self._beta0 * Nk / self.beta) *
(delta @ delta.T))
self.L = cholesky.from_matrix(Winv) # (10.62)
# Update any variables that don't change during a log-likelihood
# estimate.
self._detW = -2.0 * np.sum(np.log(np.diag(self.L)))
def __init__(self, mu=None, Sigma=None):
'''
Parameters
----------
mu : numpy.ndarray
distribution mean
Sigma : numpy.ndarray
distribution covariance
'''
super().__init__()
if mu is None:
mu = np.zeros((1, 1))
if Sigma is None:
Sigma = np.eye(1)
self.L = cholesky.from_matrix(Sigma)
self.mu = mu.copy()
detL = 2.0 * np.sum(np.log(np.diag(self.L)))
self._const = -(np.log(2 * np.pi) * self.mu.shape[0] + detL) / 2.0
def test_invalid_decomp():
'''Invalid Cholesky decomposition raises an exception.'''
S = np.zeros((3, 3))
with pytest.raises(LinearAlgebraError):
cholesky.from_matrix(S)