def _contour_t(mu, Cov, nu, axes=None, scale=4, transpose=False, colors='k'):
"""
"""
if axes is None:
axes = plt.gca()
if np.shape(mu) != (2, ) or np.shape(Cov) != (2, 2) or np.shape(nu) != ():
print(np.shape(mu), np.shape(Cov), np.shape(nu))
raise ValueError("Only 2-d t-distribution allowed")
if transpose:
mu = mu[[1, 0]]
Cov = Cov[np.ix_([1, 0], [1, 0])]
s = np.sqrt(np.diag(Cov))
x0 = np.linspace(mu[0] - scale * s[0], mu[0] + scale * s[0], num=100)
x1 = np.linspace(mu[1] - scale * s[1], mu[1] + scale * s[1], num=100)
X0X1 = misc.grid(x0, x1)
Y = X0X1 - mu
L = linalg.chol(Cov)
logdet_Cov = linalg.chol_logdet(L)
Z = linalg.chol_solve(L, Y)
Z = linalg.inner(Y, Z, ndim=1)
lpdf = random.t_logpdf(Z, logdet_Cov, nu, 2)
p = np.exp(lpdf)
shape = (np.size(x0), np.size(x1))
X0 = np.reshape(X0X1[:, 0], shape)
X1 = np.reshape(X0X1[:, 1], shape)
P = np.reshape(p, shape)
return axes.contour(X0, X1, P, colors=colors)
def gaussian_mixture_logpdf(x, w, mu, Sigma):
# Shape(x) = (N, D)
# Shape(w) = (K,)
# Shape(mu) = (K, D)
# Shape(Sigma) = (K, D, D)
# Shape(result) = (N,)
# Dimensionality
D = np.shape(x)[-1]
# Cholesky decomposition of the covariance matrix
U = linalg.chol(Sigma)
# Reshape x:
# Shape(x) = (N, 1, D)
x = np.expand_dims(x, axis=-2)
# (x-mu) and (x-mu)'*inv(Sigma)*(x-mu):
# Shape(v) = (N, K, D)
# Shape(z) = (N, K)
v = x - mu
z = np.einsum('...i,...i', v, linalg.chol_solve(U, v))
# Log-determinant of Sigma:
# Shape(ldet) = (K,)
ldet = linalg.chol_logdet(U)
# Compute log pdf for each cluster:
# Shape(lpdf) = (N, K)
lpdf = misc.gaussian_logpdf(z, 0, 0, ldet, D)
def gaussian_mixture_logpdf(x, w, mu, Sigma):
# Shape(x) = (N, D)
# Shape(w) = (K,)
# Shape(mu) = (K, D)
# Shape(Sigma) = (K, D, D)
# Shape(result) = (N,)
# Dimensionality
D = np.shape(x)[-1]
# Cholesky decomposition of the covariance matrix
U = linalg.chol(Sigma)
# Reshape x:
# Shape(x) = (N, 1, D)
x = np.expand_dims(x, axis=-2)
# (x-mu) and (x-mu)'*inv(Sigma)*(x-mu):
# Shape(v) = (N, K, D)
# Shape(z) = (N, K)
v = x - mu
z = np.einsum('...i,...i', v, linalg.chol_solve(U, v))
# Log-determinant of Sigma:
# Shape(ldet) = (K,)
ldet = linalg.chol_logdet(U)
# Compute log pdf for each cluster:
# Shape(lpdf) = (N, K)
lpdf = misc.gaussian_logpdf(z, 0, 0, ldet, D)
def _contour_t(mu, Cov, nu, axes=None, scale=4, transpose=False, colors='k'):
"""
"""
if axes is None:
axes = plt.gca()
if np.shape(mu) != (2,) or np.shape(Cov) != (2,2) or np.shape(nu) != ():
print(np.shape(mu), np.shape(Cov), np.shape(nu))
raise ValueError("Only 2-d t-distribution allowed")
if transpose:
mu = mu[[1,0]]
Cov = Cov[np.ix_([1,0],[1,0])]
s = np.sqrt(np.diag(Cov))
x0 = np.linspace(mu[0]-scale*s[0], mu[0]+scale*s[0], num=100)
x1 = np.linspace(mu[1]-scale*s[1], mu[1]+scale*s[1], num=100)
X0X1 = misc.grid(x0, x1)
Y = X0X1 - mu
L = linalg.chol(Cov)
logdet_Cov = linalg.chol_logdet(L)
Z = linalg.chol_solve(L, Y)
Z = linalg.inner(Y, Z, ndim=1)
lpdf = random.t_logpdf(Z, logdet_Cov, nu, 2)
p = np.exp(lpdf)
shape = (np.size(x0), np.size(x1))
X0 = np.reshape(X0X1[:,0], shape)
X1 = np.reshape(X0X1[:,1], shape)
P = np.reshape(p, shape)
return axes.contour(X0, X1, P, colors=colors)
def compute_fixed_moments(self, Lambda, gradient=None):
""" Compute moments for fixed x. """
L = linalg.chol(Lambda, ndim=self.ndim)
ldet = linalg.chol_logdet(L, ndim=self.ndim)
u = [Lambda, ldet]
if gradient is None:
return u
du0 = gradient[0]
du1 = (misc.add_trailing_axes(gradient[1], 2 * self.ndim) *
linalg.chol_inv(L, ndim=self.ndim))
du = du0 + du1
return (u, du)
def compute_fixed_moments_and_f(self, Lambda, mask=True):
r"""
Compute u(x) and f(x) for given x.
.. math:
u(\Lambda) =
\begin{bmatrix}
\Lambda
\\
\log |\Lambda|
\end{bmatrix}
"""
k = np.shape(Lambda)[-1]
ldet = linalg.chol_logdet(linalg.chol(Lambda))
u = [Lambda, ldet]
f = -(k + 1) / 2 * ldet
return (u, f)
def compute_fixed_moments(self, Lambda, gradient=None):
""" Compute moments for fixed x. """
L = linalg.chol(Lambda, ndim=self.ndim)
ldet = linalg.chol_logdet(L, ndim=self.ndim)
u = [Lambda,
ldet]
if gradient is None:
return u
du0 = gradient[0]
du1 = (
misc.add_trailing_axes(gradient[1], 2*self.ndim)
* linalg.chol_inv(L, ndim=self.ndim)
)
du = du0 + du1
return (u, du)
def compute_fixed_moments_and_f(self, Lambda, mask=True):
r"""
Compute u(x) and f(x) for given x.
.. math:
u(\Lambda) =
\begin{bmatrix}
\Lambda
\\
\log |\Lambda|
\end{bmatrix}
"""
k = np.shape(Lambda)[-1]
ldet = linalg.chol_logdet(linalg.chol(Lambda))
u = [Lambda,
ldet]
f = -(k+1)/2 * ldet
return (u, f)
def compute_moments_and_cgf(self, phi, mask=True):
r"""
Return moments and cgf for given natural parameters
.. math::
\langle u \rangle =
\begin{bmatrix}
\phi_2 (-\phi_1)^{-1}
\\
-\log|-\phi_1| + \psi_k(\phi_2)
\end{bmatrix}
\\
g(\phi) = \phi_2 \log|-\phi_1| - \log \Gamma_k(\phi_2)
"""
U = linalg.chol(-phi[0])
k = np.shape(phi[0])[-1]
#k = self.dims[0][0]
logdet_phi0 = linalg.chol_logdet(U)
u0 = phi[1][...,np.newaxis,np.newaxis] * linalg.chol_inv(U)
u1 = -logdet_phi0 + misc.multidigamma(phi[1], k)
u = [u0, u1]
g = phi[1] * logdet_phi0 - special.multigammaln(phi[1], k)
return (u, g)
def compute_moments_and_cgf(self, phi, mask=True):
r"""
Return moments and cgf for given natural parameters
.. math::
\langle u \rangle =
\begin{bmatrix}
\phi_2 (-\phi_1)^{-1}
\\
-\log|-\phi_1| + \psi_k(\phi_2)
\end{bmatrix}
\\
g(\phi) = \phi_2 \log|-\phi_1| - \log \Gamma_k(\phi_2)
"""
U = linalg.chol(-phi[0])
k = np.shape(phi[0])[-1]
#k = self.dims[0][0]
logdet_phi0 = linalg.chol_logdet(U)
u0 = phi[1][..., np.newaxis, np.newaxis] * linalg.chol_inv(U)
u1 = -logdet_phi0 + misc.multidigamma(phi[1], k)
u = [u0, u1]
g = phi[1] * logdet_phi0 - special.multigammaln(phi[1], k)
return (u, g)
def compute_fixed_moments(self, Lambda):
""" Compute moments for fixed x. """
ldet = linalg.chol_logdet(linalg.chol(Lambda))
u = [Lambda,
ldet]
return u
def compute_fixed_moments(self, Lambda):
""" Compute moments for fixed x. """
ldet = linalg.chol_logdet(linalg.chol(Lambda))
u = [Lambda, ldet]
return u
