def test_tangent_space(self):
A = self.rand_cases[0]
pr, vL = A.left_null_projector(True)
V = vL.reshape(-1, vL.shape[-1])
self.assertTrue(allclose(eye_like(V.T), V.conj().T @ V))
#def H(λ): return [-4*Sz12@Sz22+2*λ*(Sx12+Sx22)]
λ = 1
def H(λ):
return [Sz12 @ Sz22 + λ * (Sx12 + Sx22)]
O, l, r, K = A.Lh(H(λ), testing=True)
fun = norm(O(concatenate([real(K.reshape(-1)), imag(K.reshape(-1))])))
self.assertTrue(
abs(
norm(O(concatenate([real(K.reshape(-1)),
imag(K.reshape(-1))])))) < 1e-5)
self.assertTrue(allclose(tr(K @ r), 0))
O, l, r, K = A.Rh(H(λ), testing=True)
self.assertTrue(
abs(
norm(O(concatenate([real(K.reshape(-1)),
imag(K.reshape(-1))])))) < 1e-5)
self.assertTrue(allclose(tr(l @ K), 0))
def E(self, op, c=None):
"""E: calculate expectation of single site operator
:param op: operator to compute expectation of
:param c: canonicalisation of current state.
Should be in ['l', 'm', 'r', None].
If None, decompose to vidal then use that.
"""
if c == 'm':
L = self.L
A = self.data[0]
return real_if_close(sum(L @ A * tensordot(op, C(A) @ L, [1, 0])))
if c == 'r':
L = self.L
A = self.data[0]
return real_if_close(sum(A * tensordot(op, L**2 @ C(A), [1, 0])))
if c == 'l':
L = self.L
A = self.data[0]
return real_if_close(sum(A @ L**2 * tensordot(op, C(A), [1, 0])))
if c is None:
G, L = self.canonicalise(to_vidal=True).data[0]
circle = tr(G.dot(L).dot(L).dot(H(G)).dot(L).dot(L), axis1=1, axis2=3)
# - L - G - L -
# | |0 | |0
# | circle |, op
# | |1 | |1
# - L - G - L -
return real_if_close(tr(circle @ op))
def left_fixed_point(self, l0=None, tol=0):
d, D = self.d, self.D
if l0 is not None:
l0 = l0.reshape(D**2)
η, l = arnoldi(self.aslinearoperator().H, k=1, v0=l0, tol=tol)
l = rotate_to_hermitian(l)/sign(l[0])
l = l.reshape(D, D)
return η*np.sqrt(tr(l.conj().T@l)), l/np.sqrt(tr(l.conj().T@l))
def transform(self, y=None):
if y is None:
y = self.y
[w, mu, sigma] = [self.w, self.mu, self.sigma]
m = tr(w).dot(w) + sigma * np.eye(w.shape[1])
m = inv(m)
x = m.dot(tr(w)).dot(y - mu)
return x
def transform(self, y=None):
if y is None:
return self.q_dist.x_mean
q = self.q_dist
[w, mu, sigma] = [q.w, q.mu, q.gamma**-1]
m = tr(w).dot(w) + sigma * np.eye(w.shape[1])
m = inv(m)
x = m.dot(tr(w)).dot(y - mu)
return x
def transform(self, y=None):
if y is None:
return self.q_dist.x_mean
q = self.q_dist
[w, mu, sigma] = [q.w, q.mu, q.gamma**-1]
m = tr(w).dot(w) + sigma * np.eye(w.shape[1])
m = inv(m)
x = m.dot(tr(w)).dot(y - mu)
return x
def transform(self, y=None):
if y is None:
return self.q_dist.z_mean
q = self.q_dist
tau_mean = q.tau_a / q.tau_b
term1 = tau_mean * np.asarray([(q.z_cov[i].dot(tr(q.w_mean))).dot(y[:,i] - q.mu_mean) for i in range(y.shape[1])])
term2 = np.asarray([q.z_cov[i].dot(tr(self.E_s[i].dot(self.E_m_E_T))) for i in range(y.shape[1])])
z_mean = tr(term1 + term2)
return z_mean
def update_w(self):
q = self.q_dist
z_cov = np.zeros((self.q, self.q))
for n in range(self.n):
x = q.z_mean[:, n]
z_cov += x[:, np.newaxis].dot(np.array([x]))
q.w_cov = np.diag(q.alpha_a / q.alpha_b) + q.tau_mean() * z_cov
q.w_cov = inv(q.w_cov)
yc = self.y - q.mu_mean[:, np.newaxis]
q.w_mean = q.tau_mean() * q.w_cov.dot(q.z_mean.dot(tr(yc)))
q.w_mean = tr(q.w_mean)
def update_w(self):
q = self.q_dist
# cov
x_cov = np.zeros((self.q, self.q))
for n in xrange(self.n):
x = q.x_mean[:, n]
x_cov += x[:, np.newaxis].dot(np.array([x]))
q.w_cov = np.diag(q.alpha_a / q.alpha_b) + q.gamma_mean() * x_cov
q.w_cov = inv(q.w_cov)
# mean
yc = self.y - q.mu_mean[:, np.newaxis]
q.w_mean = q.gamma_mean() * q.w_cov.dot(q.x_mean.dot(tr(yc)))
q.w_mean = tr(q.w_mean)
def test_pure_QR(m):
random.seed(1302 * m)
A = random.randn(m, m) + 1j * random.randn(m, m)
A0 = 1.0 * A
A2 = cla_utils.pure_QR(QR, maxit=10, tol=1.0e-100)
#check it is still Hermitian
assert (np.linalg.norm(A2 - np.conj(A2).T) < 1.0e-4)
#check for orthogonality
x0 = random.randn(m)
assert (np.linalg.norm(np.dot(A2, x0)) - np.linalg(np.dot(A0, x0)) <
1.0e-6)
#check for conservation of trace
assert (np.abs(np.tr(A0) - np.tr(A2)) < 1.0e-6)
def update_w(self):
q = self.q_dist
# cov
x_cov = np.zeros((self.q, self.q))
for n in xrange(self.n):
x = q.x_mean[:, n]
x_cov += x[:, np.newaxis].dot(np.array([x]))
q.w_cov = np.diag(q.alpha_a / q.alpha_b) + q.gamma_mean() * x_cov
q.w_cov = inv(q.w_cov)
# mean
yc = self.y - q.mu_mean[:, np.newaxis]
q.w_mean = q.gamma_mean() * q.w_cov.dot(q.x_mean.dot(tr(yc)))
q.w_mean = tr(q.w_mean)
def nnCostFunction(nn_params, input_layer_size, hidden_layer_size, num_labels, X, y, lam): ''' Given NN parameters, layer sizes, number of labels, data, and learning rate, returns the cost of traversing NN. ''' Theta1 = (reshape(nn_params[:(hidden_layer_size*(input_layer_size+1))],(hidden_layer_size,(input_layer_size+1)))) Theta2 = (reshape(nn_params[((hidden_layer_size*(input_layer_size+1))):],(num_labels, (hidden_layer_size+1)))) m = X.shape[0] n = X.shape[1] #forward pass y_eye = eye(num_labels) y_new = np.zeros((y.shape[0],num_labels)) for z in range(y.shape[0]): y_new[z,:] = y_eye[int(y[z])-1] y = y_new a_1 = c_[ones((m,1)),X] z_2 = tr(Theta1.dot(tr(a_1))) a_2 = tr(sigmoid(Theta1.dot(tr(a_1)))) a_2 = c_[ones((a_2.shape[0],1)), a_2] a_3 = tr(sigmoid(Theta2.dot(tr(a_2)))) J_reg = lam/(2.*m) * (sum(sum(Theta1[:,1:]**2)) + sum(sum(Theta2[:,1:]**2))) J = (1./m) * sum(sum(-y*log(a_3) - (1-y)*log(1-a_3))) + J_reg #Backprop d_3 = a_3 - y d_2 = d_3.dot(Theta2[:,1:])*sigmoidGradient(z_2) Theta1_grad = 1./m * tr(d_2).dot(a_1) Theta2_grad = 1./m * tr(d_3).dot(a_2) #Add regularization Theta1_grad[:,1:] = Theta1_grad[:,1:] + lam*1.0/m*Theta1[:,1:] Theta2_grad[:,1:] = Theta2_grad[:,1:] + lam*1.0/m*Theta2[:,1:] #Unroll gradients grad = tr(c_[Theta1_grad.swapaxes(1,0).reshape(1,-1), Theta2_grad.swapaxes(1,0).reshape(1,-1)]) #return statement for function testing: #return Theta1,Theta2, y, a_1, z_2, d_3, a_3, J, grad, Theta1_grad, Theta2_grad #optimize.fmin expects a single value, so cannot return grad return J
def f(W, *args):
N, Y, sigma = args
_W_ = W.reshape(10, 2)
T0 = np.dot(_W_, np.transpose(_W_)) + sigma**2 * np.ones_like(
np.dot(_W_, np.transpose(_W_)))
T1 = np.dot(np.dot(Y, pinv(T0)), np.transpose(Y))
return N * det(T0) / 2 + tr(T1) / 2
def getOneEllipse(A):
evals = eig(A)[0]
f1 = evals[0]
f2 = evals[1]
center = 1 / 2 * tr(A)
x0 = center.real
y0 = center.imag
majAxVec_x = f2.real - f1.real
majAxVec_y = f2.imag - f1.imag
theta = cmath.phase(complex(majAxVec_x,
majAxVec_y)) / (2 * np.pi) * 360 # in DEGREES
b = np.sqrt(
tr(np.matmul(A, np.transpose(np.conjugate(A)))) - abs(f1)**2 -
abs(f2)**2).real
a = np.sqrt(b**2 + abs(f1 - f2)**2).real
elParam = [(x0, y0), a, b, theta]
return elParam
def update_z(self, s, T, m):
E_s = s
self.E_s = E_s
scale, dof = T
scale_inv = np.asarray([pinvh(sc) for sc in scale])
E_m = m
E_T = np.asarray([df * sc for sc,df in zip(scale_inv,dof)])
E_s_E_T = np.tensordot(E_s, E_T, axes=([1],[0]))
E_m_E_T = np.asarray([tr(em).dot(et) for em,et in zip(E_m,E_T)])
self.E_m_E_T = E_m_E_T
q = self.q_dist
tau_mean = q.tau_a / q.tau_b
q.z_cov = inv(tau_mean * tr(q.w_mean).dot(q.w_mean) + E_s_E_T)
tr_w_mean = tr(q.w_mean)
term1 = tau_mean * np.asarray([(q.z_cov[i].dot(tr_w_mean)).dot(self.y[:,i] - q.mu_mean) for i in range(self.n)])
term2 = np.asarray([q.z_cov[i].dot(tr(E_s[i].dot(E_m_E_T))) for i in range(self.n)])
q.z_mean = tr(term1 + term2)
def E(self, op):
"""E: TOTEST
"""
G, L = self.data[0]
circle = tr(G.dot(L).dot(L).dot(H(G)).dot(L).dot(L), axis1=1, axis2=3)
# - L - G - L -
# | |0 | |0
# | circle |, op
# | |1 | |1
# - L - G - L -
return real_if_close(tensordot(circle, op, [[0, 1], [0, 1]]))
def __fit_em(self, maxit=20):
w = np.random.rand(self.p, self.q)
mu = np.mean(self.y, 1)[:, np.newaxis]
sigma = self.prior_sigma
ll = self.__ell(w, mu, sigma)
yy = self.y - mu
s = self.n**-1 * yy.dot(tr(yy))
for i in xrange(maxit):
m = inv(tr(w).dot(w) + sigma * np.eye(self.q))
t = inv(sigma * np.eye(self.q) + m.dot(tr(w)).dot(s).dot(w))
w_new = s.dot(w).dot(t)
sigma_new = self.p**-1 * np.trace(s -
s.dot(w).dot(m).dot(tr(w_new)))
ll_new = self.__ell(w_new, mu, sigma_new)
print "{:3d} {:.3f}".format(i + 1, ll_new)
w = w_new
sigma = sigma_new
ll = ll_new
return (w, mu, sigma)
def energy(self, H):
"""energy of sum of two site terms
"""
d, D = self.d, self.D
h = H[0].reshape(d, d, d, d)
A = self.data[0]
_, l, r = self.eigs()
C = ncon([h]+[A, A], [[-1, -2, 1, 2], [1, -3, 3], [2, 3, -4]]) # HAA
K = ncon([[email protected](), A.conj()]+[C], [[1, 3, 4], [2, 4, -2], [1, 2, 3, -1]]) #AAHAA
self.e = tr(K@r)
return real(self.e)
def quantiles(x, qlist=[2.5, 25, 50, 75, 97.5]):
"""Returns a dictionary of requested quantiles from array"""
# Make a copy of trace
x = x.copy()
# For multivariate node
if x.ndim>1:
# Transpose first, then sort, then transpose back
sx = tr(sort(tr(x)))
else:
# Sort univariate node
sx = sort(x)
try:
# Generate specified quantiles
quants = [sx[int(len(sx)*q/100.0)] for q in qlist]
return dict(zip(qlist, quants))
except IndexError:
print "Too few elements for quantile calculation"
def update_gamma(self):
q = self.q_dist
q.gamma_a = self.hyper.gamma_a + 0.5 * self.n * self.p
q.gamma_b = self.hyper.gamma_b
w = q.w_mean
ww = tr(w).dot(w)
for n in range(self.n):
y = self.y[:, n]
x = q.x_mean[:, n]
q.gamma_b += y.dot(y) + q.mu_mean.dot(q.mu_mean)
q.gamma_b += np.trace(ww.dot(x[:, np.newaxis].dot([x])))
q.gamma_b += 2.0 * q.mu_mean.dot(w).dot(x[:, np.newaxis])
q.gamma_b -= 2.0 * y.dot(w).dot(x)
q.gamma_b -= 2.0 * y.dot(q.mu_mean)
def update_tau(self):
q = self.q_dist
q.tau_a = self.hyper.tau_a + 0.5 * self.n * self.p
q.tau_b = self.hyper.tau_b
w = q.w_mean
ww = tr(w).dot(w)
for n in range(self.n):
y = self.y[:, n]
x = q.z_mean[:, n]
q.tau_b += (y.dot(y) + q.mu_mean.dot(q.mu_mean)) / 2
q.tau_b += (np.trace(ww.dot(x[:, np.newaxis].dot([x])))) / 2
q.tau_b += (2.0 * q.mu_mean.dot(w).dot(x[:, np.newaxis])) / 2
q.tau_b -= (2.0 * y.dot(w).dot(x)) / 2
q.tau_b -= (2.0 * y.dot(q.mu_mean)) / 2
def update_gamma(self):
q = self.q_dist
q.gamma_a = self.hyper.gamma_a + 0.5 * self.n * self.p
q.gamma_b = self.hyper.gamma_b
w = q.w_mean
ww = tr(w).dot(w)
for n in xrange(self.n):
y = self.y[:, n]
x = q.x_mean[:, n]
q.gamma_b += y.dot(y) + q.mu_mean.dot(q.mu_mean)
q.gamma_b += np.trace(ww.dot(x[:, np.newaxis].dot([x])))
q.gamma_b += 2.0 * q.mu_mean.dot(w).dot(x[:, np.newaxis])
q.gamma_b -= 2.0 * y.dot(w).dot(x)
q.gamma_b -= 2.0 * y.dot(q.mu_mean)
def __ollh(self, A, mu, sigma2):
"""Observed data log likelihood"""
ll = 0.0
for i in np.arange(self.n):
s = self.S[:, i]
nobs = int(sum(s))
m = mu[s == 1]
y = self.Y[s == 1, i] - m
a = A[s == 1, :]
if nobs > 0:
C = np.dot(a, tr(a)) + sigma2 * np.eye(nobs)
logDetC = np.log(linalg.det(C))
#logDetC = sum(np.log(linalg.eigvals(C)))
ll = ll - 0.5 * nobs * np.log(
2 * np.pi) - 0.5 * logDetC - 0.5 * tr(y).dot(
pinv(C)).dot(y)
return ll
def eigs(self, l0=None, r0=None):
A = self.A
if l0 is not None:
l0 = l0.reshape(self.shape[1])
if r0 is not None:
r0 = r0.reshape(self.shape[1])
_, r = arnoldi(self.aslinearoperator(), k=1, v0=r0)
eta, l = arnoldi(self.aslinearoperator().H, k=1, v0=l0)
r, l = (rotate_to_hermitian(r.reshape(A.shape[1:]))/sign(r[0]),
rotate_to_hermitian(l.reshape(A.shape[1:]))/sign(l[0]))
n = tr(l @ r)
q = 1#tr(l@l)
return real_if_close(eta), l*sqrt(q)/sqrt(n), r/sqrt(q*n)
def hpd(x, alpha):
"""Calculate highest posterior density (HPD) of array for given alpha. The HPD is the
minimum width Bayesian credible interval (BCI).
:Arguments:
x : Numpy array
An array containing MCMC samples
alpha : float
Desired probability of type I error
"""
# Make a copy of trace
x = x.copy()
# For multivariate node
if x.ndim > 1:
# Transpose first, then sort
tx = tr(x, list(range(x.ndim)[1:]) + [0])
dims = shape(tx)
# Container list for intervals
intervals = np.resize(0.0, (2, ) + dims[:-1])
for index in make_indices(dims[:-1]):
try:
index = tuple(index)
except TypeError:
pass
# Sort trace
sx = sort(tx[index])
# Append to list
intervals[0][index], intervals[1][index] = calc_min_interval(
sx, alpha)
# Transpose back before returning
return array(intervals)
else:
# Sort univariate node
sx = sort(x)
return array(calc_min_interval(sx, alpha))
def hpd(x, alpha):
"""Calculate highest posterior density (HPD) of array for given alpha. The HPD is the
minimum width Bayesian credible interval (BCI).
:Arguments:
x : Numpy array
An array containing MCMC samples
alpha : float
Desired probability of type I error
"""
# Make a copy of trace
x = x.copy()
# For multivariate node
if x.ndim > 1:
# Transpose first, then sort
tx = tr(x, list(range(x.ndim)[1:]) + [0])
dims = shape(tx)
# Container list for intervals
intervals = np.resize(0.0, (2,) + dims[:-1])
for index in make_indices(dims[:-1]):
try:
index = tuple(index)
except TypeError:
pass
# Sort trace
sx = sort(tx[index])
# Append to list
intervals[0][index], intervals[1][index] = calc_min_interval(sx, alpha)
# Transpose back before returning
return array(intervals)
else:
# Sort univariate node
sx = sort(x)
return array(calc_min_interval(sx, alpha))
def compute_fid(model, data_set1, data_set2):
activation1 = model.predict(data_set1)
activation2 = model.predict(data_set2)
(mean1, mean2) = activation1.mean(axis=0), activation2.mean(axis=0)
(sigma1, sigma2) = cov(activation1, rowvar=False), cov(activation2,
rowvar=False)
squared_diff = np.sum(mean1 - mean2)**2.0
covariance_mean = sqrtm(sigma1.dot(sigma2))
if iscomplexobj(covariance_mean):
covariance_mean = covariance_mean.real
fid = squared_diff + tr(sigma1 + sigma2 - 2.0 * covariance_mean)
fid = round(fid, 2)
return fid
def update(self, H, δt):
"""mixed gauge update (inverse free) as in verstraeten notes
:param H: hamiltonian
:param δt: timestep
"""
raise NotImplementedError('Not implemented properly yet')
self.canonicalise('r')
d, D = self.d, self.D
h = H[0].reshape(d, d, d, d)
A = self.data[0]
_, l, r = self.eigs()
C = ncon([h]+[A, A], [[-1, -2, 1, 2], [1, -3, 3], [2, 3, -4]]) # HAA
K = ncon([[email protected](), A.conj()]+[C], [[1, 3, 4], [2, 4, -2], [1, 2, 3, -1]]) #AAHAA
self.e = tr(K@r)
pr = self.left_null_projector()
Lh = self.Lh(H)
Rh = self.Rh(H)
AL, AR, C = self.mixed()
AL, AR, C = AL[0], AR[0], C
AC = AL@C
G1 = -1j*zl(A)
G1 += ncon([ncon([h]+[AC, AR], [[-1, -2, 1, 2], [1, -3, 3], [2, 3, -4]]),
c(AR)], [[-1, 2, -2, 1], [2, -3, 1]])
G1 += ncon([ncon([h]+[AL, AC], [[-1, -2, 1, 2], [1, -3, 3], [2, 3, -4]]),
c(AL)], [[2, -1, 1, -3], [2, 1, -2]])
G1 += AC@Rh
G1 += Lh@AC
#G2 = ncon([G1, c(AC)], [[2, 1, -1], [2, 1, -2] ])
#print(tr(G2), self.e)
#raise Exception
G2 = ncon([G1, c(AL)], [[2, 1, -1], [2, 1, -2] ])
print(C1.shape, C2.shape, C3.shape, C4.shape)
raise Exception
def __ell(self, w, mu, sigma, norm=True):
m = inv(tr(w).dot(w) + sigma * np.eye(w.shape[1]))
mw = m.dot(tr(w))
ll = 0.0
for i in xrange(self.n):
yi = self.y[:, i][:, np.newaxis]
yyi = yi - mu
xi = mw.dot(yyi)
xxi = sigma * m + xi.dot(tr(xi))
ll += 0.5 * np.trace(xxi)
if sigma > 1e-5:
ll += (2 * sigma)**-1 * float(tr(yyi).dot(yyi))
ll -= sigma**-1 * float(tr(xi).dot(tr(w)).dot(yyi))
ll += (2 * sigma)**-1 * np.trace(tr(w).dot(w).dot(xxi))
if sigma > 1e-5:
ll += 0.5 * self.n * self.p * np.log(sigma)
ll *= -1.0
if norm:
ll /= float(self.n)
return ll
def crispness(alpha, T, right_eigenvectors, square_map, pi):
"""Return the crispness PCCA+ objective function.
Parameters
----------
alpha : ndarray
Parameters of objective function (e.g. flattened A)
T : csr sparse matrix
Transition matrix
right_eigenvectors : ndarray
The right eigenvectors.
square_map : ndarray
Mapping from square indices (i,j) to flat indices (k).
pi : ndarray
Equilibrium Populations of transition matrix.
Returns
-------
obj : float
The objective function
Notes
-------
Tries to make crisp state decompostion. This function is
defined in [3].
"""
A, chi, mapping = calculate_fuzzy_chi(alpha, square_map,
right_eigenvectors)
# If current point is infeasible or leads to degenerate lumping.
if (len(np.unique(mapping)) != right_eigenvectors.shape[1] or
has_constraint_violation(A, right_eigenvectors)):
return -1.0 * np.inf
obj = tr(dot(diag(1. / A[0]), dot(A.transpose(), A)))
return obj
def crispness(alpha, T, right_eigenvectors, square_map, pi):
"""Return the crispness PCCA+ objective function.
Parameters
----------
alpha : ndarray
Parameters of objective function (e.g. flattened A)
T : csr sparse matrix
Transition matrix
right_eigenvectors : ndarray
The right eigenvectors.
square_map : ndarray
Mapping from square indices (i,j) to flat indices (k).
pi : ndarray
Equilibrium Populations of transition matrix.
Returns
-------
obj : float
The objective function
Notes
-------
Tries to make crisp state decompostion. This function is
defined in [3].
"""
A, chi, mapping = calculate_fuzzy_chi(alpha, square_map,
right_eigenvectors)
# If current point is infeasible or leads to degenerate lumping.
if (len(np.unique(mapping)) != right_eigenvectors.shape[1] or
has_constraint_violation(A, right_eigenvectors)):
return -1.0 * np.inf
obj = tr(dot(diag(1. / A[0]), dot(A.transpose(), A)))
return obj
def dA_dt(self, H):
d, D = self.d, self.D
h = H[0].reshape(d, d, d, d)
A = self.data[0]
_, l, r = self.eigs()
C = ncon([h]+[A, A], [[-1, -2, 1, 2], [1, -3, 3], [2, 3, -4]]) # HAA
K = ncon([[email protected](), A.conj()]+[C], [[1, 3, 4], [2, 4, -2], [1, 2, 3, -1]]) #AAHAA
self.e = tr(K@r)
pr = self.left_null_projector()
R = ncon([pr, A], [[-3, -4, 1, -2], [1, -1, -5]])
K = self.Lh(H)
B = -1j*zl(A)
B += -1j*ncon([l@C@r, pr, inv(r)@A.conj()], [[3, 4, 1, 2], [-1, -2, 3, 1], [4, -3, 2]])
B += -1j*ncon([[email protected](), pr]+[C], [[1, 3, 4], [-1, -2, 2, 4], [1, 2, 3, -3]])
B += -1j *ncon([K, R], [[1, 2], [1, 2, -1, -2, -3]])
B = iMPS([B])
B.l, B.r, B.vL = l, r, self.vL
return B
def hpd(x, alpha):
"""Calculate HPD (minimum width BCI) of array for given alpha"""
# Make a copy of trace
x = x.copy()
# For multivariate node
if x.ndim>1:
# Transpose first, then sort
tx = tr(x, range(x.ndim)[1:]+[0])
dims = shape(tx)
# Container list for intervals
intervals = np.resize(0.0, dims[:-1]+(2,))
for index in make_indices(dims[:-1]):
try:
index = tuple(index)
except TypeError:
pass
# Sort trace
sx = sort(tx[index])
# Append to list
intervals[index] = calc_min_interval(sx, alpha)
# Transpose back before returning
return array(intervals)
else:
# Sort univariate node
sx = sort(x)
return array(calc_min_interval(sx, alpha))
means=numpy.average(samples.data,axis=0)
stds = numpy.std(samples.data,axis=0)
samples.data=( samples.data - means ) / stds
#[weights,sigma2,x_mean] = ppca(samples.data,2)
#weights=[weights]
#sigma2=[sigma2]
#x_mean=[x_mean]
#gausian_weight=[1.]
[weights,sigma2,x_mean,gausian_weight] = mppca(samples.data)
m = []
for i in range(len(x_mean)):
a = tr(weights[i]).dot(weights[i]) + float(sigma2[i]) * numpy.eye(weights[i].shape[1])
m.append(numpy.linalg.inv(a))
targets = samples.target
norm_sum=0.
for i,x in enumerate(numpy.matrix(samples.data)):
#new_sample = numpy.matrix(numpy.zeros([weights[0].shape[1],1]))
#for j in range(len(x_mean)):
# new_sample += gausian_weight[j]*(numpy.linalg.inv(weights[j].T*weights[j])*weights[j].T*x.T)
#new_sample2 = numpy.matrix(numpy.zeros([weights[0].shape[0],1]))
#for j in range(len(x_mean)):
# new_sample2 += gausian_weight[j]*(weights[j]*new_sample)
new_sample=weights[0]*numpy.linalg.inv(weights[0].T*weights[0])*weights[0].T*x.T
new_sample2=weights[1]*numpy.linalg.inv(weights[1].T*weights[1])*weights[1].T*x.T
new_sample3=weights[2]*numpy.linalg.inv(weights[2].T*weights[2])*weights[2].T*x.T
#new_sample4=weights[3]*numpy.linalg.inv(weights[3].T*weights[3])*weights[3].T*x.T
def Wishart(old, new):
#First get the expectation of the log of the determinant of rho:
d = old['tau'].shape[1]
i = np.arange(d)
ER = -log(det(old['tau']) / 2) + sum(psi((old['alpha'] - i + 1)/2))
#Now compute the overall KL divergence:
return (old['alpha']-new['alpha']) / 2 * ER - old['alpha'] * d / 2 + old['alpha'] / 2. * tr(new['tau'] * inv(old['tau'])) + log(Z(new['alpha'], new['tau'])) - log(Z(old['alpha'], old['tau']))
def structure(self, val):
self._structure = val
self.bmat = tr(inv(val.cell))
self.point_group = val.get_spacegroup(self.prec)["Number"]
self._assign_path()
def update_x(self):
q = self.q_dist
gamma_mean = q.gamma_a / q.gamma_b
q.x_cov = inv(np.eye(self.q) + gamma_mean * tr(q.w_mean).dot(q.w_mean))
q.x_mean = gamma_mean * q.x_cov.dot(tr(q.w_mean)).dot(self.y - q.mu_mean[:, np.newaxis])
trK += Ktmp[counter, j]
counter += 1
yK = np.array(yK).reshape(1, n_subj)
XX = dot(t(X), X)
XXinv = np.linalg.inv(XX)
Z = dot(X, XXinv)
yZ = dot(t(y), Z)
yPy = dot(t(y), y) - dot(dot(yZ, t(X)), y)
yZXK = dot(yZ, XK)
XKZ = dot(XK, Z)
yPKPy = dot(yK, y) - 2 * dot(yZXK, y) + dot(dot(yZXK, X), t(yZ))
trPK = trK - tr(XKZ)
trPKPK = trKK - 2 * tr(dot(dot(XXinv, XK), t(XK))) + tr(XKZ * XKZ)
S = np.array([trPKPK, trPK, trPK, n_subj - n_cov]).reshape(2, 2)
q = np.array([yPKPy, yPy]).reshape(2, 1)
Vc = dot(np.linalg.inv(S), q)
Vc[Vc < 0] = 0
s = trPKPK - trPK * trPK / (n_subj - n_cov)
h2 = np.asscalar(max(min(Vc[0] / sum(Vc), 1), 0))
se = pow(2 / s, 0.5)
print h2, se
print("--- %s seconds ---" % (time.time() - start_time))
def PPCA(Y_mat, d=20):
"""
Implements probabilistic PCA for data with missing values,
using a factorizing distribution over hidden states and hidden observations.
Args:
Y: (N by D ) input numpy ndarray of data vectors
d: ( int ) dimension of latent space
dia: (boolean) if True: print objective each step
Returns:
ss: ( float ) isotropic variance outside subspace
C: (D by d ) C*C' + I*ss is covariance model, C has scaled principal directions as cols
M: (D by 1 ) data mean
X: (N by d ) expected states
Ye: (N by D ) expected complete observations (differs from Y if data is missing)
Based on MATLAB code from J.J. VerBeek, 2006. http://lear.inrialpes.fr/~verbeek
"""
Y = Y_mat.copy()
N, D = shape(
Y
) # N observations in D dimensions (i.e. D is number of features, N is samples)
threshold = 1E-4 # minimal relative change in objective function to continue
hidden = isnan(Y)
missing = hidden.sum()
if (missing > 0):
M = nanmean(Y, axis=0)
else:
M = average(Y, axis=0)
Ye = Y - repmat(M, N, 1)
if (missing > 0):
Ye[hidden] = 0
# initialize
C = normal(loc=0.0, scale=1.0, size=(D, d))
CtC = mm(C.T, C)
X = mm(mm(Ye, C), inv(CtC))
recon = mm(X, C.T)
recon[hidden] = 0
ss = np.sum((recon - Ye)**2) / (N * D - missing)
count = 1
old = np.inf
# EM Iterations
while (count):
Sx = inv(eye(d) + CtC / ss) # E-step, covariances
ss_old = ss
if (missing > 0):
proj = mm(X, C.T)
Ye[hidden] = proj[hidden]
X = mm(mm(Ye, C), Sx / ss) # E-step: expected values
SumXtX = mm(X.T, X) # M-step
C = mm(mm(mm(Ye.T, X), (SumXtX + N * Sx).T),
inv(mm((SumXtX + N * Sx), (SumXtX + N * Sx).T)))
CtC = mm(C.T, C)
ss = (np.sum((mm(X, C.T) - Ye)**2) + N * np.sum(CtC * Sx) +
missing * ss_old) / (N * D)
# transform Sx determinant into numpy float128 in order to deal with high dimensionality
Sx_det = np.min(Sx).astype(np.float64)**shape(Sx)[0] * det(
Sx / np.min(Sx))
objective = N * D + N * (D * log(ss) + tr(Sx) - log(Sx_det)) + tr(
SumXtX) - missing * log(ss_old)
rel_ch = np.abs(1 - objective / old)
old = objective
count = count + 1
if (rel_ch < threshold and count > 5):
count = 0
# if (dia == True):
# print('Objective: %.2f, Relative Change %.5f' % (objective, rel_ch))
# C = orth(C)
# covM = cov(mm(Ye, C).T)
# vals, vecs = eig(covM)
# ordr = np.argsort(vals)[::-1]
# vals = vals[ordr]
# vecs = vecs[:, ordr]
# C = mm(C, vecs)
# X = mm(Ye, C)
# add data mean to expected complete data
Ye = Ye + repmat(M, N, 1)
# return C, ss, M, X, Ye
return Ye