def get_l2_measures(w_matrix, sequences):
"""
Computes the squared L2-norm between the visible units representing a MRI
sequence and each hidden unit.
Parameters
----------
w_matrix: numpy array.
Weights matrix of the RBM.
sequences: list.
List containing the name of the MRI sequences
Returns
-------
numpy array.
Array [n_sequences X n_hidden_units] containing the squared L2-norm
between each hidden unit and the MRI sequences
"""
l2_measures = np.zeros((len(sequences), w_matrix.shape[1]))
n_visibles_voxels = int(float(w_matrix.shape[0]) / float(len(sequences)))
for i in xrange(0, len(sequences)):
l2_measures[i, :] = \
l2(w_matrix[i * n_visibles_voxels: (i + 1) * n_visibles_voxels, :],
axis=0)
return l2_measures
def L_r(Tt, ut, bf, c):
if Tt.ndim<=1:
Tt = np.atleast_2d(Tt).T
ut = np.atleast_2d(ut).T
cut = c * ut
Id = np.eye(Tt.shape[0])
if bf <= 1e-10:
return 0*Id, bf * np.ones((Id.shape[0], 1)), (-Tt + cut) / l2(-Tt + cut)
_M = M(Tt, cut, bf)
alpha = -Tt.T.dot(-Tt + cut) / (l2(-Tt) * l2(-Tt + cut))
delta = min(l2(-Tt) / bf, 1)
if alpha < 0:
beta = 1 / (1 - alpha * delta)
else:
beta = 1
IdM_inv = np.linalg.inv(Id - beta * _M)
v = IdM_inv.dot(-Tt + cut) / l2(-Tt + cut)
return c * (IdM_inv - Id), -IdM_inv.dot(hf(Tt, cut, bf)), v
def mean_field_hs(Vs, K):
"""
Pj(xj) = 1/Z0 *exp(-beta*hj(xj)), where
hj(xj) = \sum_{<j,jp>} \sum_{xjp \in jp} V(xj,xjp)*Pjp(xjp)
We assume a Potts model of m variables x0...xj...xm-1 where each
variable can take on K states 0...i...K-1. Mean field functions h
are represented as a matrix hss where each row gives the values
hj(i). [Note that i,j are reversed from the usual row-column
convention.]
Input is a matrix Vs of pairwise contributions to the hamiltonian
where Vs[j][jp] is a function V(xj,xjp)
"""
M = len(Vs)
jpairs = pairs(range(M))
hs = [[1 for i in range(K)] for j in range(M)]
def Pj(xj, j):
# print xj,j
return exp(-beta * hs[j][xj]) / sum(exp(-beta * hs[j][xjp]) for xjp in range(K))
old_hs = matcopy(hs)
while True:
for j in range(M):
for i in range(K):
hs[j][i] = sum(sum(Vs[j][jp](i, ip) * Pj(ip, jp) for ip in range(K)) for jp in range(j + 1, M)) + sum(
sum(Vs[jp][j](ip, i) * Pj(ip, jp) for ip in range(K)) for jp in range(0, j - 1)
)
print l2(concat(hs), concat(old_hs))
if old_hs == hs:
break
else:
old_hs = matcopy(hs)
print hs
return hs
def baum_welch(obs,L):
"""Given sequence and bs length L, approximate MLE parameters for
emission probabilities,transition rate a01 (background->site).
TODO: non-uniform background frequencies"""
states = range(L+1)
a01 = random.random()
start_p = make_start_p(a01)
trans_p = make_trans_p(a01)
emit_p = [simplex_sample(4) for state in states]
hidden_states = [random.choice(states) for ob in obs]
iterations = 0
while True:
# compute hidden states, given probs
prob,hidden_states_new = viterbi(obs, states, start_p, trans_p, emit_p)
# compute probs, given hidden states
# first compute a01
a01_new = estimate_a01(hidden_states_new)
start_p_new = make_start_p(a01_new)
trans_p_new = make_trans_p(a01_new)
emit_p_new = estimate_emit_p(obs,hidden_states_new,states)
if (start_p_new == start_p and
trans_p_new == trans_p and
emit_p_new == emit_p and
hidden_states_new == hidden_states):
break
else:
print iterations,a01,l2(start_p,start_p_new),
print l2(concat(trans_p),concat(trans_p_new)),
print l2((hidden_states),hidden_states_new)
a01 = a01_new
start_p = start_p_new
trans_p = trans_p_new
emit_p = emit_p_new
hidden_states = hidden_states_new
iterations += 1
return start_p,trans_p,emit_p,hidden_states
def mean_field_hs():
"""Following derviation on wikipedia's mean field theory page..."""
def V(xj,xjp):
if xj == xjp > 0:
retval = 10**10
else:
retval = (eps[xj] + eps[xjp])/choose(J,2) # divide by choose(J,2) since we're summing over pairs
if random.random() < 0:
print "V(%s,%s) = %s" % (xj,xjp,retval)
return retval
# because each term appears J-1 times. self-consistency equation
# for mean field approximation is:
# Pj(xj) = 1/Z0 *exp(-beta*hj(xj)), where
# hj(xj) = \sum_{<j,jp>} \sum_{xjp \in jp} V(xj,xjp)*Pjp(xjp)
# In this case, the graph is fully connected and all variables xj
# are exchangeable, so sum over pairs reduces to (J-1).
# Moreover, due to exchangeability hj is the same for each
# variable, so we can update a single function h(x) for all
# variables. h(x) is a function with G+1 possible input values,
# so we can represent h as an array of size G+1 such that h[i]
# stores the value h(i).
# Initialize it arbitrarily
h_cur = [1] * (G+1)
h_next = [0] * (G+1)
def P(i):
"""Return probability at time t that x takes on value i"""
return exp(-beta*h_cur[i])/sum(exp(-beta*h_cur[ip]) for ip in range(G+1))
while True:
for i in range(G+1):
terms = [V(i,ip)*P(ip) for ip in range(G+1) if not i == ip]
#print i,terms
h_next[i] = (J-1)*sum(terms)
if l2(h_next,h_cur) < 10**-10:
break
h_cur = h_next[:]
print h_cur
return h_cur
def Ff(Tt, cut, bf):
denominator = max(bf, l2(-Tt)) * l2(-Tt + cut)
# To avoid issues with shapes we multiply the scalar denominator (scalar for each face)
# into the Lagraniane multiplyer
numerator = -Tt.dot((-Tt + cut).T)
return numerator / denominator
def ef(Tt, cut, bf):
return bf / l2(-Tt + cut)
def active_sliding(Tt, ut, bf, ct):
return l2(-Tt + ct * ut) - bf > 1e-10
def fit(self):
# sparse group lasso selection on kappa
distance = 1.
nb_iter = 1
while distance > self.rtol and nb_iter <= self.max_iter:
coefs_old = self.coef_.copy()
for gr in self.groups[0][1:]:
# 1- Should the group be zero-ed out?
tmp_coefs_gr = self.coef_.copy()
tmp_coefs_gr[gr] = 0.
if discard_group(self.y, self.knots, tmp_coefs_gr, self.splrep,
self.proj_matrix, self.u, self.lbda1,
self.alpha1, gr):
self.coef_[gr] = 0.
# 2- If the group is not zero-ed out, update each component
else:
self.coef_[gr] = block_wise_descent_fitting(
self.coef_, self.y, self.knots, self.splrep,
self.proj_matrix, self.u, self.lbda1, self.alpha1, gr)
# sparse group lasso selection on tau
for gr in self.groups[1][1:]:
# 1- Should the group be zero-ed out?
tmp_coefs_gr = self.coef_.copy()
tmp_coefs_gr[gr] = 0.
if discard_group(self.y, self.knots, tmp_coefs_gr, self.splrep,
self.proj_matrix, self.u, self.lbda2,
self.alpha2, gr):
self.coef_[gr] = 0.
# 2- If the group is not zero-ed out, update each component
else:
self.coef_[gr] = block_wise_descent_fitting(
self.coef_, self.y, self.knots, self.splrep,
self.proj_matrix, self.u, self.lbda2, self.alpha2, gr)
# estimate beta_0
ind_beta0 = int(self.groups[0][0])
beta0_old = self.coef_[ind_beta0]
coef_excl_beta0 = get_coef_exclude_ind(self.coef_, ind_beta0)
beta0_new, ignored1, ignored2 = fmin_l_bfgs_b(
func=gradient_f_beta0,
x0=beta0_old,
args=(coef_excl_beta0, ind_beta0, self.y, self.knots,
self.splrep, self.proj_matrix, self.u),
approx_grad=True)
self.coef_[ind_beta0] = beta0_new
# estimate theta_0
ind_theta0 = int(self.groups[1][0])
theta0_old = self.coef_[ind_theta0]
coef_excl_theta0 = get_coef_exclude_ind(self.coef_, ind_theta0)
theta0_new, ignored3, ignored4 = fmin_l_bfgs_b(
func=gradient_f_theta0,
x0=theta0_old,
args=(coef_excl_theta0, ind_theta0, self.y, self.knots,
self.splrep, self.proj_matrix, self.u),
approx_grad=True)
self.coef_[ind_theta0] = theta0_new
# update_nb_iter
nb_iter += 1
# update_distance
distance = l2(self.coef_ - coefs_old)
return self
