def backward_sample(self, steps=1):
"""
Generate state sequence using distributions:
.. math:: p(\theta_{t} | \theta_{t + k} D_t)
"""
from statlib.distributions import rmvnorm
if steps != 1:
raise Exception('only do one step backward sampling for now...')
T = self.nobs
# Backward sample
mu_draws = np.zeros((T + 1, self.ndim))
m = self.mu_mode
C = self.mu_scale
a = self.mu_forc_mode
R = self.mu_forc_scale
mu_draws[T] = rmvnorm(m[T], C[T])
# initial values for smoothed dist'n
for t in xrange(T - 1, -1, -1):
# B_{t} = C_t G_t+1' R_t+1^-1
B = chain_dot(C[t], self.G.T, LA.inv(R[t + 1]))
# smoothed mean
ht = m[t] + np.dot(B, mu_draws[t + 1] - a[t + 1])
Ht = C[t] - chain_dot(B, R[t + 1], B.T)
mu_draws[t] = rmvnorm(ht, np.atleast_2d(Ht))
return mu_draws.squeeze()
def backward_sample(self, steps=1):
"""
Generate state sequence using distributions:
.. math:: p(\theta_{t} | \theta_{t + k} D_t)
"""
from statlib.distributions import rmvnorm
if steps != 1:
raise Exception('only do one step backward sampling for now...')
T = self.nobs
# Backward sample
mu_draws = np.zeros((T + 1, self.ndim))
m = self.mu_mode
C = self.mu_scale
a = self.mu_forc_mode
R = self.mu_forc_scale
mu_draws[T] = rmvnorm(m[T], C[T])
# initial values for smoothed dist'n
for t in xrange(T-1, -1, -1):
# B_{t} = C_t G_t+1' R_t+1^-1
B = chain_dot(C[t], self.G.T, LA.inv(R[t+1]))
# smoothed mean
ht = m[t] + np.dot(B, mu_draws[t+1] - a[t+1])
Ht = C[t] - chain_dot(B, R[t+1], B.T)
mu_draws[t] = rmvnorm(ht, np.atleast_2d(Ht))
return mu_draws.squeeze()
def _update_mu(v, w, phi, lam):
# FFBS
# allocate result arrays
mode = np.zeros((T + 1, p))
a = np.zeros((T + 1, p))
C = np.zeros((T + 1, p))
R = np.zeros((T + 1, p))
# simple priors...
mode[0] = 0
C[0] = np.eye(p)
# Forward filter
Ft = m_([[1]])
for i, obs in enumerate(y):
t = i + 1
at = phi * mode[t - 1] if t > 1 else mode[0]
Rt = phi**2 * C[t - 1] + w if t > 1 else C[0]
Vt = lam[t - 1] * v
Qt = chain_dot(Ft.T, Rt, Ft) + Vt
At = np.dot(Rt, Ft) / Qt
# forecast theta as time t
ft = np.dot(Ft.T, at)
err = obs - ft
# update mean parameters
mode[t] = at + np.dot(At, err)
C[t] = Rt - np.dot(At, np.dot(Qt, At.T))
a[t] = at
R[t] = Rt
# Backward sample
mu = np.zeros((T + 1, p))
# initial values for smoothed dist'n
fR = C[-1]
fm = mode[-1]
for t in xrange(T + 1):
if t < T:
# B_{t} = C_t G_t+1' R_t+1^-1
B = np.dot(C[t] * phi, la.inv(np.atleast_2d(R[t + 1])))
# smoothed mean
fm = mode[t] + np.dot(B, mode[t + 1] - a[t + 1])
fR = C[t] + chain_dot(B, C[t + 1] - R[t + 1], B.T)
mu[t] = dist.rmvnorm(fm, np.atleast_2d(fR))
return mu.squeeze()
def _update_mu(v, w, phi, lam):
# FFBS
# allocate result arrays
mode = np.zeros((T + 1, p))
a = np.zeros((T + 1, p))
C = np.zeros((T + 1, p))
R = np.zeros((T + 1, p))
# simple priors...
mode[0] = 0
C[0] = np.eye(p)
# Forward filter
Ft = m_([[1]])
for i, obs in enumerate(y):
t = i + 1
at = phi * mode[t - 1] if t > 1 else mode[0]
Rt = phi ** 2 * C[t - 1] + w if t > 1 else C[0]
Vt = lam[t - 1] * v
Qt = chain_dot(Ft.T, Rt, Ft) + Vt
At = np.dot(Rt, Ft) / Qt
# forecast theta as time t
ft = np.dot(Ft.T, at)
err = obs - ft
# update mean parameters
mode[t] = at + np.dot(At, err)
C[t] = Rt - np.dot(At, np.dot(Qt, At.T))
a[t] = at
R[t] = Rt
# Backward sample
mu = np.zeros((T + 1, p))
# initial values for smoothed dist'n
fR = C[-1]
fm = mode[-1]
for t in xrange(T + 1):
if t < T:
# B_{t} = C_t G_t+1' R_t+1^-1
B = np.dot(C[t] * phi, la.inv(np.atleast_2d(R[t+1])))
# smoothed mean
fm = mode[t] + np.dot(B, mode[t+1] - a[t+1])
fR = C[t] + chain_dot(B, C[t+1] - R[t+1], B.T)
mu[t] = dist.rmvnorm(fm, np.atleast_2d(fR))
return mu.squeeze()