def _filter_xs(self, s_0, s_ts, pi_0, sigma_0, As, Cs, Qs, Rs):
"""
Computes x_1|1, P_1|1, ..., x_T|T, P_T|T given observations, parameters and values of switching
variable at each time.
:returns:
-n_LF x T numpy array
Smoothed means
-T x n_LF x n_LF numpy array
Smoothed covariances
-T-1 x n_LF x n_LF numpy array
Smoothed lagged covariances (ie, cov[x_t, x_t-1])
"""
# Initialize Kalman filter using values of parameters at t = 0, even though they're never used
kf = Kalman(mu_0=pi_0.copy(), sigma_0=sigma_0.copy(), A=As[s_0], B=None, C=Cs[s_0], D=None,
Q=Qs[s_0], R=Rs[s_0])
# x_t|t, P_t|t, t = 1, ..., T
x_filts = np.zeros([self.T, self.n_LF])
P_filts = np.zeros([self.T, self.n_LF, self.n_LF])
# x_t|t-1, P_t|t-1. t = 1, ..., T. Indexed by t.
x_pred = np.zeros([self.T, self.n_LF])
P_pred = np.zeros([self.T, self.n_LF, self.n_LF])
# Filtering step
for t in range(0, self.T): # corresponds to t = 1, ..., T
# Change parameters. Never need to use A_{s_0}, etc.
kf.A = As[s_ts[t]]
kf.C = Cs[s_ts[t]]
kf.Q = Qs[s_ts[t]]
kf.R = Rs[s_ts[t]]
# Compute x_{t|t-1}, P_{t|t-1}
kf.predict()
x_pred[t] = kf.mu
P_pred[t] = kf.sigma
# Compute x_{t|t}, P_{t|t}
kf.update(self.y_ts[t])
x_filts[t] = kf.mu
P_filts[t] = kf.sigma
# TODO: run smoother to fill in missing data!!!
return x_filts, P_filts, x_pred, P_pred
def kalman_smooth(Y, U, V, pi0, sigma0, A, B, C, D, Q, R, n_LF):
"""
Runs Kalman filtering and smoothing step of dynamic factor model estimator.
Arguments
-Y: N x T
Observations
-U: L x T
State controls
-V: P x T
Observation controls
-pi0: n_LF x 1
-sigma0: n_LF x n_LF
-A: n_LF x n_LF
-B: n_LF x L
-C: N x n_LF
-D: N x M
-Q: n_LF x n_LF
-R: N x N
Returns
-n_LF x T numpy array
Smoothed means
-T x n_LF x n_LF numpy array
Smoothed covariances
-T-1 x n_LF x n_LF numpy array
Smoothed lagged covariances (ie, cov[x_t, x_t-1])
"""
N, T = Y.shape
# Initialize Kalman filter
kf = Kalman(mu_0=pi0.copy(),
sigma_0=sigma0.copy(),
A=A,
B=B,
C=C,
D=D,
Q=Q,
R=R)
# sigma_t|t, mu_t|t
sigma_filt = np.zeros([T, n_LF, n_LF])
sigma_filt[0] = sigma0.copy()
mu_filt = np.zeros([T, n_LF])
mu_filt[0] = pi0.copy()
# sigma_t|t-1, mu_t|t-1. NOTE: indexed by t-1!!!
sigma_pred = np.zeros([T - 1, n_LF, n_LF])
mu_pred = np.zeros([T - 1, n_LF])
# Filtering step
for t in range(1, T):
### Prediction step
kf.predict(U[:, t])
# Save mu_t|t-1 and sigma_t|t-1
sigma_pred[t - 1, :, :] = kf.sigma
mu_pred[t - 1] = kf.mu
### Update step
kf.C = C.copy()
kf.D = D.copy()
kf.R = R.copy()
# Need to change observation, observation control and observation noise matrices if observation is
# incomplete! See Shumway and Stoffer page 314.
# First, figure out which sensors don't have observations:
nan_sensors = np.where(np.isnan(Y[:, t]))[0]
# If some observations were present, go through with the update step
if nan_sensors.size < N:
# Zero those observations
y_t = np.nan_to_num(Y[:, t])
# Modify parameters
kf.C[nan_sensors, :] = 0.0
kf.D[nan_sensors, :] = 0.0
kf.R[nan_sensors, :] = 0.0
kf.R[:, nan_sensors] = 0.0
kf.R[nan_sensors, nan_sensors] = 1.0
kf.update(y_t, V[:, t])
# Save filtered mean, covariance
sigma_filt[t, :, :] = kf.sigma
mu_filt[t] = kf.mu
# sigma_t|T, mu_t|T
sigma_smooth = np.zeros((T, n_LF, n_LF))
mu_smooth = np.zeros((T, n_LF))
# Initialize: sigma_T|T = sigma_T|T(filtered)
sigma_smooth[-1] = sigma_filt[-1]
# mu_T|T = mu_T|T(filtered)
mu_smooth[-1] = mu_filt[-1]
# Lagged covariance. Indexed by t-1.
sigma_lag_smooth = np.zeros((T - 1, n_LF, n_LF))
# sigmaLag_{T,T-1} = (1 - K_T C) A V_{T-1|T-1}, where K_T is Kalman gain at last timestep.
# TODO: unclear what to do here if last observation contains missing components
K_T = np.dot(sigma_pred[-1], np.dot(kf.C.T, np.linalg.pinv(np.dot(kf.C, \
np.dot(sigma_pred[-1], kf.C.T)) + kf.R)))
sigma_lag_smooth[-1] = np.dot(
np.dot((np.identity(n_LF) - np.dot(K_T, kf.C)), kf.A), sigma_filt[-2])
# Backwards Kalman gain
J = np.zeros((T - 1, n_LF, n_LF))
# Smoothing step. Runs from t=T-1 to t=0.
for t in range(T - 2, -1, -1):
# Backward Kalman gain matrix
J[t] = np.dot(np.dot(sigma_filt[t], kf.A.T),
np.linalg.pinv(sigma_pred[t]))
# Smoothed mean
mu_smooth[t] = mu_filt[t] + np.dot(J[t], mu_smooth[t + 1] - mu_pred[t])
# Smoothed covariance. This is explicity symmetric.
sigma_smooth[t, :, :] = sigma_filt[t] + np.dot(
np.dot(J[t], sigma_smooth[t + 1] - sigma_pred[t]), J[t].T)
# Lagged smoothed covariance (NOT SYMMETRIC!)
for t in range(T - 3, -1, -1):
sigma_lag_smooth[t, :, :] = np.dot(sigma_filt[t+1], J[t].T) \
+ np.dot(np.dot(J[t+1], (sigma_lag_smooth[t+1] - np.dot(kf.A, sigma_filt[t+1]))), J[t].T)
# Fill in missing Y values
Y_imp = Y.copy()
nanRows, nanCols = np.where(np.isnan(Y_imp))
for s, t in zip(nanRows, nanCols):
Y_imp[s,
t] = np.dot(C[s, :], mu_smooth[t, :]) + np.dot(D[s, :], V[:, t])
return mu_smooth.T, sigma_smooth, sigma_lag_smooth, Y_imp, sigma_filt