Merge pull request #202 from QuantEcon/whitener

Whitener

quantecon/kalman.py

@@ -9,6 +9,7 @@
 import numpy as np
 from numpy import dot
 from scipy.linalg import inv
+from quantecon.lss import LinearStateSpace
 from quantecon.matrix_eqn import solve_discrete_riccati
 
 
@@ -83,6 +84,70 @@ def __str__(self):
  """
  return dedent(m.format(n=self.ss.n, k=self.ss.k))
 
+ def whitener_lss(self):
+ r"""
+ This function takes the linear state space system
+ that is an input to the Kalman class and it converts
+ that system to the time-invariant whitener represenation
+ given by
+
+ \tilde{x}_{t+1}^* = \tilde{A} \tilde{x} + \tilde{C} v
+ a = \tilde{G} \tilde{x}
+
+ where
+
+ \tilde{x}_t = [x+{t}, \hat{x}_{t}, v_{t}]
+
+ and
+
+ \tilde{A} = [A 0 0
+ KG A-KG KH
+ 0 0 0]
+
+ \tilde{C} = [C 0
+ 0 0
+ 0 I]
+
+ \tilde{G} = [G -G H]
+
+ with A, C, G, H coming from the linear state space system 
+ that defines the Kalman instance
+
+
+ Returns
+ -------
+ whitened_lss : LinearStateSpace
+ This is the linear state space system that represents
+ the whitened system
+ """
+ # Check for steady state Sigma and K
+ if self.K_infinity is None:
+ Sig, K = self.stationary_values()
+ self.Sigma_infinity = Sig
+ self.K_infinity = K
+ else:
+ K = self.K_infinity
+
+ # Get the matrix sizes
+ n, k, m, l = self.ss.n, self.ss.k, self.ss.m, self.ss.l
+ A, C, G, H = self.ss.A, self.ss.C, self.ss.G, self.ss.H
+
+ Atil = np.vstack([np.hstack([A, np.zeros((n, n)), np.zeros((n, l))]),
+ np.hstack([dot(K, G), A-dot(K, G), dot(K, H)]),
+ np.zeros((l, 2*n + l))])
+
+ Ctil = np.vstack([np.hstack([C, np.zeros((n, l))]),
+ np.zeros((n, m+l)),
+ np.hstack([np.zeros((l, m)), np.eye(l)])])
+
+ Gtil = np.hstack([G, -G, H])
+
+ whitened_lss = LinearStateSpace(Atil, Ctil, Gtil)
+ self.whitened_lss = whitened_lss
+
+ return whitened_lss
+
+
  def prior_to_filtered(self, y):
  r"""
  Updates the moments (x_hat, Sigma) of the time t prior to the