kalman

Filename: kalman.py Reference: http://quant-econ.net/py/kalman.html

Implements the Kalman filter for a linear Gaussian state space model.

class quantecon.kalman.Kalman(ss, x_hat=None, Sigma=None)

Bases: object

Implements the Kalman filter for the Gaussian state space model

x_{t+1} = A x_t + C w_{t+1} y_t = G x_t + H v_t.

Here x_t is the hidden state and y_t is the measurement. The shocks w_t and v_t are iid standard normals. Below we use the notation

Q := CC’ R := HH’

Parameters:

ss : instance of LinearStateSpace

An instance of the quantecon.lss.LinearStateSpace class

x_hat : scalar(float) or array_like(float), optional(default=None)

An n x 1 array representing the mean x_hat and covariance matrix Sigma of the prior/predictive density. Set to zero if not supplied.

Sigma : scalar(float) or array_like(float), optional(default=None)

An n x n array representing the covariance matrix Sigma of the prior/predictive density. Must be positive definite. Set to the identity if not supplied.

References

http://quant-econ.net/py/kalman.html

Attributes

Sigma, x_hat	(as above)
Sigma_infinity	(array_like or scalar(float)) The infinite limit of Sigma_t
K_infinity	(array_like or scalar(float)) The stationary Kalman gain.

Methods

filtered_to_forecast()	Updates the moments of the time t filtering distribution to the
prior_to_filtered(y)	Updates the moments (x_hat, Sigma) of the time t prior to the time t filtering distribution, using current measurement y_t.
set_state(x_hat, Sigma)
stationary_coefficients(j[, coeff_type])	Wold representation moving average or VAR coefficients for the steady state Kalman filter.
stationary_innovation_covar()
stationary_values()	Computes the limit of Sigma_t as t goes to infinity by solving the associated Riccati equation.
update(y)	Updates x_hat and Sigma given k x 1 ndarray y.
whitener_lss()	This function takes the linear state space system

filtered_to_forecast()

Updates the moments of the time t filtering distribution to the moments of the predictive distribution, which becomes the time t+1 prior

prior_to_filtered(y)

Updates the moments (x_hat, Sigma) of the time t prior to the time t filtering distribution, using current measurement y_t.

The updates are according to

x_{hat}^F = x_{hat} + Sigma G’ (G Sigma G’ + R)^{-1}

(y - G x_{hat})

Sigma^F = Sigma - Sigma G’ (G Sigma G’ + R)^{-1} G

Sigma

Parameters:

y : scalar or array_like(float)

The current measurement

set_state(x_hat, Sigma)

stationary_coefficients(j, coeff_type='ma')

Wold representation moving average or VAR coefficients for the steady state Kalman filter.

Parameters:

j : int

The lag length

coeff_type : string, either ‘ma’ or ‘var’ (default=’ma’)

The type of coefficent sequence to compute. Either ‘ma’ for moving average or ‘var’ for VAR.

stationary_innovation_covar()

stationary_values()

Computes the limit of Sigma_t as t goes to infinity by solving the associated Riccati equation. Computation is via the doubling algorithm (see the documentation in matrix_eqn.solve_discrete_riccati).

Returns:

Sigma_infinity : array_like or scalar(float)

The infinite limit of Sigma_t

K_infinity : array_like or scalar(float)

The stationary Kalman gain.

update(y)

Updates x_hat and Sigma given k x 1 ndarray y. The full update, from one period to the next

Parameters:

y : np.ndarray

A k x 1 ndarray y representing the current measurement

whitener_lss()

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