# kalman¶

Filename: kalman.py Reference: https://lectures.quantecon.org/py/kalman.html

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

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

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

https://lectures.quantecon.org/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()[source]

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)[source]

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

$\hat{x}^F = \hat{x} + \Sigma G' (G \Sigma G' + R)^{-1} (y - G \hat{x}) \Sigma^F = \Sigma - \Sigma G' (G \Sigma G' + R)^{-1} G \Sigma$
Parameters: y : scalar or array_like(float) The current measurement
set_state(x_hat, Sigma)[source]
stationary_coefficients(j, coeff_type='ma')[source]

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()[source]
stationary_values()[source]

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)[source]

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()[source]

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

$\begin{split}\tilde{A} = \begin{bmatrix} A & 0 & 0 \\ KG & A-KG & KH \\ 0 & 0 & 0 \\ \end{bmatrix}\end{split}$
$\begin{split}\tilde{C} = \begin{bmatrix} C & 0 \\ 0 & 0 \\ 0 & I \\ \end{bmatrix}\end{split}$
$\begin{split}\tilde{G} = \begin{bmatrix} G & -G & H \\ \end{bmatrix}\end{split}$

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