2
$\begingroup$

There is an Autoregressive model of order 1 (AR(1)) that is excited by a non-linear signal as the input: $$x_t = \rho x_{t-1} + u_t \tag{1}$$ The time series $u_t$ is generated from a nonlinear map, $$u_t = f(u_{t-1},\mathbf{w}) \tag{2}$$ where $f$ is the nonlinear function. The observations are $$y_t = x_t + v_t \tag{3}$$ where $v_t$ is the measurement noise that is an Additive White Gaussian Noise.

Q1: Can I re-write the model (1) as a state space in the following way:

$$x_t = Ax_{t-1} + f(u_{t-1},\mathbf{w}) $$

$$y_t = x_t+ v_t \tag{4}$$

Is the above representation correct? If not then I shall be grateful for the correct technique to represent it.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Does it matter how $u_t$ is generated? Why are you applying the Unscented KF? What parameters are you interested in estimating? Your simple substitution probably isn't quite there. You will probably have to include $u$ as part of the state, and then $A$ will instead be a nonlinear operation on $x_{t-1}$ and $u_{t-1}$. $\endgroup$ – Peter K.♦ Aug 7 '15 at 17:38
  • $\begingroup$ Not sure if my answer is clear; please comment on it and I'll try to update it. $\endgroup$ – Peter K.♦ Aug 11 '15 at 0:08
2
$\begingroup$

Rather than write

$$ x_t = Ax_{t-1} + f(u_{t-1},\mathbf{w}) $$

as the state update equation, I'd write:

$$ \xi = \left[ \begin{array}{c} x_t\\ u_t \end{array} \right] $$

and then $$ \xi_t = g(\xi_{t-1},A,\mathbf{w}) $$

so that $$ y_t = \left[ 1\ \ \ 0 \right] \xi_t + v_t $$

Then you could apply the non-additive noise formulation of the EKF to get your filter equations.

Improve this answer
$\endgroup$
2
  • $\begingroup$ The representation makes sense but I am unsure about the non additive part. I need to do augmented state representation so how can I do augmented state representation where the process noise $f$ and the measurement noise are augmented states? $\xi$ appears to be the augmented state for signal state and process noise = $[\mathbf{x^T_{t-1},u_t^T}]$;. Please correct me if wrong $\endgroup$ – Srishti M Aug 12 '15 at 17:02
  • 1
    $\begingroup$ Well, your output equation is already linear, so you don't need to augment the state for the measurement noise; it's just the process noise that is problematic. Yes, $\xi$ as above is an augmented state. $\endgroup$ – Peter K.♦ Aug 12 '15 at 17:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.