# Autoregressive model with observable noise

The classical autoregressive model is a linear model for the dynamic variable $$x$$, where the added noise $$\epsilon$$ is directly affecting the dynamics of the model $$x_{t} = \sum_i \alpha_i x_{t-i} + \epsilon_t$$ I am considering a related model, where the noise is only due to imperfect observation of the underlying variable, but does not directly affect the dynamics. Then, the dynamics of a hidden variable $$x$$ is given by

$$x_t = \sum_i \alpha_i x_{t-i}$$

and of the observable $$y$$

$$y_t = x_t + \nu_t$$

where $$\nu_t$$ is the observable noise.

Questions:

• Under what name is the "observable noise AR model" known in the literature?
• Under what name is the composition of the above two models known in the literature? That is, a model that possesses both dynamic and observational noise
• In your first equation, $\epsilon_i$ is supposed to be $\epsilon_t$ – Firebug Oct 19 at 15:57
• You seem to be looking for state space models. – Chris Haug Oct 19 at 17:23
• @Firebug, it was a typo, thanks for spotting – Aleksejs Fomins Oct 20 at 7:07

Consider the model $$\begin{cases} x_t = \sum_i \alpha_i x_{t-i}\\ y_t = x_t + \nu_t \end{cases}$$

Substituting $$y_t = \sum_i \alpha_i x_{t-i} + \nu_t\\ y_t = \sum_i \alpha_i (y_{t-i}-\nu_{t-i}) + \nu_t\\$$

This is an ARMA model in $$y$$ with a very specific constraint.

$$\begin{cases} y_t = \sum_i \phi_i y_{t-i} + \sum_i \theta_i \nu_{t-i} + \nu_t\\ \theta_i=-\phi_i \end{cases}$$

• Nice observation. What about the composite model? I have tried the same math as you did for composite model, and it looks almost like ARMA, but there are two sources of noise, one window-averaged and another instantaneous – Aleksejs Fomins Oct 20 at 7:22
• @AleksejsFomins how did you implement the other model given that $x$ is hidden? – Firebug Oct 20 at 11:16
• The point is to infer the hidden variables $x$ from observations $y$. The variables form a closed system, given a prior for $x_0$, $\alpha$ and the noise model. In case the observable is obtained by means of a function of the hidden variable, not just the noise, the model is called Dynamical Causal Model, however it is only popular in neuroscience community. I was wondering if I can find analogues of this model in regular statistical literature. – Aleksejs Fomins Oct 20 at 11:27
• Also, could you suggest a good read for learning about different existing time-series statistical models like ARMA. – Aleksejs Fomins Oct 20 at 11:32
• @AleksejsFomins the DCM assumes continuous time (I'm a neuroscientist). It has a stochastic variant with neuronal noise as well (Stochastic DCM). Having said that, in DCM $x$ is latent, but indeed partially observable (due to the assumption that it's zero at the onset of sessions). – Firebug Oct 20 at 12:38