Modeling AR 1 Term with Missing Observations

Modeling
1

Hello,

I am trying to fit a simple AR 1 model, as shown below. I read over the missing values section in the Stan manual, but it is not clear to me how to fit this model if I have missing observations in my data. Would it be as straight forward as setting \tau = 0 if there is a missing observation at time i-1? Or draw another value of \tau_0 at the time point before each missing observation? In some sense, treating it again as requiring an initial value.

Y_{i} = f(\theta)_{i} + \tau_{i}
\tau_i = \rho \tau_{i-1} + \epsilon_i
\epsilon_i \sim N(0, \sigma^2)
\tau_0 \sim N(0, a)

Thank you,
Colin

2

The easiest way to do do it is to declare in the parameters block a vector, say y_miss whose size is equal to the number of missing observations. Then, weave together a “complete data” vector in temporal order that consists of the observed observations and y_miss. The “complete data” vector then has an AR1 “complete data likelihood”.

2 Likes
3

Hi Ben,

Thanks for you reply. As long as my y’s are in the right temporal order, it looks like I can add indexing to identify the responses that are missing, and then combine them into one big vector in the transformed parameters block.

int<lower = 1, upper = N_obs + N_mis> ii_obs[N_obs]; 
int<lower = 1, upper = N_obs + N_mis> ii_mis[N_mis];

transformed parameters{
real y[N]; 
y[ii_obs] = y_obs; 
y[ii_mis] = y_mis; 
}

One more quick follow up, I am looping over the all observations rather than doing nested loops for each time series. I have a time vector that that repeats the time indices–e.g., 1 2 3 1 2 3…1 2 3 (if I had 3 time points for each series). Without the AR term it looks basically like the following (assume x[i] indicates the time point).

model{
for(i in 1:N){
y[i] ~ normal(f[x[i]], sigma)
}

Is there simple way to let the model know that at each time point i = 1, we need to reset the AR process for the next series? I was thinking something like if x[i]==1 let \eta \sim N(0, a); otherwise, \eta = \rho\eta_{i-1}. Does this make sense?

Thank you,
Colin

4

You can put in if {} else {} statements but for observations where i = 1, the unconditional variance is not sigma but rather sigma / sqrt(1 - square(rho)).