AR processes for binned time points

[wds15]

I am working on a problem which requires to model binned time-points as a AR(1) process... but brms does not let me do that as I get the error Error: Time points within groups must be unique.. I don't quite see why it would not make sense to have an AR process defined on time points which are over binned time points? For example, let's say we have a process measured on days, but I want the AR process to work on weeks, for example:

# Load necessary library
set.seed(5678678) # For reproducibility

# Parameters
n <- 100 # Length of the time series
phi <- 0.7 # AR(1) coefficient
sigma <- 1 # Standard deviation of the primary white noise
sigma_eta <- 0.5 # Standard deviation of additional white noise

# Initialize the time series
X <- numeric(n)
X[1] <- rnorm(1, mean = 0, sd = sigma) # Initial value

# Simulate AR(1) process with additional white noise
for (t in 2:n) {
  e_t <- rnorm(1, mean = 0, sd = sigma) # Primary white noise
  eta_t <- rnorm(1, mean = 0, sd = sigma_eta) # Additional white noise
  X[t] <- phi * X[t-1] + e_t + eta_t
}

# Plot the time series
## plot(X, type = 'l', main = 'AR(1) Time Series with Additional White Noise', xlab = 'Time', ylab = 'X_t')

# Display the first few values
## head(X)

ar_example <- data.frame(y=X, time=1:length(X)) |> transform(week=as.integer((floor(time/7))))

library(brms)

## works ok:
day_ar <- brm(bf(y ~ ar(time, p=1)), data=ar_example)

## not possible to do?
week_ar <- brm(bf(y ~ ar(week,p=1)), data=ar_example)

[wds15]

Basically, the AR process should be allowed to be defined to go over time bins rather than data rows.. or is that somehow possible with the gr argument?

[paul-buerkner]

Should be possible. Not sure how much effort it would be though. Can you tell me the math model these binned time points would imply?