With the parameter estimates having been defined, we're finally in a position to use our model for forecasting. We've actually already written most of the code we need to do this: we have an arma function that's capable of generating an autoregressive moving-average series based on some seed data and the model parameters p and q. The seed data will be our measured values of y from the airline data, and the values of p and q will be the parameters that we calculated using the Nelder-Mead method.

Let's plug those numbers into our ARMA model and generate a sequence of predictions for y:

(defn ex-9-32 []
  (let [data (i/log (airline-passengers))
        diff-1  (difference 1 data)
        diff-12 (difference 12 diff-1)
        forecast (->> (arma (take 9 (reverse diff-12))
                       []
                       (:ar params)
                       (:ma params) 0)
                      (take 100)
                      (undifference 12 diff-1)
            ...