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I've got this prediction problem for daily data across several years. My data has both yearly and weekly seasonality. It's also stationary.

I tried using the following recurrence:(which I just came up with, from nowhere if you like) xn = 1/4(xn-738 + xn-364 + xn-7 + 1/6(xn-1+xn-2+xn-3+xn-4+xn-5+xn-6)

Basically, I am taking into consideration some of the previous days in the week before the day I am trying to predict and also the corresponding day a year and two years earlier. I am doing an average over them.

I got the following histogram of residuals: enter image description here

Why or Why not would such a formula be a good idea and can I improve this FOR example by adding some terms sampled from a distribution or in another way? Thanks!

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  • $\begingroup$ Stationary series can't be seasonal. Please double check your data. $\endgroup$
    – Skander H.
    Commented Jun 14, 2018 at 16:30
  • $\begingroup$ Yeah, I didn’t realise that stationarity had such a technical definition...I didn’t really check the stationarity But maybe people could help me regardless of that condition? Thanks! $\endgroup$
    – Sorin Tirc
    Commented Jun 14, 2018 at 20:34

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Based on the recurrence formula you've come up with, you seem to be trying to reinvent an Auto-Regressive model AR(p), which is essentially any model of the form:

$X_n = a_1 X_{n-1} + a_2 X_{n-2} + a_3 X_{n-3} + ... a_p X_{n-p}$.

Although your intuition is correct and your approach is on the correct path, there is a more systematic way of arriving at the same type of model that you are trying to achieve, using ARIMA and Seasonal ARIMA models.

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  • $\begingroup$ Hi Alex and thanks for the answer! I tried using ARIMA and Seasonal ARIMA but the problem is that ARIMA cannot capture double seasonality, so I tried to come up with something myself. Any idea how one could find the optimal a1,...,ap ? Optimal wrt to mean squared residual error for example $\endgroup$
    – Sorin Tirc
    Commented Jun 15, 2018 at 7:32
  • $\begingroup$ @SorinTirc I don't know of any ARIMA model that can capture multiple seasonalities (In theory a really high order ARIMA model could, but in practice that would lead to bad results). You should try TBATs or Facebook Prophet - but those models use completely different approaches from the auto-regressive models. $\endgroup$
    – Skander H.
    Commented Jun 15, 2018 at 8:15

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