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Sometimes forecasting with ARIMA loses precision due attenuation. In my case I have: enter image description here

And the ACF plot for this time series: enter image description here

Where we can see periods.

The question is why attention happens and is there any way to improve precision? And the second question is why amplitude of the forecast is so low?

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  • $\begingroup$ This is not an attenuation case. Draw the confidence bands to see for yourself. There's a lot of noise in the data. ARIMA forecast will have interesting features only in first few lags, then it just straightens into the long run mean $\endgroup$ – Aksakal Nov 2 '16 at 17:34
  • $\begingroup$ @Aksakal, the confidence bands are already there, aren't they? $\endgroup$ – Richard Hardy Nov 2 '16 at 17:35
  • $\begingroup$ @RichardHardy, right :) so the attenuation is not the case. it's just the signal is buried in the noise, and the forecast reflects it $\endgroup$ – Aksakal Nov 2 '16 at 17:37
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Edit: After the edit of the OP, my aswer addresses the following part:

why amplitude of the forecast is so low?

The best forecast (in minimum mean square sense) need not look like an extrapolation of the past. For example, the best forecast for a random walk is a straight horizontal line starting from the last observed value. This is nothing like the observed past, but nevertheless any other forecast (even if more similar in shape) would be worse in expectation.

Perhaps you have the same situation here. The model treats most of the fluctuations as unpredictable noise and finds only a weak signal that is extrapolated into the future as the little-wiggly blue line.

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  • $\begingroup$ But obviously in the past there is no such values as predicted. How to make forecast looking more similar to original time series? Predict residuals between ARIMA and time sires with xhboost and then add predicted reseduntals to forecast? $\endgroup$ – user135437 Nov 2 '16 at 17:51
  • $\begingroup$ @user135437, Think again about the random walk. Having a forecast resembling any typical trajectory of a random walk would be nice for the eye but poor from the expected error point of view. Any sensible loss function that I know of would tell you to choose a straight horizontal line over anything else. It might seem counterintuitive, but if you acknowledge noise for what it is, you are left with forecasts looking quite different than the past observations. $\endgroup$ – Richard Hardy Nov 2 '16 at 17:54
  • $\begingroup$ Ok, now I see... But does it means that if I have in my time series strongly periodical data (for example ideal 50 periods, sin or something like this) and only one or few periods with significant outliers. In such case ARIMA will produce weak periods in a forecast. Is it very sensitive to noise in other words? $\endgroup$ – user135437 Nov 2 '16 at 18:01
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    $\begingroup$ @user135437, perhaps you should consider seasonally integrated SARIMA models then? Because simple, stationary ARMA will not generate persistent seasonal patterns, you will have regression towards the mean. $\endgroup$ – Richard Hardy Nov 2 '16 at 18:17
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The acf of the residuals suggest a possible seasonal structure perhaps being dampened by all the anomalies/level shifts/time trends etc that might be in the noise process. Secondarily think about a random process that has a large variance. It's history/plot will "look" similar to yours. The forecast for this data would then be a constant with commensurately wide limits. If you create bootstrap forecasts (via monte carlo resampling the model residuals ) then one can obtain a family of forecasts for each future period. This family of forecasts can then lead to possible traces (via simulation) for the future ... all of which will average to the baseline forecast.

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