I have a couple of series of numbers (sequences/time series) $X_j = \{x_{1,j}, ... , x_{N_{j}, j}\}$ which exhibit a high autocorrelation and a dynamic cyclical feature. Below are some illustrations of some of these series.

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I am looking at ways to capture this dynamic cyclical behaviour and predict the next number $\hat{x}_{N_{j}+1, j}$ in the sequence $X_j$.

What are ways to capture this behavior systematically?

I tried a linear regression on the first lag, i.e. $AC(1)$, since the autocorrelation is close to 0.99, but I am faced with a high MSE as the numbers $x_{i,j}$ have a high nominal value; thus being a slight off is already not accurate enough.

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    $\begingroup$ Could you forecast the absolute value of $x,$ and then add the alternating sign? $\endgroup$ – Nir Dec 10 '16 at 10:43
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    $\begingroup$ Also be aware that ARMA models with several lags can model a surprisingly broad set of behavior. AR(1) will decay towards the mean but AR(2) can give you cycles. $\endgroup$ – Matthew Gunn Dec 10 '16 at 10:49
  • $\begingroup$ why don't you post your data and I will try and help you . $\endgroup$ – IrishStat Dec 10 '16 at 11:49

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