Supposed that we have a given autocorrelation function. The question is, how to build a signal (any signal, I know that the solution is not unique) that has the given autocorrelation function?
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$\begingroup$ You might want to provide more specifics (including if relevant the software package you want to implement this in) to increase the chances of getting the kind of response you want. For example, if you want to generate an autoregressive series (at discrete time intervals - or do you want time being continuous?) you can start with $x_1$ distributed as $N(0,1)$ and $x_i=\rho x_{i-1}+\epsilon_i*\sqrt{1-\rho^2}$ and $\epsilon_i$ distributed as $N(0,1)$. If you have a particular software package in mind, then maybe there's something already written. $\endgroup$– JimBCommented Sep 17, 2015 at 15:00
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$\begingroup$ In the future down the line I would like to implement this in Python. But I think this is just a general procedure so I asked it that way. Isn't out there a Mathematical general way of answering that?. $\endgroup$– Heberto MayorquinCommented Sep 17, 2015 at 15:35
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$\begingroup$ You can use LR decomposition to create a vector of (auto)correlated random variables $\endgroup$– Ggjj11Commented Oct 26 at 14:13
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Given an auto-correlation function you can derive the partial auto-correlation and thusly the appropriate ADL since the partial auto-correlation can suggest coefficients of the model. Simulation can then proceed.
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1$\begingroup$ given that a series is stationary, there's a one to one between the ACF and the corresponding ARIMA series with that ACF. So, if you're given an autocorrelation function, you try to map it to the autocorrelation function of a some chosen ARIMA series. But, unless the autocorrelation function really comes from some ARIMA series, you're not going to get a perfect match. $\endgroup$– mloftonCommented Sep 30, 2020 at 15:28
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$\begingroup$ which is precisely why one has to do this in a robust manner allowing for pulses/level/step shifts,local time trends and seasonal pulses AND changes in error variance and parameters over time ala AUTOBOX $\endgroup$ Commented Oct 3, 2020 at 19:49