For the past month I have been trying to understand the math behind the autocorrelation function and partial autocorrelation function for time-series project I have been working on. However, I am only able to find loads of articles which answer questions like How you can generate ACF & PACF plot in Python or in R, How to understand ACF and PACF plot? or How to obtain p and q values from ACF or PACF plot? Nowhere I am able to find something which tells me the exact math behind them!

I am looking for something that derives this comprehensively enough, in hopes of trying to replicate derivation myself(am a computer science graduate). Can anyone help me out with the same? Any resource would do! Or maybe list down all the steps so that I can try researching in pieces.

  • $\begingroup$ This paper on the characterization of the pacf might be of some help to you. $\endgroup$
    – Stochastic
    Sep 1, 2020 at 9:56
  • $\begingroup$ Thank you @Stochastic for the resource. Will definitely check this out. $\endgroup$ Sep 1, 2020 at 17:26
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    $\begingroup$ You will find all the math in the Brockwell & Davis book "Introduction to Time Series and Forecasting" $\endgroup$ Feb 1, 2022 at 17:06

1 Answer 1


The lectures provided here, based on the book Time Series Analysis are comprehensive. They might require a bit of background in control theory though.

  • $\begingroup$ Thank you @Akylas, will check the resource out! The lectures seem to be comprehensive. Will try and add an answer here, if I am able to churn up any of the derivations! $\endgroup$ Sep 1, 2020 at 17:28

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