Partial autocorrelation and Autocorrelation difference

I have been trying to understand better the PACF and ACF, but I'm literally struggling.

Have been using a series of articles like: https://machinelearningmastery.com/gentle-introduction-autocorrelation-partial-autocorrelation/

And even looking at the method signature of statmodels for PACF and ACF I am still not able to get it.

Can somebody be so kind to make a "for dummies" explanation with some example of what is happening when data a considered?

Is it correct saying, among the other things, that

• ACF is the plot of the AutoRegressive (AR) modelling whereas
• PACF does the same but for the Moving Average (MA)

Or is it the other way round?

• ACF is the plot of the Moving Average (MA) modelling whereas
• PACF does the same but for the AutoRegressive (AR).

Good evening,

Let's begin with a simple example: you have a time series process, for example some process with correlation of up to lag 4.

An order (p) process is time series process where current value depends on previous p values. You can think for examples of weekly sales in a supermarket, and your AR(4) model in this case shows that your current week sales depend on the previous 4 weeks of sales (on week 1, week 2, week 3, and week 4).

Now, let's get into ACF and PACF.

You have your AR(4) model, which basically tells you that points are correlated up to lag 4, meaning that your correlation between wk 1 and wk 5 is the same as between wk2 and wk 6 (note that the "distance" between them is 4 lags). Similarly, for any number of lags <4 this rule works the same way: corr wk 1 and wk 2 is same as wk 2 and wk 3 and so on. Assume that correlation at lag 1 (wk 1 and wk 2) is stronger than lag 2 (wk1 and wk 3), this means that as number of lags increases, the autocorrelation will also decrease. At each further lag the information from previous lags carries over. Here is a visualization.

PACF is a completely different concept. What it primarily focuses on is finding out the correlation between two points at a particular lag. In your case, say you want to find the "independent" correlation between wk4 and wk3, this is exactly what PACF will show you. Here is a visualization.

All together: ACF data for wk4 will include all the information up to wk3 (wk1, wk2, wk3). PACF data for wk4 will include "independent" (partial) correlations between wk3 and wk4 only (meaning it will by correlation between these points which wasn't explained by their mutual correlations.

To your point about AR and MA processes, it's the other way around. The lag at which ACF becomes very small is order of MA process. And the lag at which PACF becomes very small is order of AR process.

Hope this helps.

• Thanks @Data Sharkie. I've almost got the concept. So ACF analyses all points, from the current up to current - lag using the AutoRegressive (AR) model. The PACF aims for 2 points only at any given lag. But what model is used here? Is the MA? Looking at the MA formula it doesn't seem to be the case. I'm completely lost on the last sentence. – Andrea Moro Apr 17 '20 at 11:57
• @AndreaMoro here is a detailed answer to how ACF and PACF are calculated: stats.stackexchange.com/questions/129052/acf-and-pacf-formula – Data Sharkie Apr 17 '20 at 20:43
• thanks again. It seems even more complicated :( though reading this I may got a better clou However, instead of just correlating with the initial time series, we put together all the lags in-between, and perform a regression analysis, so that the variance explained by the previous time points can be excluded (controlled).. So if I say I want a lag 5 ... I will get 5 previous time serie data but not the current one from where I am starting. Would that be correct my understanding? Sorry, but I'm neither a mathematician or a statistician – Andrea Moro Apr 18 '20 at 8:54
• question for you @Data Sharkie, if you are still around. Was reading again the answer above. When you say ... "You have your AR(4) ..." and then you say "PACF is a completely different story".... but isn't the PACF looking at the AR (AutoRegressive) part? Shouldn't that be ACF? – Andrea Moro Aug 27 '20 at 13:20