The Autocorrelation function is one of the widest used tools in timeseries analysis. It is used to determine stationarity and seasonality.

Stationarity:

This refers to whether the series is “going anywhere” over time. Stationary series have a constant value over time.

Below is what a non-stationary series looks like. Note the changing mean.

And below is what a stationary series looks like. This is the first difference of the above series, FYI. Note the constant mean (long term).
The above time series provide strong indications of (non) stationary, but the ACF helps us ascertain this indication.

If a series is non-stationary (moving), its ACF may look a little like this:

The above ACF is “decaying”, or decreasing, very slowly, and remains well above the significance range (dotted blue lines). This is indicative of a non-stationary series.

On the other hand, observe the ACF of a stationary (not going anywhere) series:

Note that the ACF shows exponential decay. This is indicative of a stationary series.

Consider the case of a simple stationary series, like the process shown below:

We do not expect the ACF to be above the significance range for lags 1, 2, … This is intuitively satisfactory, because the above process is purely random, and therefore whether you are looking at a lag of 1 or a lag of 20, the correlation should be theoretically zero, or at least insignificant.

Thank you Abbas for simple and well explained topic.
My question is in non-stationary data how can we find auto correlation? is partial auto correlation is a good alternative?

Hi Bahaa, thanks for the kind words. An autocorrelation for a non-stationary series would look funny, kinda of like here: https://coolstatsblog.files.wordpress.com/2013/08/berlin2.jpeg. Are you trying to prove that the realizations/values are correlated?

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Hi Good simple explanation – I’ve always believed if you can explain simply – the person has understood it thoroughly 🙂 Came across the term an hour or so ago (ACF term) and was looking for a simpler explanation

I agree, Rajesh. I think the best part about understanding something fully is that you can take control of the language around it, and therefore simplify it. Thanks for visiting!

Thanks for this clarifying post!
The criteria for a stationary time series are (1) constant mean, (2) constant variance, (3) the covariance between today’s independent variable and tomorrow’s independent variable is not a function of time. In exactly what way does autocorrelation (correlation in the error terms) violate these three criteria?

thank you very much…..

Thank you Abbas for simple and well explained topic.

My question is in non-stationary data how can we find auto correlation? is partial auto correlation is a good alternative?

Hi Bahaa, thanks for the kind words. An autocorrelation for a non-stationary series would look funny, kinda of like here: https://coolstatsblog.files.wordpress.com/2013/08/berlin2.jpeg. Are you trying to prove that the realizations/values are correlated?

You’re welcome Amin 🙂

Hi Can you explain relation between Auto correlation and Confidence Interval with same intuitive explanation

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you’re talking about! Bookmarked. Please also talk over with my web site =).

We will have a hyperlink alternate arrangement between us

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It’s simple, yet effective. A lot of times it’s

tough to get that “perfect balance” between user friendliness and visual appeal.

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on Opera. Excellent Blog!

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Hi Good simple explanation – I’ve always believed if you can explain simply – the person has understood it thoroughly 🙂 Came across the term an hour or so ago (ACF term) and was looking for a simpler explanation

And after a few hits – here it is 🙂

Rajesh

I agree, Rajesh. I think the best part about understanding something fully is that you can take control of the language around it, and therefore simplify it. Thanks for visiting!

Hi Abbas,

Just a non scientific comment to edit the post:

The word autocorrelation on the title is missspelled and needs a “L” 🙂

Thanks!

Can the acf be used to provide at least five comments about a series? If it is possible pls give me five of them

I am interested in knowing how do we assign the blue line in stationary series data

Hello,

a slight correction needed: MA(1) process is Y(t)=u(t)+b*u(t-1).

What you gave an example of above is a MA(0) process.

Good spot, Pranjal! It has been corrected. Thanks.

thanx sir, how can i get a pdf paper for this subject.

Thanks for this clarifying post!

The criteria for a stationary time series are (1) constant mean, (2) constant variance, (3) the covariance between today’s independent variable and tomorrow’s independent variable is not a function of time. In exactly what way does autocorrelation (correlation in the error terms) violate these three criteria?

awesome explanation… Thank you sir….