# What does it mean for a time series to be autocorrelated?

I am trying to become more familiar with time series analysis. I am reading through Dangers and uses of cross-correlation in analyzing time series in perception, performance, movement, and neuroscience: The importance of constructing transfer function autoregressive models by Dean and Dunsmuir, and they mention time series that are "individually autocorrelated". What does this term mean? I am familiar with computing the autocorrelation function of a time series as a function of time lag, but I am not sure what it means for a time series to itself be autocorrelated. Is there just some criteria you check with the autocorrelation function to say whether or not the whole time series is autocorrelated?

• It is not clear what you are asking about. Your question is about understanding the definition of autocorrelation? What is unclear about it? Commented May 17, 2023 at 21:00
• @SextusEmpiricus What does it mean for a whole time series to be autocorrelated? Commented May 17, 2023 at 21:02
• It means that there is a correlation between the samples in the time series. Commented May 17, 2023 at 21:04
• I would claim the phrasing "the time-series are autocorrelated" is somewhat non-standard and I would personally avoid it because it can be confusing. It should be spelled out what exactly they mean. As the OP's question illustrates one can know what an auto-correlation is and be somewhat stumped by this way of phrasing. Commented May 20, 2023 at 20:22

Take the time series without the first observation, $$X_2, \dots, X_T$$, and the time series without the last observation, $$X_1, \dots, X_{T-1}$$. You have two vectors of length $$T-1$$. Calculate their correlation. The result is the lag 1 autocorrelation.

Similarly, you can calculate the lag 2 autocorrelation, which is the correlation between $$X_3, \dots, X_T$$ and $$X_1, \dots, X_{T-2}$$, and more generally any lag $$k$$ autocorrelation.

A series is "autocorrelated" if any one of these is "large enough". Of course, all the sample autocorrelations will typically be nonzero, so one usually checks if any one is significantly larger or smaller than zero.

#### "Auto-correlation" is correlation "with the self" at different points in time

The prefix "auto" means self or same (from the Greek "autós") so "auto-correlation" refers to correlation of a variable with itself, when observed at different times. If you have a set of values $$X_1,...X_T$$ that are measurements of the same essential quantity at different times then the correlation between these would be referred to as "auto-correlation". If there is non-zero correlation between values at different times then we would say that they are "auto-correlated".

I don't really agree with the other answer here. If there is any non-zero correlation at all for a time-series of values then they are auto-correlated, though of course the true correlation value may be unknown and may need to be inferred from observation (and so in that sense a low observed sample correlation may suggest that the true auto-correlation is zero). Moreover, while we often model auto-correlation as a function of time-lags between variables, that is just one way to do it, so that is a non-essential aspect of the concept.

• "though of course the true correlation value may be unknown and may need to be inferred from observation" because of noise? Commented May 18, 2023 at 3:06
• It's not clear from your answer what you “don't agree” with?
– Tim
Commented May 18, 2023 at 4:37
• @Tim It looks like the disagreement is with "A series is 'autocorrelated' if any one of these is 'large enough'" Commented May 18, 2023 at 4:51
• It seems like the difference is that you discuss the underlying autocorrelation of the data generating process, whereas I discuss the sample autocorrelation of the observed values, yes? I could indeed have been more explicit. Commented May 18, 2023 at 14:13
• @StephanKolassa: Yes, I think that's the main difference. Perhaps it's just a foible on my part, but I have always considered moments like correlation to refer to the distributional property and I don't think of the "sample correlation" as a correlation at all --- I think of it as an estimator of correlation.
– Ben
Commented May 18, 2023 at 22:04

Autocorrelation as a function: is that function random, deterministic?

It's from the point of view of the autocorrelation function and its nature that one could shed some light on the idea of autocorrelated time series.

When I first read the notion brought in the question, I thought that, maybe like the OP, talking about a time series being autocorrelated was awkward at best, both mathematically and from a semantic point of view.

Autocorrelation, as has been reminded by the other answer's authors, refers to the correlation of that series with a time- (or sample-) shifted copy of itself. How could 2 identical signals not be correlated?

In a long gone (and missed) past I worked in physical optics, where one measures the duration of ultra fast laser pulses using an autocorrelation operation; it was only possible because each laser pulses were exactly correlated with one another (they were copies of themselves).

So that a time series (or more generally a signal or a function) be correlated with itself sounded obvious: there's no difference between it and itself.

But as hinted, that is ignoring that autocorrelation should more accurately be referred to as an autocorrelation function.

If that function is predictable (or even, deterministic) then the signal can be said to be autocorrelated.

The article actually provides a good definition:

When a time series is autocorrelated, this means that the current value of the series parameter is dependent on preceding values, and can be predicted (at least in part) on the basis of knowledge of those values.

I'd like to illustrate the idea with few plots of time series and their autocorrelation function plots.

• A pure sine wave

• the same sine wave with added amplitude Gaussian noise

• the same pure sine wave with added same amplitude Gaussian noise and a Gaussian phase noise.

As expected, the autocorrelation function has its peak when the shift (lag) is 0, for all versions of the sine wave (pure and noisy ones). For the pure sine wave, the autocorrelation has an attenuated cosine waveform.

The interesting observation is for the autocorrelation function of the noisy sine waves: while the plots are also noisy, they do also have an attenuated cosine waveform, which could be predicted (with proper filtering) and/or approximated; this shows that samples in the time series at a time t have a certain degree of correlation with samples at earlier times.

Following the definition of autocorrelated time series proposed in the article, one can say the 3 sine wave time series shown above are all autocorrelated.

For a final visual illustration, here is the autocorrelation of the pseudo-random amplitude noise that was added to the pure sine wave. The autocorrelation is very low, this time series is not autocorrelated.