Autocorrelation

To some extent it’s what it says: how much something correlates with itself but what on earth does that mean, surely something must correlate perfectly with itself? This is actually a pretty fundamental part of time series analysis. It assumes we have a series of observations made over time, in my worked example in my Rblog, here, I use a series of daily values for my waking and overnight lowest pulse rates each of those is a vector of values. Autocorrelation is about the correlation within either with later values of the same variable, for example how strongly the values correlated with those a day later (“lag 1” autocorrelation in the jargon as it’s the correlation across just one unit step forward).

Details #

Let’s look at the first 10 values of my waking pulse to make this easy enough to show in the glossary format. These values were: 58, 58, 53, 75, 60, 76, 57, 56, 59 and 56. This shows the table of the observed values and their lagged values, laggged by one day.

	PulseRate	lag1PR
1	58	NA
2	58	58
3	53	58
4	75	53
5	60	75
6	76	60
7	57	76
8	56	57
9	59	56
10	56	59

The correlation against the nine pairs of values there is -.231, that’s the lag1 autocorrelation for the first 10 waking pulse rates. It’s actually for nine not ten values because you haven’t got a next value for the last of the 10 observations). That tells us about the stability of this variable across that set of observations for that participant.

Exploring the autocorrelation across the values for any time series variable is generally a staring point for time series analyses and fundamental to cross-lagged correlation analysis (q.v.) and it goes a long way, given other conditions, to allow statistical, i.e. probablistic, analyses of time series data despite the obvious violation of independence of observations (these are all observations from one person, as it happens, me!).

In principle you can only look at autocorrelation and conduct most time series analyses if you have no missing values. In reality particularly for within therapy observations, missing values are often unavoidable and are dealt with by interpolation but the more missing values we have the more cautious we have to be about interpreting time series analyses including autocorrelation values.