Interpreting a negative autocorrelation (created 2009-02-16)

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I have two questions regarding autocorrelation:

  1. if there is negative autocorrelation is it correct to say that "past values decreasingly influence future values?
  2. Why is positive auto-correlation considered more important by most statisticians.

Autocorrelation is the tendency for observations made at adjacent time points to be related to one another. The NIST Engineering Statistics Handbook has a nice description of autocorrelation in section

Your interpretation is troublesome. For ANY autocorrelation past values decreasingly influence future values in the sense that the strength of the correlation dies down as the separation in time increases.

A negative autocorrelation changes the direction of the influence. A negative autocorrelation implies that if a particular value is above average the next value (or for that matter the previous value) is more likely to be below average. If a particular value is below average, the next value is likely to be above average.

It's not difficult to show that values separated by two time periods rather than one are positively correlated for the same reason that a negative number squared is positive.

Here are the examples of negative autocorrelation that I provide in my classes.

  1. If you've ever seen a row of cabbages growing in a garden, you'll frequently notice an alternating pattern--big cabbage, little cabbage, big cabbage, little cabbage, etc. This happens because one cabbage might have a slight edge in growth. It extends into its neighbor's space, stealing water and nutrition for itself. Because of this slight competitive edge, the one cabbage grows even bigger at the expense of the neighboring cabbage.
  2. If you are looking at the amount of time a doctor spends with successive patients, if the first patient finished faster than expected, you are more likely to adopt a leisurely approach with the second patient. If the first patient takes longer than expected, you are more likely to rush with the second patient, trying to get back on schedule.
  3. In an assembly line process where small pieces are cut from a single large piece, if the first piece is a bit long, the next piece is likely to be a bit short and vice versa.

Negative autocorrelation is a violation of independence but it is generally less worrisome because (a) it seems to appear less frequently than positive autocorrelation, and (b) it actually produces greater precision in the average than an independent series would. The alternating pattern in a negative autocorrelation insures that a series will be more likely to bracket the true mean. Still it represents a lost opportunity to model the correlation and get a better estimate of confidence limits.