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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic

1.3.3.1.

Autocorrelation Plot
Purpose:
Check randomness 		Autocorrelation plots (Box and Jenkins, pp. 28-32) are a commonly-used tool for checking randomness in a data set. This randomness is ascertained by computing autocorrelations for data values at varying time lags. If random, such autocorrelations should be near zero for any and all time lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero.

In addition, autocorrelation plots are used in the model identification stage for Box-Jenkins autoregressive, moving average time series models.

Sample Plot:
Autocorrelations should be near-zero for randomness. Such is not the case in this example and thus the randomness assumption fails 		A sample autocorrelation plot

This sample autocorrelation plot shows that the time series is not random, but rather has a high-degree of auto-correlation between adjacent and near-adjacent observations.

Definition:
r(h) versus h 		Autocorrelation plots are formed by
  • Vertical axis: Autocorrelation coefficient

    where Ch is the autocovariance function

    Note--Rh is between -1 and +1.

  • Horizontal axis: Time lag h (h = 1, 2, 3, ...)
Questions The autocorrelation plot can provide answers to the following questions:
  1. Is the data random?
  2. Is an observation related to an adjacent observation?
  3. Is an observation related to an observation twice-removed? (etc.)
  4. Is the observed time series white noise?
  5. Is the observed time series sinusoidal?
  6. Is the observed time series autoregressive?
  7. What is an appropriate model for the observed time series?
  8. Is the model     Y = constant + error     valid and sufficient?
  9. Is the formula valid?
Importance:
Ensure validity of engineering conclusions

Randomness (along with fixed model, fixed variation, and fixed distribution) constitute one of the four assumptions that typically underlie all measurement processes. The randomness assumption is critically important for the following three reasons:

  1. Most standard statistical tests depend on randomnness. The validity of the test conclusions is directly linked to the validity of the randomness assumption.

  2. Many commonly-used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard deviation of the sample mean:

    where is the standard deviation of the data. Although heavily used, the results from this formula are of no value unless the randomness assumption holds.

  3. For univariate data, the default model is

        Y = constant + error

    If the data is not random, this model is incorrect and invalid, and the estimates for the parameters (such as the constant) become non-sensical and invalid.

In short, if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The autocorrelation plot is an excellent way of checking for such randomness.
Examples
  1. Random (= White Noise)
  2. Weak autocorrelation
  3. Strong autocorrelation and autoregressive model
  4. Sinusoidal model
Related Techniques Partial Autocorrelation Plot
Lag Plot
Spectral Plot
Seasonal Subseries Plot
Case Study The autocorrelation plot is demonstrated in the beam deflection data case study.
Software Autocorrelation plots are available in most general purpose statistical software programs including Dataplot.
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