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. 2832)
are a commonlyused 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 timelag separations. If
nonrandom, then one or more of the autocorrelations will be
significantly nonzero.
In addition, autocorrelation plots are used in the model
identification stage for
BoxJenkins autoregressive, moving average time series models.

Autocorrelation is Only One Measure of Randomness

Note that uncorrelated does not necessarily mean random.
Data that has significant autocorrelation is not random. However,
data that does not show significant autocorrelation can still
exhibit nonrandomness in other ways. Autocorrelation is just one
measure of randomness. In the context of model validation (which is the
primary type of randomness we dicuss in the Handbook), checking for
autocorrelation is typically a sufficient test of randomness since the
residuals from a poor fitting models tend to display nonsubtle
randomness. However, some applications require a more rigorous
determination of randomness. In these cases, a battery of tests,
which might include checking for autocorrelation, are applied since
data can be nonrandom in many different and often subtle ways.
An example of where a more rigorous check for randomness is needed
would be in testing random number generators.

Sample Plot:
Autocorrelations should be nearzero for randomness. Such is
not the case in this example and thus the randomness assumption
fails

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

Definition:
r(h) versus h

Autocorrelation plots are formed by
 Vertical axis: Autocorrelation coefficient
\[ R_{h} = C_{h}/C_{0} \]
where C_{h} is the autocovariance
function
\[ C_{h} = \frac{1}{N}\sum_{t=1}^{Nh}(Y_{t} 
\bar{{Y}})(Y_{t+h}  \bar{{Y}}) \]
and C_{0} is the variance function
\[ C_{0} = \frac{\sum_{t=1}^{N}(Y_{t}  \bar{Y})^2}{N} \]
Note that R_{h} is between 1 and +1.
Note that some sources may use the following formula for
the autocovariance function
\[ C_{h} = \frac{1}{Nh}\sum_{t=1}^{Nh}(Y_{t} 
\bar{{Y}})(Y_{t+h}  \bar{{Y}}) \]
Although this definition has less bias, the (1/N)
formulation has some desirable statistical properties and
is the form most commonly used in the statistics literature.
See pages 20 and
4950 in Chatfield for details.
 Horizontal axis: Time lag h (h = 1, 2, 3, ...)
 The above line also contains several horizontal reference
lines. The middle line is at zero. The other four lines
are 95 % and 99 % confidence bands. Note that there are
two distinct formulas for generating the confidence bands.
 If the autocorrelation plot is being used to test for
randomness (i.e., there is no time dependence in the
data), the following formula is recommended:
\[ \pm \frac{z_{1\alpha/2}} {\sqrt{N}} \]
where N is the sample size, z is the
cumulative distribution function of the standard normal
distribution and
\( \alpha \)
is the significance level. In this case, the confidence bands
have fixed width that depends on the sample size. This is the
formula that was used to generate the confidence bands in
the above plot.
 Autocorrelation plots are also used in the model
identification stage for fitting
ARIMA models.
In this case, a moving average model is assumed for the data
and the following confidence bands should be generated:
\[ \pm z_{1\alpha/2} \sqrt{\frac{1}{N}
(1 + 2 \sum_{i=1}^{k}{y_{i}^2})} \]
where k is the lag, N is the sample size,
z is the cumulative distribution function of the
standard normal distribution and
\( \alpha \)
is the significance level. In this case, the confidence
bands increase as the lag increases.

Questions

The autocorrelation plot can provide answers to the following
questions:
 Are the data random?
 Is an observation related to an adjacent observation?
 Is an observation related to an observation twiceremoved?
(etc.)
 Is the observed time series white noise?
 Is the observed time series sinusoidal?
 Is the observed time series autoregressive?
 What is an appropriate model for the observed time series?
 Is the model
valid and sufficient?
 Is the formula
\[ s_{\bar{{Y}}} = s/\sqrt{N} \]
valid?

Importance:
Ensure validity of engineering conclusions

Randomness (along with fixed model, fixed variation, and fixed
distribution) is one of the four assumptions that typically
underlie all measurement processes. The randomness assumption is
critically important for the following three reasons:
 Most standard statistical tests depend on randomness. The
validity of the test conclusions is directly linked to the
validity of the randomness assumption.
 Many commonlyused statistical formulae depend on the
randomness assumption, the most common formula being the
formula for determining the standard deviation of the sample
mean:
\[ s_{\bar{{Y}}} = s/\sqrt{N} \]
where s is the standard
deviation of the data. Although heavily used, the results
from using this formula are of no value unless the
randomness assumption holds.
 For univariate data, the default model is
If the data are not random, this model is incorrect
and invalid, and the estimates for the parameters (such as
the constant) become nonsensical 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

Examples of the autocorrelation plot for several common
situations are given in the following pages.
 Random (= White Noise)
 Weak autocorrelation
 Strong autocorrelation and
autoregressive model
 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.
