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TIME SERIES ANALYSIS:

AUTOCORRELATION

Chatfield (1996) Chapter 2 (example: Rebel et al., 2012)

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Autocorrelation

• The correlation of data with itself at different ‘lags’

• Lag: difference between points in time

1 year lag between values

2 year lag between values

6 month lag

Autocorrelation

• The correlation of data with itself at different ‘lags’

• Lag: difference between points in time

1 year lag between values

Pair is a given value and the lagged value. Look at ALL PAIRS in the time series.

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Autocorrelation

• Autocorrelation Function (ACF):

• Autocorrelation coefficients measure the correlation

(if any) between observations at different distances

(lags in time) apart

��� �, � − � = ∑ �� � −�̅� � �� −�̅�

�� ���(� − 1)

Autocorrelation

• Autocorrelation Function (ACF):

• Pearson Correlation (r):

��� �, � − � = ∑ �� � −�̅� � �� −�̅�

�� ���(� − 1)

����������� � = ∑ ��� −��̅ ��� −��̅

� − 1

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Autocorrelation Plots

• An autocorrelation plot is a plot of autocorrelation

for multiple lags

• Autocorrelation coefficients are plotted against the lag

• Correlation coefficients are calculated for

observations and observations at multiple lags

Autocorrelation Plots

Time

Lag

Val

ueA

CF

Original

Time Series

Autocorrelation

Plot

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Autocorrelation Example

Autocorrelation Example

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Autocorrelation Plots

• If the time series oscillates either side of its mean,

so the autocorrelation plot oscillates

• If the time series contains a trend, correlations will

not reduce to zero

• If the time series contains seasonal variation, the

autocorrelation will oscillate at the same frequency

• Autocorrelation plots are useful in identifying a

suitable type of model for the data

• e.g. AR, MA, ARMA, ARIMA, etc.