Autocorrelation is a measure of persistence within a data set, which can be defined as the tendency for successive data points to be similar (Wilks, 2011). In atmospheric science temporal autocorrelation can be a helpful tool for model evaluation. Temporal autocorrelation is also a fundamental concept for synthetic weather generation (for more detail see Julie’s fantastic series of blog posts on synthetic weather generation here). Calculating autocorrelation within a sample data set can also be a helpful for assessing the applicability of classical statistical methods requiring independence of data points within a sample. Should a data set prove to be strongly persistent, such methods will likely yield inaccurate results.

Autocorrelation is commonly computed by making a copy of the original data set, shifting the copy k points forward (where k is the *lag* over which you would like to compute the autocorrelation) and computing the Pearson correlation coefficient between the original data set and the copy.

Where:

The calculation of autocorrelation for a number of different lags at once is known as the autocorrelation function. Plotting the autocorrelation graphically can be a helpful tool for quickly assessing the presence of autocorrelation within a data set.

You can generate such plots in Matlab using the simple command shown below:

autocorr(T,k)
% T is your data set and k is the number of lags you would like to compute

The command generates a plot of the autocorrelation function. Below are two examples, the first is the autocorrelation function of a set of observed temperature values in Des Moines Iowa, the second is autocorrelation function of the temperature values at the same location as modeled by the MM5I regional climate model:

Figure 1: Temporal autocorrelation function of temperature observations from Des Moines Iowa (temperatures reported at 3 hour intervals)

Figure 2: Temporal autocorrelation function of temperature produced by the MM5I regional climate model for Des Moines Iowa (temperatures reported at 3 hour intervals)

Note the cyclical nature of the autocorrelation functions, this is a reflection of the daily temperature cycle. The autocorrelations function of the maximum or minimum temperatures would show more constant persistence.

Sources:

Wilks, D. S. (2011). *Statistical methods in the atmospheric sciences*. Burlington, MA: Academic Press.

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