The formulas for s 0, s 2, and r 2 from Definition 2 are shown in cells G8, G11, and G12 (along with an alternative formula in G13). The formulas for calculating s 2 and r 2 using the usual COVARIANCE.S and CORREL functions are shown in cells G4 and G5. ExampleĮxample 1: Calculate s 2 and r 2 for the data in range B4:B19 of Figure 1. For values of n which are large with respect to k, the difference will be small. Observation: The definition of autocovariance given above is a little different from the usual definition of covariance between respectively. See Correlogram for information about the standard error and confidence intervals of the r k, as well as how to create a correlogram including the confidence intervals. A plot of r k against k is known as a correlogram.
The autocorrelation function ( ACF) at lag k, for k ≥ 0, of the time series is defined by The autocovariance function at lag k, for k ≥ 0, of the time series is defined by Note that γ 0 is the variance of the stochastic process.ĭefinition 2: The mean of a time series y 1, …, y n is Definition 1: The autocorrelation function ( ACF) at lag k, denoted ρ k, of a stationary stochastic process, is defined as ρ k = γ k/ γ 0 where γ k = cov(y i, y i+k) for any i.
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