The autocorrelation function (ACF) provides some information about the distribution of hills and valleys across the surface. The normalized ACF, ρ(β), of a profile Z(x) is defined as

L being the sampling length and β the displacement along the surface (Fig. 1.9). When β is zero, the value of the normalized ACF ρ(0) is a maximum and equal to unity. As β tends to infinity, the extent of correlation decreases and ρ(β) tends to zero. If ρ(β) is plotted against β, the curve decays from a value of unity to zero asymptotically at large values of β. For many real surfaces the ACF may be approximated by an exponential decay function. The form of the decay curve provides some information on the horizontal distribution of roughness. Sometimes a correlation length l is defined as the value of β at which ρ(β) equals 0.1. The value of this l is significantly higher in the case of an open texture surface than in a closed one (Fig. 1.10). It is suggested that the simple exponential decay function given by ρ(β) = exp (− 2.3β/l) is a good fit for many surfaces with randomness.

The autocorrelation function (ACF) defines how data points in a time series are related, on average, to the preceding data points (Box, Jenkins, & Reinsel, 1994). In other words, it measures the self-similarity of the signal over different delay times. Accordingly, the ACF is a function of the delay or lag τ, which determines the time shift taken into the past to estimate the similarity between data points. For instance, in a structured process where nearby measurements have similar values but distant points have no relation, the autocorrelation decreases as the lag τ increases. Conversely, the autocorrelation of an unstructured processes like white noise is, in theory, equal to zero for all values of τ > 0 because there is no effect from one time point on another. This fact is exploited to determine the significance of the autocorrelation values. This significance can be estimated by comparing the autocorrelation of a given time series X with the standard error of the autocorrelation of a white noise series with the same variance and number of points as in X. A value is considered significant if its magnitude exceeds the standard error of the white noise (Box et al., 1994). A positive autocorrelation value for a particular lag τ can be interpreted as a measure of persistence of data points separated by this lag to stay above and/or below the mean value of the signal. A negative autocorrelation indicates that data points separated by this lag tend to alternate about the mean value. An important piece of information provided by the ACF is the maximum lag τmax that still has a significant value. This lag indicates the “memory” or temporal persistence of the fluctuation series. Data points separated by time lags greater than τmax are uncoupled. The ACF is often redundantly plotted for positive and negative values of τ, although by definition it is symmetric about τ = 0. Of note, the ACF can also be computed in space. In spatial autocorrelation, the lag τ is then interpreted as a distance between data points. In either case, the characteristics of the temporal and spatial autocorrelation of a signaling process help us to understand the scale at which the pathway operates. These scales help us to define appropriate sampling and provide information on the spatiotemporal characteristics of the associated signal transduction network.