One way to detect CSD is to measure the rate at which a system returns to its initial state following known disturbances. A resilient system with strong restoring feedbacks will return to its initial state faster than one which is near to a tipping point (Wissel, 1984). However, this method requires the occurrence of well-defined perturbations, as well as clear knowledge of when the equilibrium state of the system has been reached again, neither of which are always clearly defined in the real world. Hence statistical techniques are often used to detect CSD behaviour in the form of resilience loss of a system (resilience defined in this chapter as the ability to return to the equilibrium state) prior to a tipping point.

As a system approaches a tipping point and its recovery slows down, the system at each time step *t* is more correlated to the previous timestep, *t-1* (as shown in Figure 1.6.2). This can be measured with ‘lag-1 autocorrelation’ (or AR(1)) which measures a system’s self-similarity through time, and tends towards 1 as a system experiences CSD prior to tipping to a different state (Scheffer et al., 2009). Visually, this can be viewed by observing a scatterplot of a section of the time series data of the system against the same section of time series lagged by one time point (Figure 1.6.2). When the system is far from tipping (left column Figure. 1.6.2), there is no relationship between the system now and with itself at the previous time step (i.e. low AR(1)). As the system approaches the tipping point, CSD means that there is a stronger correlation between the system now and with itself at the previous time step (and thus a higher AR(1)). Larger deviations in the red section of the time series can be seen, further showing this slowing down and increase in AR(1).

Similarly, as resilience is lost, a given perturbation will cause a greater movement of the ‘ball’ from Figure 1.6.1, meaning the variability (measured as ‘variance’) of the system is expected to increase. The system can sample more of its ‘state space’ (all the possible states the system can be in) due to the shallower well (Scheffer et al., 2009). Theory shows that because of CSD, AR(1) and variance should increase together in a characteristic fashion where their ratio remains constant. Hence an increase in both variance and AR(1) should be sought for robustness. However, there are other factors which can lead to a change in variance, such as a change in the variance of the system’s external forcing (Ditlevsen and Johnson, 2010).

For these time-based measures to be robust they require the time series data of the system to be stationary (i.e. the data’s statistical properties do not depend on the time at which the data is observed). Multiple methods exist for removing non-stationarity, typically by detrending the time series (i.e. subtracting any trends). The EWS is then calculated on this detrended time series on a short section that shifts one time point at a time before recalculating (a ‘moving window’) (Figure 1.6.2).

In our example tipping point, as the current steady state of the system is losing resilience and the probability to shift to an alternative state increases, the distribution of states of the system is expected to become increasingly skewed toward the alternative state (because of the increasing asymmetry of the potential well, see Figure 1.6.2). This can be quantified by the ‘skewness’ of the system, again measured on a moving window as described above. A change in skewness itself is not linked to CSD, but can be used as an EWS. ‘Flickering’ may also be observed before a tipping point, where sufficient noise can push a system temporarily into an alternative state before returning to the original with increasing likelihood as the system is approaching tipping (Wang et al., 2012; Dakos et al., 2013).

Once an EWS indicator has been calculated across a time series dataset, its tendency can be measured to determine if there is a movement towards tipping. Kendall’s tau correlation coefficient is a common way to estimate trends. It is 1 if the time series of the indicator is always increasing (every value in the time series is higher than all of the previous values), -1 if it is always decreasing, and 0 if there is no overall trend. The significance of this trend can be calculated using null models that resample the time series such that the statistical properties of the system are maintained (e.g. mean and overall variance) but the memory of the system, such as changes in AR(1) and variance over time, are destroyed (Dakos et al., 2008). From these, the significance or p-value of an observed trend can be calculated. In practice, different detrending techniques and window lengths are used to test the robustness of EWS.

Particularly for temporal methods, there is a need to carefully consider the different timescales of the system. For EWS to work, a slow external forcing should move the system towards tipping, while faster ‘noise’ processes push the system away from equilibrium in either direction (noise can be thought of as short-term weather events in the climate system). This allows us to measure the return rate and other indicators on a time series from the system as it is monitored somewhere in between these timescales. The indicators may fail if, for example, the system is forced too fast towards tipping. This is further detailed in the Limitations section (1.6.1.6).