b) multivariate early warning signals
The second module contained in EWSmethods is the expansion of EWSs to multivariate data. The benefit of using multivariate techniques over univariate is that assessments of stability and proximity to tipping points can be performed at the system/community level rather than being constrained to the population level. Many of these multivariate EWSs have been tested and supported by Weinans et al. (2021) but open-source tools to calculate them remain unavailable.EWSmethods consequently provides multivariate EWS calculation via the multiEWS function.
There are two forms of EWS indicators appropriate for multivariate data: those averaged across all time series representing the system of interest (Dakos 2018), and those calculated from a dimension reduction (Held and Kleinen 2004, Weinans et al. 2019). The former is a simple technique to implement using just uniEWS but can be influenced by outlier time series, whereas the latter can display informative properties not identifiable in individual time series (Weinans et al. 2021). Unfortunately, their theoretical relationship with CSD is less well understood. EWSmethods and the multiEWS function therefore provides 12 multivariate indicators across both averaging and dimension reduction forms, each of which is described in Table 3.
Parameterisation of multiEWS is identical to uniEWSapart from the lack of capability for composite EWSs. This is due to it being currently unknown how combining multivariate EWS indicators influences their prediction reliability. Rolling and expanding windows are still available for multivariate EWSs and their interpretation remains the same as their univariate equivalents.