Spatial convergent cross mapping to detect causal relationships from short time series

Abstract

Recent developments in complex systems analysis have led to new techniques for detecting causal relationships using relatively short time series, on the order of 30 sequential observations. Although many ecological observation series are even shorter, perhaps fewer than ten sequential observations, these shorter time series are often highly replicated in space (i.e., plot replication). Here, we combine the existing techniques of convergent cross mapping (CCM) and dewdrop regression to build a novel test of causal relations that leverages spatial replication, which we call multispatial CCM. Using examples from simulated and real-world ecological data, we test the ability of multispatial CCM to detect causal relationships between processes. We find that multispatial CCM successfully detects causal relationships with as few as five sequential observations, even in the presence of process noise and observation error. Our results suggest that this technique may constitute a useful test for causality in systems where experiments are difficult to perform and long time series are not available. This new technique is available in the multispatialCCM package for the R programming language.

A 3000 year record of Caribbean reef urchin communities reveals causes and consequences of long‐term decline in Diadema antillarum

“…CCM tests for significant causal relationships by recognizing that the observed values of a forcing process should be significantly better explained by observed values of a response process than expected by chance, and that the accuracy of that explanation should improve with increasing time series length (Sugihara et al 2012, Clark et al 2015, Ye et al 2015. We used a variant of CCM, multispatial CCM, which is appropriate for spatially replicated time series that individually contain short series (∼ 30) of sequential ecological observations of systems that share similar dynamics (Clark et al 2015), such as samples (5 cm slices) from our four individual cores.…”

“…parrotfish abundance, urchin abundance, accretion rate), 2) test for nonlinearity and stochastic noise to ensure that none of the processes were purely random and that stochastic noise was not so large that causal links could not be recovered; stochasticity and nonlinearity were tested by visually assessing whether predictive power was reasonably high for short time steps and decreased with increasing prediction time, respectively (Sugihara et al 2012, Clark et al 2015, 3) calculate the ability of two processes to describe each other's dynamics using CCM by confirming that predictive skill (Pearson correlation coefficient, rho) increased with greater sample size, 4) use bootstrapping with replacement to leverage spatial information to reshuffle the order of spatial replicates and calculate rho, and 5) use nonparametric bootstrapping to test whether predictions indicated a significant causal relationship by determining whether calculated rho was significantly greater than zero and whether it increased significantly with sample size. One hundred bootstrap iterations were performed for steps 4 and 5.…”

Relating seasonal dynamics of enhanced vegetation index to the recycling of water in two endorheic river basins in north-west China

“…It is always possible to get bi-directional convergence when variables are strongly forced by an external third variable, resulting in synchrony between variables being assessed. Synchrony should be tested for convergent cross-mapping to determine bi-directional pairing (Sugihara et al, 2012;Clark et al, 2015). When synchrony exists, it can sometimes be minimised by processing the "first difference" of cross-correlated variables by subtracting previous observations (at time t − 1) from current observations (at t) in the original time series prior to performing the analysis (Granger and Newbold, 1974).…”

“…Dotted lines on either side of the predictive-skill curves represent the ±standard error of estimate assessed from bootstrapping based on 3000 iterations. Convergent cross-mapping is based on procedures written in the R-programming language initially developed by Clark et al (2015). The feature that assures causality in variables is the convergence in the predictive curves as record length increases.…”

Sparse Causality Network Retrieval from Short Time Series

“…There are alternative methods to extract causality networks from short time series, in particular Multispatial CCM [26,27] appears to perform well for short time series. A comparison between different approaches and the application of these methods to real data will be extremely interesting.…”