  Inferring nonlinear and asymmetric causal relationships between multivariate
longitudinal data is a challenging task with wide-ranging application areas
including clinical medicine, mathematical biology, economics and environmental
research. A number of methods for inferring causal relationships within complex
dynamic and stochastic systems have been proposed but there is not a unified
consistent definition of causality in this context. We evaluate the performance
of ten prominent bivariate causality indices for time series data, across four
simulated model systems that have different coupling schemes and
characteristics. In further experiments, we show that these methods may not
always be invariant to real-world relevant transformations (data availability,
standardisation and scaling, rounding error, missing data and noisy data). We
recommend transfer entropy and nonlinear Granger causality as likely to be
particularly robust indices for estimating bivariate causal relationships in
real-world applications. Finally, we provide flexible open-access Python code
for computation of these methods and for the model simulations.
