Fig. 5. Model-free methods, but not our method, make a false prediction due to the presence of synchrony and indirect effect.

a–d We apply our method and popular model-free methods (i.e., GC, CCM, and PCM) to various experimental time-series data obtained from the prey–predator system merged with the genetic oscillator (a); repressilator (b); cofactors at the pS2 promoter (c); and air pollutants and cardiovascular disease (d). For the air pollutants and cardiovascular disease data, we test the methods on three years of data (d gray) and on two years of data (d purple). Source data are provided as a Source Data file.