Forecast error with different integration steps.

Based on phase space reconstruction theory, the root mean square error is used as a quantitative criterion for identifying the appropriate embedding dimension and time step and selecting the optimal configuration for these factors. The phase space is then reconstructed, and the convergent cross-mapping algorithm is applied to analyse the causality between time series. The causality among the variables in the Lorenz equation is first discussed, and the response of this causality to the integration step of numerical solutions to the Lorenz equation is analyzed. We conclude that changes in the integration step do not alter the causality but will affect its strength. Variables X and Y drive each other, whereas variable Z drives variables X and Y in a unidirectional manner. Second, meteorological data from 1948–2022 are used to analyse the effect of the Southern Hemisphere annular mode on the East Asian summer monsoon index and surface air temperature driving capacity. From a dynamic perspective, it is concluded that the Southern Hemisphere annular mode is the driving factor affecting the East Asian summer monsoon index and surface air temperature. Based on ideal test results and the observation data, the collaborative selection of the embedding dimension and time step is more reliable in terms of determining causality. This provides the ability to determine causality between climate indices and theoretically guarantees the selection of climate predictors.