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The empirical evaluating formulae of the stochastic differential equation (SDE) are considered in this study, in order to reconstruct the information of the dynamics by the use of time-series itself. In this framework, we regard the time-series as a realisation of a solution of the SDE. By using the solution-coefficient relation studied in the probability theory, we can evaluate the drift vector and the diffusion matrix of the SDE, which are calculated by the first and the second order variations per unit time. The drift vector represents the deterministic bahaviour of the motion and the diffusion matrix play the role of stochastic effect. Utilising the statistically evaluated SDE, one can obtain the predictability estimates in terms of the ensemble variance. The results indicate that the deterministic components are significant for the short-term prediction. Moreover, we need the stochastic components to obtain better predictability estimates by the use of the numerical simulation of the SDE.

Time-series analysis and predictability estimates by empirical SDE modelling