Causal analysis of time series data for modeling nonlinear phenomena

In typhoon forecasting, air-only and coupled air-sea models have similar accuracy in predicting typhoon trajectories. However, air-sea interactions must be considered to accurately forecast typhoon intensity [1]. Although coupling between multiple modules, including turbulence, waves, ecosystem, and chemistry, has been suggested to improve forecast accuracy, the modules and their individual model equations for typhoon forecasting are still determined empirically. Accurate modeling of the interactions between phenomena across multiple modules is an essential determinant of simulation accuracy. To determine critical factors within each module, parameterizations should be determined quantitatively, not empirically. However, it is challenging to impose preconditions on models that accurately capture the many complex interactions between air and sea.

In this study, we propose a modeling method to identify these critical factors using a causal analysis based on information theory. The causality of typical causal network models depends on the precondition network shape, but by using information theory, it is possible to extract causality comprehensively without preconditions. This allows for a quantitative assessment of causality without making the assumptions necessary for causal networks, such as Bayesian networks. In the proposed method, the information flux T, also known as transfer entropy, is defined as the difference in the Shannon entropy for multi-elements Q over two timesteps tⁿ and tⁿ⁺¹ [2], as follows

T_J→I = H(Q_jⁿ⁺¹Q_≠iⁿ ) − H(Q_jⁿ⁺¹Qⁿ),

= ∑_i,j p(iⁿ⁺¹,iⁿ,jⁿ) log p(iⁿ⁺¹iⁿ,jⁿ) / p(iⁿ⁺¹iⁿ),

where H(Q) = Σ p(q) log p(q) is Shannon entropy, and we define Q as containing two elements Q = (I,J). Information flux quantifies the causality and amount of information flow between two time series. The magnitude of T corresponds to the parameter value indicating the interactions within and between the models. For example, recently, this method of quantifying causality was also applied to turbulence [3], which is one of the most chaotic phenomena, and used to clarify the causality of interactions between scales in the transport of scales in developed turbulence [4]. As a first step, we apply this method to a simplified non-linear model, and try to reconstruct its original model equation for test cases of the Lotka-Volterra model and the Lorenz model. For combinations of time series data for multiple variables generated by the models as multi-dimensional ordinary differential equations, we calculated the information flux according to the equation to extract the causal relationships of combinations with high T values. Then, by selectively rebuilding the model with only the variables of the elements that cause a high T_{cause→effect} value as the basis of the model function, the cost of parameter optimization is reduced, and the optimal parameter values are determined by fitting with the original time series data. In the presentation, we will discuss possibilities of the proposed method and its potential applications in climate simulations.

 

References
[1] L. R. Schade and K. A. Emanuel, J. Atmos. Sci. 56, pp. 642–651 (1999).
[2] T. Schreiber, Phys. Rev. Lett. 85, pp. 461–464 (2000).
[3] A. Lozano-Durán and G. Arranz, Phys. Rev. Res. 4, 023195 (2022).
[4] R. Araki, A. Vela-Martín, and A. Lozano-Durán, J. Phys.: Conf. Ser. 2753, 012001 (2024).