Delay-coordinate reconstruction and conditional-moment causal diagnostics in stochastic systems

Jun Ohkubo

Partial observation and delay-coordinate reconstruction are rooted in deterministic dynamical-systems theory, whereas there are many systems with intrinsic stochasticity. We propose a conditional-moment interpretation of delay-coordinate reconstruction for stochastic systems, in which a delay vector is used to reconstruct conditional moments of a future distribution rather than a unique future sample path. Two complementary arguments motivate this viewpoint. First, the probability density of a stochastic differential equation obeys a deterministic Fokker-Planck equation and, under certain assumptions, is represented by an infinite deterministic hierarchy of moments. Hence, a finite-moment closure suggests a Takens-like finite-dimensional approximation. Second, a discussion based on the Koopman operator theory clarifies that the time evolution of an observable in the Mori-Zwanzig formalism yields a conditional expectation in stochastic systems. Then, the orthogonal "noise" term in the coefficient-space Mori-Zwanzig equation vanishes in the stochastic cases; this result is consistent with the moment-based argument. As an application of this stochastic delay-reconstruction…