Delay-coordinate reconstruction reveals causal links in random systems
Delay-coordinate reconstruction and conditional-moment causal diagnostics in stochastic systems
Machine Learning
Summary
Understanding cause and effect in systems that have random elements is tricky because you can't predict exact outcomes. This paper shows how using past data points together (delay-coordinate reconstruction) can help estimate the likely future behaviors of such systems, not just single outcomes. The authors explain this with math describing probabilities and moments, which gives a way to explore causal connections even when things are noisy and unpredictable. They then use this idea to improve a method for detecting cause and effect in systems influenced by randomness.
What this means in practice
- •For data scientists in complex systems: Improve detection of causal relationships in noisy, stochastic models to better understand interactions in finance, biology, or climate data.
- •For control engineers: Use delay-coordinate reconstruction based on conditional moments to design controllers for systems subject to unpredictable disturbances.
Tested on simulated data.
Authors
Jun Ohkubo
Abstract
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 viewpoint, we revisit convergent cross mapping (CCM) for diagnosing certain causal relationships. Although CCM based on the embedding theorem cannot generally be applied to stochastic systems, it is possible to examine certain types of causal relationships by using conditional moments. Using coupled logistic systems with additive and multiplicative coupling mechanisms, we discuss how causal relationships are embedded in stochastic systems.