Observability-based causal inference in linear stochastic systems with hidden inputs

Causal inference from neural time series is confounded by correlations induced by hidden common inputs, a problem that is especially acute because the full set of variables shaping neural activity is rarely accessible. Prediction based measures such as Granger causality can therefore falsely attribute such dependencies to causal influence. Delay coordinate reconstruction, which is also closely connected to observability in control theory, provides a complementary route: in deterministic systems, downstream delays can reconstruct upstream causes, whereas the reverse does not generally hold. This asymmetry provides a signature of causal direction, as exploited by convergent cross mapping. In stochastic dynamics, however, dynamical and observation noise enter delay coordinates and prevent exact trajectory reconstruction. Here, we develop data driven criteria that extend this observability principle to partially observed stochastic systems. First, we show that observability can be tested by regression of time lagged covariance matrices, which removes direct noise contributions while preserving the observability relation. Second, we show that causal direction can be tested through inclusion of stochastic Koopman eigenvalues estimated from upstream and downstream delay coordinates, because downstream observations generically contain the dynamical components observable upstream. Numerical examples show that the proposed criteria recover causal direction and reject spurious dependencies induced by unobserved common inputs, including cases in which Granger causality and convergent cross mapping fail to distinguish common input from causal influence. Although motivated by neural recordings, the framework applies broadly to partially observed stochastic dynamical systems beyond neuroscience.