Using mutual information to measure time lags from nonlinear processes in astronomy

Measuring time lags between time series or light curves at different wavelengths from a variable or transient source in astronomy is an essential probe of physical mechanisms causing multiwavelength variability. Time lags are typically quantified using discrete correlation functions (DCFs), which are appropriate for linear relationships. However, in variable sources such as x-ray binaries, active galactic nuclei (AGNs), and other accreting systems, the radiative processes and the resulting multiwavelength light curves often have nonlinear relationships. For such systems it is more appropriate to use nonlinear information-theoretic measures of causation such as mutual information, routinely used in other disciplines. We demonstrate with toy models the limitations of using the standard DCF and show improvements when using a discrete mutual information function (DMIF). For nonlinear correlations, the latter accurately and sharply identifies the lag components as opposed to the DCF, which can be erroneous. Following that, we apply the DMIF to the multiwavelength light curves of AGN NGC 4593. We find that x-ray fluxes are leading UVW2 fluxes by ∼0.2 days, closer to model predictions from reprocessing by the accretion disk than the DCF estimate. The uncertainties with the current light curves are too large, though, to rule out negative lags. Additionally, we find another delay component at approximately 1 day, i.e., UVW2 leading x rays, consistent with inward propagating fluctuations in the accretion disk scenario. This is not detected by the DCF. Keeping in mind the nonlinear relation between x ray and UVW2, this is worthy of further theoretical investigation. From both the toy models and real observations, it is clear that the mutual-information-based estimator is highly sensitive to complex nonlinear relations. With sufficiently high temporal resolution and signal-to-noise ratio, we will precisely detect each of the lag features corresponding to these relations.