A geometric method of analysis for the light curves of active galactic nuclei

We propose a geometric method of analysis for the light curves of active galactic nuclei (AGNs). The time series of flux ratio is modelled by possibly non-linear random oscillation without specifying the function form. Based on the model, we map the dynamic behaviour of flux ratio to a vector field on a manifold, and then analyse the vector field to retrieve information on the dynamic properties closely linked with the activity of AGNs. While the function form of the model is unspecified, the vector fields and those associated quantities can be estimated by applying a non-parametric filtering method. We illustrate the proposed analysis with an application to light curves of two AGNs supplied by the Kepler satellite. The application shows that the vector field, its derivative, and their combination will be used as the tools of picking up various signals that help understanding of the activity of AGNs. In addition, from a technical viewpoint, the non-parametric filtering method allows the estimation to be robust against outliers. The proposed analysis could be used as an alternative time series analysis of the optical variability other than the analysis by spectral densities or structure functions.