Delay time of GCR modulation with respect to the OF, derived using mutual information, for protons measured by IMP-8, SOHO/EPHIN, AMS-02 and electrons measured by AMS-02 in different time periods (panels (a)–(c), respectively). The delay time and associated uncertainty derived from the modulation potential are shown with markers and error bars. The shaded red and blue areas in panel (c) represent the delay time derived directly from AMS-02 proton and electron data, respectively. The dashed lines in panels (a) and (b) and the dash-dotted lines in panel (c) are the fit delays obtained using the format of κ_R in Eq. (19), detailed in Sect. 5.2.