  Oscillators are ubiquitous in nature, and usually associated with the
existence of an asymptotic phase that governs the long-term dynamics of the
oscillator. % We show that asymptotic phase can be estimated using a carefully
chosen series expansion which directly computes the phase response curve and
provide an algorithm for estimating the co-efficients of this series. Unlike
all previously available data driven phase estimation methods, our algorithm
can: (i) use observations that are much shorter than a cycle; (ii) recover
phase within any forward invariant region for which sufficient data are
available; (iii) recover the phase response curves (PRC-s) that govern weak
oscillator coupling; (iv) show isochron curvature, and recover nonlinear
features of isochron geometry. Our method may find application wherever models
of oscillator dynamics need to be constructed from measured or simulated
time-series.
