  Dripping water from a faucet is a typical example exhibiting rich nonlinear
phenomena. For such a system, the time stamps at which water drops separate
from the faucet can be directly observed in real experiments, and the time
series of intervals \tau_n between drop separations becomes a subject of
analysis. Even if the mass m_n of a drop at the onset of the n-th separation,
which cannot be observed directly, exhibits perfectly deterministic dynamics,
it sometimes fails to obtain important information from time series of \tau_n.
This is because the return plot \tau_n-1 vs. \tau_n may become a multi-valued
function, i.e., not a deterministic dynamical system. In this paper, we propose
a method to construct a nonlinear coordinate which provides a "surrogate" of
the internal state m_n from the time series of \tau_n. Here, a key of the
proposed approach is to use ISOMAP, which is a well-known method of manifold
learning. We first apply it to the time series of $\tau_n$ generated from the
numerical simulation of a phenomenological mass-spring model for the dripping
faucet system. It is shown that a clear one-dimensional map is obtained by the
proposed approach, whose characteristic quantities such as the Lyapunov
exponent, the topological entropy, and the time correlation function coincide
with the original dripping faucet system. Furthermore, we also analyze data
obtained from real dripping faucet experiments which also provides promising
results.
