  The estimation of equations from data is of interest in physics. One of the
famous methods is the sparse identification of nonlinear dynamics (SINDy),
which utilizes sparse estimation techniques to estimate equations from data.
Recently, a method based on the Koopman operator has been developed; the
generator extended dynamic mode decomposition (gEDMD) estimates a time
evolution generator of dynamical and stochastic systems. However, a naive
application of the gEDMD algorithm cannot work well for stochastic differential
equations because of the noise effects in the data. Hence, the estimation based
on conditional expectation values, in which we approximate the first and second
derivatives on each coordinate, is practical. A naive approach is the usage of
locally weighted expectations. We show that the naive locally weighted
expectation is insufficient because of the nonlinear behavior of the underlying
system. For improvement, we apply the clustering method in two ways; one is to
reduce the effective number of data, and the other is to capture local
information more accurately. We demonstrate the improvement of the proposed
method for the double-well potential system with state-dependent noise.
