  Inferring dynamics from time series is an important objective in data
analysis. In particular, it is challenging to infer stochastic dynamics given
incomplete data. We propose an expectation maximization (EM) algorithm that
iterates between alternating two steps: E-step restores missing data points,
while M-step infers an underlying network model of restored data. Using
synthetic data generated by a kinetic Ising model, we confirm that the
algorithm works for restoring missing data points as well as inferring the
underlying model. At the initial iteration of the EM algorithm, the model
inference shows better model-data consistency with observed data points than
with missing data points. As we keep iterating, however, missing data points
show better model-data consistency. We find that demanding equal consistency of
observed and missing data points provides an effective stopping criterion for
the iteration to prevent overshooting the most accurate model inference. Armed
with this EM algorithm with this stopping criterion, we infer missing data
points and an underlying network from a time-series data of real neuronal
activities. Our method recovers collective properties of neuronal activities,
such as time correlations and firing statistics, which have previously never
been optimized to fit.
