Inference of stochastic time series with missing data.

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 from the restored data. Using synthetic data of 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 going beyond the most accurate model inference. Using the EM algorithm and the stopping criterion together, we infer missing data points from a time-series data of real neuronal activities. Our method reproduces collective properties of neuronal activities such as correlations and firing statistics even when 70% of data points are masked as missing points.