Memberships, distance and proper-motion of the open cluster NGC 188 based on a machine learning method

In this paper, we present an investigation of the memberships, distance and proper motion for the old open cluster NGC 188 using a machine-learning-based method. This method combines two widely used algorithms: spectral clustering (SC) and random forest (RF). The former one is used to construct a reliable training set, the membership probabilities are calculated based on the latter one. This method only depends on reliable training set, no prior knowledge about the cluster is needed. This method is based on the basic assumption that most if not all the information about the cluster members and field stars are contained in a reliable training set, this makes it highly suitable for handling high-dimensional data sets. We use this method to investigate the likely memberships of the old open cluster NGC 188 based on the high-precision astrometry and photometry from the $\text{Gaia}$ Data Release 2 ($\text{Gaia}$ DR2). Based on seven-dimensional features (positions, parallax, proper motions and color-magnitude) of 3780 sample stars in the region of NGC 188, 645 likely members with high membership probabilities ($\ge 0.75$) are obtained. Further analysis confirms the effectiveness of our membership determination. Based on these high-probability memberships, the distance and proper motion of the cluster are determined to be $1866\pm 4$ pc and $(\overline{{\mu }_{\alpha }},\overline{{\mu }_{\delta }})=(-2.33\pm 0.01,-0.97\pm 0.01)$ mas/yr, respectively.