The Implementation of Binned Kernel Density Estimation to Determine Open Clusters' Proper Motions: Validation of the Method

Abstract: Stellar membership determination of an open cluster is an important process to do before further analysis. Basically, there are two classes of membership determination method: parametric and non-parametric. In this study, an alternative of non-parametric method based on Binned Kernel Density Estimation that accounts measurements errors (simply called BKDE-e) is proposed. This method is applied upon proper motions data to determine cluster's membership kinematically and estimate the average proper motions of the cluster. Monte Carlo simulations show that the average proper motions determination using this proposed method is statistically more accurate than ordinary Kernel Density Estimator (KDE). By including measurement errors in the calculation, the mode location from the resulting density estimate is less sensitive to non-physical or stochastic fluctuation as compared to ordinary KDE that excludes measurement errors. For the typical mean measurement error of 7 mas/yr, BKDE-e suppresses the potential of miscalculation by a factor of two compared to KDE. With median accuracy of about 93%, BKDE-e method has comparable accuracy with respect to parametric method (modified Sanders algorithm). Application to real data from The Fourth USNO CCD Astrograph Catalog (UCAC4), especially to NGC 2682 is also performed. The mode of member stars distribution on Vector Point Diagram is located at $\mu_{\alpha}\cos\delta=-9.94\pm0.85$ mas/yr and $\mu_{\delta}=-4.92\pm0.88$ mas/yr. Although the BKDE-e performance does not overtake parametric approach, it serves a new view of doing membership analysis, expandable to astrometric and photometric data or even in binary cluster search.