Application of the Gaussian mixture model in pulsar astronomy -- pulsar classification and candidates ranking for {it Fermi} 2FGL catalog

Machine learning, algorithms to extract empirical knowledge from data, can be used to classify data, which is one of the most common tasks in observational astronomy. In this paper, we focus on Bayesian data classification algorithms using the Gaussian mixture model and show two applications in pulsar astronomy. After reviewing the Gaussian mixture model and the related Expectation-Maximization algorithm, we present a data classification method using the Neyman-Pearson test. To demonstrate the method, we apply the algorithm to two classification problems. Firstly, it is applied to the well known period-period derivative diagram, where we find that the pulsar distribution can be modeled with six Gaussian clusters, with two clusters for millisecond pulsars (recycled pulsars) and the rest for normal pulsars. From this distribution, we derive an empirical definition for millisecond pulsars as $frac{dot{P}}{10^{-17}} leq3.23(frac{P}{100 textrm{ms}})^{-2.34}$. The two millisecond pulsar clusters may have different evolutionary origins, since the companion stars to these pulsars in the two clusters show different chemical composition. Four clusters are found for normal pulsars. Possible implications for these clusters are also discussed. Our second example is to calculate the likelihood of unidentified textit{Fermi} point sources being pulsars and rank them accordingly. In the ranked point source list, the top 5% sources contain 50% known pulsars, the top 50% contain 99% known pulsars, and no known active galaxy (the other major population) appears in the top 6%. Such a ranked list can be used to help the future follow-up observations for finding pulsars in unidentified textit{Fermi} point sources.