Obstructive sleep Apnea (OSA) is a form of sleep disordered breathing characterized by frequent episodes of upper airway collapse during sleep. Pediatric OSA occurs in 1-5% of children and can related to other serious health conditions such as high blood pressure, behavioral issues, or altered growth. OSA is often diagnosed by studying the patient's sleep cycle, the pattern with which they progress through various sleep states such as wakefulness, rapid eye-movement, and non-rapid eye-movement. The sleep state data is obtained using an overnight polysomnography test that the patient undergoes at a hospital or sleep clinic, where a technician manually labels each 30 second time interval, also called an "epoch", with the current sleep state. This process is laborious and prone to human error. We seek an automatic method of classifying the sleep state, as well as a method to analyze the sleep cycles. This article is a pilot study in sleep state classification using two approaches: first, we'll use methods from the field of topological data analysis to classify the sleep state and second, we'll model sleep states as a Markov chain and visually analyze the sleep patterns. In the future, we will continue to build on this work to improve our methods.