In this paper, we address the detection of engagement decrease of users spontaneously interacting with a socially assistive robot in a public space. We first describe the UE-HRI dataset that collects spontaneous Human-Robot Interactions following the guidelines provided by the Affective Computing research community to collect data "in-the-wild". We then analyze the users' behaviors focusing on proxemics, gaze, head motion, facial expressions and speech during interactions with the robot. Engaged behaviors versus signs of engagement decrease exhibited by the users were annotated and analyzed. Finally, we investigate the use of deep leaning techniques (Recurrent and Deep Neural Networks) to detect user engagement decrease in real-time. The results of this work particularly highlight the relevance of taking into account temporal dynamics of the user's behavior. Allowing 1 to 2 seconds as buffer delay improves the performance of taking a decision on user engagement.