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Students often drift in and out of focus during class. Effective teachers recognize this and re-engage them when necessary. With the shift to remote learning, teachers have lost the visual feedback needed to adapt to varying student engagement. We propose using readily available front-facing video to infer attention levels based on movements of the eyes, head, and face. We train a deep learning model to predict a measure of attention based on overt eye movements. Specifically, we measure Inter-Subject Correlation of eye movements in ten-second intervals while students watch the same educational videos. In 3 different experiments (N=83) we show that the trained model predicts this objective metric of attention on unseen data with $R^2$=0.38, and on unseen subjects with $R^2$=0.26-0.30. The deep network relies mostly on a student's eye movements, but to some extent also on movements of the brows, cheeks, and head. In contrast to Inter-Subject Correlation of the eyes, the model can estimate attentional engagement from individual students' movements without needing reference data from an attentive group. This enables a much broader set of online applications. The solution is lightweight and can operate on the client side, which mitigates some of the privacy concerns associated with online attention monitoring. GitHub implementation is available at https://github.com/asortubay/timeISC, Comment: 10 pages, 3 figures

Complex systems, such as brains, markets, and societies, exhibit internal dynamics influenced by external factors. Disentangling delayed external effects from internal dynamics within these systems is often challenging. We propose using a Vector Autoregressive model with eXogenous input (VARX) to capture delayed interactions between internal and external variables. While this model aligns with Granger's statistical formalism for testing "causal relations", the connection between the two is not widely understood. Here, we bridge this gap by providing fundamental equations, user-friendly code, and demonstrations using simulated and real-world data from neuroscience, physiology, sociology, and economics. Our examples illustrate how the model avoids spurious correlation by factoring out external influences from internal dynamics, leading to more parsimonious explanations of the systems. We also provide methods for enhancing model efficiency, such as L2 regularization for limited data and basis functions to cope with extended delays. Additionally, we analyze model performance under various scenarios where model assumptions are violated. MATLAB, Python, and R code are provided for easy adoption: https://github.com/lcparra/varx