• Anomaly Detection to Predict Relapse Risk in Schizophrenia:
    The integration of technology in clinical care is growing rapidly and has become especially relevant during the global COVID-19 pandemic. Smartphone-based digital phenotyping, or the use of integrated sensors to identify patterns in behavior and symptomatology, has shown potential in detecting subtle moment-to-moment changes. These changes, often referred to as anomalies, represent significant deviations from an individual’s baseline, may be useful in informing the risk of relapse in serious mental illness. Our investigation of smartphone-based anomaly detection resulted in 89% sensitivity and 75% specificity for predicting relapse in schizophrenia. These results demonstrate the potential of longitudinal collection of real-time behavior and symptomatology via smartphones and the clinical utility of individualized analysis. Future studies are necessary to explore how specificity can be improved, just-in-time adaptive interventions utilized, and clinical integration achieved.

  • On Marshall Hall’s Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves:
    For a non-square positive integer x, let k_x denote the distance between x^3 and the perfect square closest to x^3. A conjecture of Marshall Hall states that the ratios r_x = (x^(1/2))/k_x, are bounded above. (Elkies has shown that any such bound must exceed 46.6.) Let {x(n)} be the sequence of “Hall numbers”: positive non-square integers for which r_x(n) exceeds 1. Extensive computer searches have identified approximately 50 Hall numbers. (It can be proved that infinitely many exist.) In this paper we study the minimum gap between consecutive Hall numbers. We prove that for all n, x(n + 1) - x(n) > (1/5)x(n)^(1/6), with stronger gaps applying when x(n) is close to perfect even or odd squares (approximately x(n)^(1/3) or x(n)^(1/4), respectively). This result has obvious implications for the minimum “horizontal gap” (and hence straight line and arc distance) between integer points (whose x-coordinates exceed k^2) on the Mordell elliptic curves x^3 - y^2 = k, a question that does not appear to have been addressed.

  • Similarity Matrix-Based Anomaly Detection for Clinical Intervention:
    The use of digital phenotyping methods in clinical care has allowed for improved investigation of spatiotemporal behaviors of patients. Moreover, detecting abnormalities in mobile sensor data patterns can be instrumental in identifying potential changes in symptomology. We propose a method that temporally aligns sensor data in order to achieve interpretable measures of similarity. These computed measures can then be used for anomaly detection, baseline routine computation, and trajectory clustering. In addition, we apply this method on a study of 695 college participants, as well as on a patient with worsening anxiety and depression. With varying temporal constraints, we find mild correlations between changes in routine and clinical scores. Furthermore, in our experiment on an individual with elevated depression and anxiety, we are able to cluster GPS trajectories, allowing for improved understanding and visualization of routines with respect to symptomology. In the future, we aim to apply this method on individuals that undergo data collection for longer periods of time, thus allowing for a better understanding of long-term routines and signals for clinical intervention.

  • Viola Jones Facial Detection Framework:
    Viola Jones implementation in Python to detect faces in pictures

  • Feed-Forward NN for Digit Classification using MNIST dataset:
    Feed-forward neural network (NN) from scratch in Python using only standard matrix/vector operators. Trained by back-propagation to solve 10-digit classification task.