Harish S. Bhat — Page 2

arXiv2018

Density Tracking by Quadrature for Stochastic Differential Equations

We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of the stochastic differential equation, resulting in a discrete-time Markov chain with continuous state space. At each time step, DTQ applies quadrature to solve the Chapman-Kolmogorov equation for this Markov chain. In this paper, we focus on a particular case of the DTQ method that arises from applying the Euler-Maruyama method in time and the trapezoidal quadrature rule in space. Our main result establishes that the density computed by DTQ converges in $L^1$ to both the exact density of the Markov chain (with exponential convergence rate), and to the exact density of the stochastic differential equation (with first-order convergence rate). We establish a Chernoff bound that implies convergence of a domain-truncated version of DTQ. We carry out numerical tests to show that the empirical performance of DTQ matches theoretical results, and also to demonstrate that DTQ can compute densities several times faster than a Fokker-Planck solver, for the same level of error.

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arXiv2017

Predicting Adolescent Suicide Attempts with Neural Networks

Though suicide is a major public health problem in the US, machine learning methods are not commonly used to predict an individual's risk of attempting/committing suicide. In the present work, starting with an anonymized collection of electronic health records for 522,056 unique, California-resident adolescents, we develop neural network models to predict suicide attempts. We frame the problem as a binary classification problem in which we use a patient's data from 2006-2009 to predict either the presence (1) or absence (0) of a suicide attempt in 2010. After addressing issues such as severely imbalanced classes and the variable length of a patient's history, we build neural networks with depths varying from two to eight hidden layers. For test set observations where we have at least five ED/hospital visits' worth of data on a patient, our depth-4 model achieves a sensitivity of 0.703, specificity of 0.980, and AUC of 0.958.

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