Estimating the distribution of random parameters in a diffusion equation forward model for a transdermal alcohol biosensor

We estimate the distribution of random parameters in a distributed parameter model with unbounded input and output for the transdermal transport of ethanol. The underlying model is a diffusion equation with input: blood alcohol concentration and output: transdermal alcohol concentration. We reformulate the dynamical system so that the random parameters are treated as additional space variables. When the distribution to be estimated is absolutely continuous with a joint density, estimating the distribution is equivalent to estimating the diffusivity in a multi-dimensional diffusion equation. Well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods may be employed. We use our technique to estimate a bivariate normal distribution based on data for multiple drinking episodes from a single subject.