My research interests are in the design of experiments and the development of statistically meaningful yet computationally tractable estimation methods for compartmental models. Compartmental models provide a valuable tool in analyzing a wide variety of problems, such as the the kinetics of drugs through a body or the flow of nutrients through an ecosystem via the design of tracer experiments. In most cases, cost and/or feasibility are issues when designing a tracer experiment since such experiment is often non-repeatable due to high cost, ethical issues, or possible damage to the organism. Nonetheless, the careful design of an experiment involving compartmental models can be accomplished through proper collaboration between the scientist who has understanding of the biological or physiological mechanism of the system and the compartmental analyst who has knowledge of the mathematical and computational issues that arise in the identifiability and stability of the parameters or flow rates which describe the system. Once a model, deemed reasonable for the process under study, has been proposed, the objective is to estimate the parameters of the model from a set of observations gathered through time. However, the commonly-occurring non-constant variance of biological data and/or the presence of outlying observations discourage standard application of ordinary least squares or equally weighted L_2 estimation. Although L_1-norm estimation would reduce the influence of outliers, for statistical reasons, it is generally not preferred. To obtain statistically optimal estimators, workers often perform weighted least squares using empirical weights, iteratively reweighted least squares, or transform the data and/or the model often using the logarithm transformation. However, in practice, the optimal weights are not known, thus application of weighted or iteratively reweighted least squares is not straightforward. Moreover, methods involving current transformation methodology, such as Box-Cox transformations, while succeeding in stabilizing the variance, often do not yield a computationally more desirable problem to solve. My immediate research consists of investigating the use and properties of integral transform methods in estimation. Such integral transform should be `natural' to a given problem; natural, in that applying such transformation should yield a problem which would be computationally no more costly than current methods but more amenable to the model, in that such transformation should stabilize the variance yet be computationally simple. Currently, I am looking at these issues with the Laplace transform applied to linear time invariant compartmental models.
