[Population Modeling] Paragraph for paper

[Matthias Chung]

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Here is my paragraph.

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Tia

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Title of paragraph: Parameter estimation for population dynamical models.

Author: Matthias Chung, Department of Mathematics, Computational Modeling & Data Analytics, Virginia Tech

Inferring information from observed population dynamics onto population interactions is inherently difficult. We consider parameter estimation methods to overcome such obstacles. 

Lets assume the dynamics of interacting species can mathematically be modelled by a generalized Lotka-Volterra system y' = diag(y)(r + A y).
Here the vector function y describes the time dependent dynamic, r captures the intrinsic growth, and A describes the interaction between species y. Notice that in higher dimensions (more than two species) dynamics of y are highly sensitive to small changes in the interaction A. Hence inferring A from longitudinal observations d is notably difficult.  

Single and multiple shooting methods are standard methods for point estimation of ordinary differential equation. However, these methods are known fail for highly sensitive equations such as population dynamical systems. To overcome this issue the underlying parameter estimation problem is reformulated as

min || m(s) - d || + a || s' - diag(s)(r + A s) ||,

where s is an adequate parameterized function approximation of y and m is a projection of that function onto the observation space. Further, || . || is the Euclidian norm and a is an appropriate regularization parameter, while we optimize over A and s. These continuous shooting methods have been shown to generate robust estimates for the inferred parameters A, [1,2].

References:

[1] J. Ramsay, Principal differential analysis: data reduction by differential operators, J R Stat Soc Series B Methodol, 58 (1996), pp. 495-508.

[2] M. Chung, J. Krueger, M. Pop, Robust Parameter Estimation for Biological Systems: A Study on the Dynamics of Microbial Communities.  ArXiv http://arxiv.org/abs/1509.06926, (2015), pp. 1-33.