Relating finite noisy count data to continuum-limit PDE solutions in practical settings

Determine the precise relationship between observed finite, noisy count data C_s^{o}(i,k), defined as the number of agents from subpopulation s in column i after k time steps on a lattice of height J, and the continuum-limit partial differential equation solution c_s(x,t) for the corresponding lattice-based random walk model when J is small (i.e., typical experimental fields of view), thereby clarifying how column-wise counts map to continuous densities outside the asymptotic regime J→∞.

Background

The paper derives a continuum-limit PDE model for lattice-based random walks and recalls a computationally verified relationship indicating that, in the limit of very large image height J, the PDE solution c_s(x,t) equals the column-averaged occupancy derived from counts. However, experimental images typically have relatively small fields of view, producing noisy counts with substantial fluctuations that challenge this asymptotic mapping.

The authors note that under practical conditions (e.g., J≈20), directly relating the finite, noisy column counts C_s^{o}(i,k) to the PDE solution c_s(x,t) is not clear, motivating the introduction of measurement error models (Gaussian and multinomial) to operationalize inference and prediction. A rigorous characterization of the mapping outside the asymptotic limit remains unresolved.