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"Interpolate" routes to daily time step #234

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@dsheldon

BirdFlow uses a weekly time step, which is too coarse for some applications.

I recently thought about a few possible ways to "downscale" the model in time to generate routes at a daily time step. The simplest idea is:

  1. First generate a route at weekly time step.
  2. Use the logic of route_between to fill in daily locations.

In more detail, using the existing BirdFlow routines, we generate the sequence $X_1, X_2, X_3, \ldots$ where $X_t$ is the location of the bird during the midpoint of week $t$. For each week $t$ we want to generate the sequence of daily locations $Y_{1}, Y_{2}, \ldots, Y_{8}$ subject to $X_t = Y_1$ and $X_{t+1} = Y_{8}$---i.e. to fill in the time gaps. Mathematically, this is the exact same problem solved by route_between. One thing is we need to have a transition matrix for each day. I thought about same ways to take the weekly transition matrix $P_{t, t+1}$ and try to factor it into daily transition matrices like

$$P_{t, t+1} = Q_1 Q_2 \cdots Q_7$$

but this seemed very challenging mathematically. Instead, one option is to just use the weekly transition matrix $P_{t, t+1}$ for each day. If you applied this without conditioning on the weekly locations it wouldn't make any sense, but I think it is a very reasonable approximation given that the starting and ending points for each week would be fixed. Other alternatives could be explored.

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