2.3. Observations

For simplicity, we assume that both the prey and predator populations — \(x(t)\) and \(y(t)\) — are directly observed, and that the observation error is distributed normally with zero mean and a known standard deviation \(\sigma_\mathrm{obs}\).

\[\begin{split}\mathbf{x_t} &= [x, y, \alpha, \beta, \delta, \gamma]^T \\ \mathcal{L}(x(t) \mid \mathbf{x_t}) &\sim \mathcal{N}(\mu=x, \sigma=\sigma_\mathrm{obs}) \\ \mathcal{L}(y(t) \mid \mathbf{x_t}) &\sim \mathcal{N}(\mu=y, \sigma=\sigma_\mathrm{obs})\end{split}\]

These observation models are implemented by pypfilt.examples.predation.ObsModel, and are used to update our beliefs about how well each particle \(\mathbf{x_t}\) explains each of the provided observations (shown below).

Example observations of \(x(t)\). date value 1 1.403933 2 1.190422 3 1.477227 4 0.916595 5 0.794995 6 0.585725 7 0.775698 8 1.000713 9 1.463746 10 1.557791 11 1.654735 12 0.794288 13 0.713317 14 0.659150 15 1.126689

Example observations of \(y(t)\). date value 1 0.456042 2 0.152997 3 0.668645 4 0.711037 5 0.837072 6 0.673318 7 0.133188 8 0.056321 9 0.264425 10 0.299256 11 0.729363 12 0.803457 13 0.731722 14 0.517535 15 0.215541