# Section 2 Theory

The brook trout occupancy model is a nested hierarchical Bayesian model that predicts probability of occupancy based on catchment characteristics and climate conditions.

Daily mean stream temperature for each catchment is assumed to be a normally distributed random variable:

\[t_{h,c,y,d} \sim \mathcal{N}(\mu_{h,c,y,d},\sigma_{[t]})\]

where \(t_{h,c,y,d}\) is the mean stream temperature on day \(d\) within year \(y\) for catchment \(c\), which is located within HUC8 \(h\). This random variable is normally distributed with an expected mean \(\mu_{h,c,y,d}\) and standard deviation \(\sigma_{[t]}\).

The expected mean is computed as:

\[ \mu_{h,c,y,d} = \left \{ \begin{array}{l l} \omega_{h,c,y,d} + \delta_{h}(t_{h,c,y,d-1} - \omega_{h,c,y,d-1}) & \quad \text{for } t_{h,c,y,d-1} \text{ is real} \\ \omega_{h,c,y,d} & \quad \text{for } t_{h,c,y,d-1} \text{ is not real} \end{array} \right. \]

where \(\delta_h\) is an autoregressive [AR(1)] coefficient and \(\omega_{h,c,y,d}\) is the expected temperature before accounting for temporal autocorrelation in the error structure.

The expected temperature is computed as a linear equation with four sets of terms:

\[\omega_{h,c,y,d} = X_{[0]} B_{[0]} + X_{h,c} B_{h,c} + X_{h} B_{h} + X_{y} B_{y}\]

where

- \(B_{[0]}\) is a vector of fixed effect coefficients
- \(B_{h,c}\) is a vector of random effect coefficients for catchment \(c\)
- \(B_{h}\) is a vector of random effect coefficients for HUC \(h\)
- \(B_{y}\) is a vector of random effect coefficients for year \(y\)

Each of these vectors is multiplied by a corresponding matrix containing the corresponding predictor values (\(X\)) of each catchment \(c\) (located within HUC \(h\)) and on each day \(d\) (within year \(y\)).

## 2.1 Fixed Effects

The fixed effects are shared among all catchments within the model domain. They include the following terms:

Variable | Description |
---|---|

`intercept` |
Intercept |

`AreaSqKM` |
Total Drainage Area (km2) |

`impoundArea` |
Impounded Drainage Area (km2) |

`agriculture` |
Agricultural Land Cover (%) |

`devel_hi` |
High Development Land Cover (%) |

`forest` |
Riparian (200 ft Buffer) Forest Cover (%) |

`prcp2` |
2-day Precipitation (mm) |

`prcp30` |
30-day Precipitation (mm) |

The fixed effects also include the following interaction terms.

Interaction Term | Description |
---|---|

`prcp2.da` |
2-day Precipitaation x Drainage Area |

`prcp30.da` |
30-day Precipitaation x Drainage Area |

`airTemp.da` |
Air Temperature x Total Drainage Area |

`airTemp.impoundArea` |
Air Temperature x Impounded Drainage Area |

`airTemp.agriculture` |
Air Temperature x Agricultural Land Cover |

`airTemp.forest` |
Air Temperature x Riparian (200 ft Buffer) Forest Cover |

`airTemp.devel_hi` |
Air Temperature x High Development Land Cover |

`airTemp.prcp2` |
Air Temperature x 2-day Precipitation |

`airTemp.prcp30` |
Air Temperature x 30-day Precipitation |

`airTemp.prcp2.da` |
Air Temperature x 2-day Precipitation x Drainage Area |

`airTemp.prcp30.da` |
Air Temperature x 30-day Precipitation x Drainage Area |

## 2.2 Catchment Random Effects

The random effects for each catchment (\(c\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |

`airTemp` |
Air Temperature (degC) |

`temp7p` |
7-day Mean Air Temperature (degC) |

## 2.3 HUC Random Effects

The random effects for each HUC (\(h\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |

`airTemp` |
Air Temperature (degC) |

`temp7p` |
7-day Mean Air Temperature (degC) |

## 2.4 Year Random Effects

The random effects for each year (\(y\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |