papers/zoo-habitat-modeling/sections/03-problem-formulation.tex

\section{Problem Formulation}

\label{sec:problem}
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\subsection{Notation}

Let $\Species = \{s_1, \ldots, s_K\}$ denote a set of $K$ focal species.
The study area is discretized into a regular grid of $N$ cells, each
represented as a node $v_i$ in a spatial graph $G = (V, E)$, where $V =
\{v_1, \ldots, v_N\}$ and $E \subseteq V \times V$.

Each node $v_i$ is associated with an environmental feature vector
$\mathbf{x}_i \in \R^d$ comprising bioclimatic variables, topographic
metrics, land cover, and vegetation indices. For a climate scenario $c
\in \Climate$ and time period $t$, the features become
$\mathbf{x}_i^{(c,t)}$.

Edge weights $w_{ij}$ encode the dispersal permeability between adjacent
cells, determined by terrain, land cover, and geographic distance.

\subsection{Objectives}

For each species $s_k$, we seek to learn a function:
\begin{equation}
  f_k: (G, \mathbf{X}^{(c,t)}) \rightarrow [0,1]^N
\end{equation}
that maps the spatial graph with time- and scenario-specific features to
a vector of habitat suitability scores across all cells. The model must:

\begin{enumerate}
  \item Accurately predict current species distributions when validated
    against independent occurrence data.
  \item Produce calibrated probability estimates under future climate
    projections.
  \item Identify spatially connected regions of suitable habitat
    (connectivity analysis).
  \item Locate optimal corridors linking disjoint habitat patches.
\end{enumerate}

\subsection{Loss Function}

Given presence-absence labels $y_i^{(k)} \in \{0, 1\}$ for species $k$
at cell $i$, we minimize a composite loss:
\begin{equation}
  \Loss = \Loss_{\text{BCE}} + \lambda_1 \Loss_{\text{spatial}} +
  \lambda_2 \Loss_{\text{connect}}
  \label{eq:loss}
\end{equation}
where $\Loss_{\text{BCE}}$ is the weighted binary cross-entropy,
$\Loss_{\text{spatial}}$ penalizes spatially incoherent predictions
(encouraging smooth habitat boundaries), and $\Loss_{\text{connect}}$
rewards predictions that maintain graph connectivity of suitable patches.


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