LEAS: Land Expansion Analysis Strategy

• LEAS is a spatially explicit approach that integrates risk (α) and trade-off (β) parameters to guide territorial expansion planning.
• It employs a truncated normal density-driven scheme to generate order weights, providing precise suitability mapping via Ordered Weighted Averaging.
• The framework clusters suitability maps and uses rank robustness metrics like Kendall’s τ to ensure decision transparency and strategic reliability.

Land Expansion Analysis Strategy (LEAS) defines a spatially explicit, multi-criteria workflow for territorial planning, with emphasis on modeling risk and trade-off attitudes in site-selection using Ordered Weighted Averaging (OWA) methods. LEAS characterizes the decision-strategy space via two continuous parameters—risk ($\alpha$) and trade-off ($\beta$)—and operationalizes these through a density-driven scheme for order-weight generation, systematic sampling of strategy space, spatial clustering of suitability maps, variance analysis, and rank robustness metrics. This approach enables transparent evaluation of how risk/trade-off affects expansion outcomes, suitable for practical GIS implementation and scenario-based urban development planning (Billaud et al., 2019).

1. Formalization of the Decision–Strategy Space

LEAS requires explicit mathematical definition of the decision-strategy space $(\alpha,\beta)$, where $\alpha\in[0,1]$ quantifies risk preference (high values favor risk-seeking, low values risk-aversion), and $\beta\in[0,1]$ encodes the degree of trade-off among criteria (higher $\beta$ yields greater compensability). Not all $(\alpha,\beta)$ are feasible. The admissible set $\mathcal{D}$ is:

$$\mathcal{D}\;=\;\Bigl\{ (\alpha,\beta)\in[0,1]^2 : \beta \;\le\; 4\,\alpha\,(1-\alpha)\Bigr\}$$

The vertices correspond to:

• $(0,0)$: list-min (extreme risk aversion)
• $(1,0)$: list-max (extreme risk seeking)
• $(0.5,1)$: full trade-off (classic weighted sum).

This parameterization underlies generation of order-weights and systematic exploration of the full spectrum of strategic behaviors intrinsic to territorial expansion modeling (Billaud et al., 2019).

2. Ordered Weighted Averaging Formulation

The core aggregation mechanism employs OWA with both criterion-importance weights ($\mathbf{V}$) and design-dependent order-weights ($\mathbf{W}$):

• $\mathbf{V}=(v_j)_{1\le j\le n}$: importance weight for each criterion, expert-elicited and normalized such that $\sum_j v_j=1$.
• $\mathbf{W}=(w_j)_{1\le j\le n}$: order-weights generated from $(\alpha,\beta)$ parameters, also normalized.

Given standardized pixel-wise suitability scores $\mathbf{z}_i=(z_{i1},\dots,z_{in})\in[0,1]^n$, the values are sorted in increasing order ($z_{i(1)} \le \cdots \le z_{i(n)}$) and $v_{(j)}$ reordered accordingly. The pixel-level aggregated suitability score is:

$$S(i) = \sum_{j=1}^n \frac{v_{(j)}\,w_j}{\sum_{k=1}^n v_{(k)}\,w_k} \;z_{i(j)}, \tag{OWA}$$

At $\beta=1$, the aggregation reduces to the classical weighted linear combination (WLC), making the scheme compatible with conventional MCDA approaches and facilitating incorporation within existing GIS decision-support systems (Billaud et al., 2019).

3. Generation of Order-Weights from $(\alpha,\beta)$

Order-weights are generated by constructing a truncated normal probability density function with mean $\alpha$ and standard deviation $\beta/2$, evaluated at $n$ equispaced nodes. The process is:

• Compute $x_j=(j-\tfrac{1}{2})/n$ for $j=1\ldots n$.
• $$w_j = \frac{p_{\text{trunc}}(x_j)}{\sum_{k=1}^n p_{\text{trunc}}(x_k)}$$,

where $p_{\text{trunc}}$ is the value of the truncated normal PDF at $x_j$. This yields a vector of order-weights for any $(\alpha,\beta)\in\mathcal{D}$, ensuring normalization and parameter-driven control over the degree of risk and trade-off encoded in the aggregation (Billaud et al., 2019).

4. Systematic Sampling and Map Generation

Efficient exploration of the decision-strategy space requires quasi-uniform sampling (e.g., Latin Hypercube with rejection) of the $(\alpha,\beta)$ domain $\mathcal{D}$. For each sampled pair,

• Generate $\mathbf{W}$,
• Compute corresponding OWA map $S^{(p)}$.

Pseudocode representation:

for p in range(N_samples): W = truncatedNormalWeights(alpha_p, beta_p, n) for i in pixels: z_sorted, v_reordered = sort(z_i), reorder(v) S_p[i] = sum((v_reordered[j]*W[j] / sum(v_reordered[k]*W[k])) * z_sorted[j] for j in range(n))

This batch process yields an ensemble of N suitability maps spanning the strategy space (Billaud et al., 2019).

5. Clustering, Variance Analysis, and Site Ranking

Suitability maps are hierarchically clustered based on pairwise Euclidean distance:

$$D_{pq} = \sqrt{ \sum_{i} (S^{(p)}(i) - S^{(q)}(i))^2 }$$

Ward’s method is used to identify K clusters (e.g., 4 clusters in Thau), each characterizing typical strategies (low-risk, medium-risk, high-risk). For each cluster:

• Mean map: $$\bar S_k(i) = \frac{1}{|C_k|}\sum_{p\in C_k} S^{(p)}(i)$$
• Pixel-wise standard deviation: $$\sigma_k(i) = \sqrt{ \frac{1}{|C_k|-1} \sum_{p\in C_k} (S^{(p)}(i) - \bar S_k(i))^2 }$$

Ranking robustness across clusters is quantified by Kendall’s $\tau$ between ranked mean suitability values. The case study exhibits strong rank-order correlation (e.g., $\tau_{1,2}=0.84$) and allows further examination of intersection among top sites under divergent risk/trade-off preferences (Billaud et al., 2019).

6. Case Study Application: Ecosystem Services and Strategic Insights

The LEAS framework was applied in the Thau region (South of France) with ten expert-weighted criteria (provisioning, regulating, cultural, disservice, connectivity), mapped to $[0,1]$, and $N=1000$ sampled strategy points. Key findings include:

• Cluster 1 (risk-averse): very selective expansion, adjacent to existing built-up area.
• Cluster 4 (risk-seeking): broad suitability, including extensive agricultural land.
• Spatial variance analysis identifies land-cover types most responsive to trade-off among criteria.
• Site ranking stability is quantifiable via rank-correlation and overlap metrics.

This approach demonstrates the impact of attitude toward risk and compromise on land expansion prioritization; enables explicit spatial sensitivity analysis; and provides planners with policy-relevant clusters and variance maps to inform transparent, multi-actor decision processes (Billaud et al., 2019).

7. Technical Implementation and Reproducibility

LEAS is immediately deployable with standard GIS tools. Required steps:

• Loop over admissible $(\alpha,\beta)$,
• Compute truncated-normal-based order weights,
• Calculate OWA at each grid cell,
• Cluster resulting maps and analyze stage-wise variance and rank robustness.

Post-processing (e.g., clustering, Kendall’s $\tau$ calculation) is readily performed with R or Python. This ensures full reproducibility and extensibility to other regions or criteria sets, provided appropriately normalized input layers and expert-derived criterion weights (Billaud et al., 2019).