---
title: 'LEAS: Land Expansion Analysis Strategy'
url: https://www.emergentmind.com/topics/land-expansion-analysis-strategy-leas
type: topic
---

# LEAS: Land Expansion Analysis Strategy

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 [1911.11460].

## 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 [1911.11460].

## 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 [1911.11460].

## 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 [1911.11460].

## 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:
```python
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 [1911.11460].

## 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 [1911.11460].

## 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 [1911.11460].

## 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 [1911.11460].

Source: https://www.emergentmind.com/topics/land-expansion-analysis-strategy-leas