GeoAI LISA — Atlanta Rent

GeoAI · Spatial Analysis

Does Your Weights Matrix Reflect Reality?

Atlanta Metro · Median Gross Rent · ACS 5-Year 2022

Most spatial analysis workflows — LISA, spatial regression, spatial econometrics — begin with a weights matrix W that defines which places count as neighbors. The standard approach is geometric: queen contiguity, rook, or k-nearest neighbors based on polygon borders or centroid distance.

The weights matrix defines the structure, but the data flowing through it carries its own uncertainty. In the ACS, every estimate comes with a margin of error. A tract's median gross rent may have a MOE of ±$800, while a neighboring tract has ±$3,200. When both neighbors receive equal weight, the noisier estimate has the same influence on spatial statistics as the reliable one — and that can shape results in ways that are difficult to detect.

This tool compares three conditions: geometric weights, a learned GAT without MOE-aware training, and a learned GAT informed by data reliability. The neighborhood definition also matters — queen, rook, and KNN already produce meaningfully different spatial patterns. Accounting for data uncertainty adds another layer of sensitivity on top of that.

Validation: MOE% vs Attention Weight

Each point represents one tract-neighbor pair — one edge in the spatial graph. The x-axis shows the neighbor's MOE%, a measure of uncertainty in the rent estimate. The y-axis shows the learned attention weight assigned to that edge.

Geometric W (light gray dashed): always flat. Uniform weights by definition — MOE% has no effect on neighbor influence.

GAT no-MOE (gray): inputs are fixed during training. The model can observe uncertainty through the MOE% feature, but has no direct training incentive to act on it. A relatively flat slope is expected.

GAT + MOE ★ (teal): rent inputs are perturbed each epoch by noise scaled by MOE / 1.645. High-MOE neighbors vary more during training. If the perturbation is working as intended, attention weights should decline as neighbor uncertainty increases — a clearly negative slope.

The slope comparison is a diagnostic. A more negative teal slope supports the interpretation that attention is responding to data reliability. It does not confirm the method is definitively correct — but it suggests something beyond random variation is happening.

Geometric W GAT no-MOE GAT + MOE ★

NN vs GNN vs GAT

Model	Spatial?	Weighted?
Neural Network	No — each tract independent	—
GNN	Yes — blends neighbors	Equally
GAT	Yes — blends neighbors	Learned

A regular NN treats tract A in isolation. A GNN lets A absorb information from its neighbors — but weights them equally. A GAT learns how much each neighbor should count. Those learned percentages become W*.

Both GAT conditions use two input features per tract: normalized rent and normalized MOE%. The difference is whether the training process gives the network a reason to act on the uncertainty signal.

The Three Conditions

Geometric W. Queen, rook, or KNN weights used directly. Every neighbor receives equal weight after row-standardization. This is the conventional approach used in most published spatial analysis.

GAT, no MOE. Each tract has two input features: normalized rent and normalized MOE%. The model can observe uncertainty, but inputs are fixed across epochs. Attention weights primarily reflect learned spatial and attribute relationships under stable inputs.

GAT + MOE ★. Same two features. Each epoch, the rent value is perturbed by Gaussian noise scaled by the tract's standard error (MOE / 1.645). High-MOE tracts are less stable during training. Low-MOE tracts remain more consistent. The model can learn to assign lower attention to neighbors whose values are less reliable across epochs.

The architecture and neighborhood graph are held constant between conditions 2 and 3. Differences in LISA outcomes between them are most plausibly attributable to the uncertainty-aware perturbation process — though this is exploratory, not a controlled experiment.

Worked Example

Tract A — Illustration

Tract A has 4 neighbors. In Condition 1 (geometric), each gets weight 0.250. In Condition 3 (GAT+MOE), neighbor C has high MOE — its estimate is unreliable. C gets down-weighted.

Neighbor B — low MOE	0.250 → 0.341
Neighbor C — high MOE	0.250 → 0.087
Neighbor D — low MOE	0.250 → 0.318
Neighbor E — low MOE	0.250 → 0.254

Tract A's LISA classification changes from NS to HH — a cluster that geometry masked because a noisy neighbor diluted the signal.

Key Concepts

Divergence

Sum of |W − W*| across all neighbors of a tract. High divergence means the GAT significantly re-weighted the geometric connections — either due to high-MOE neighbors or adjacencies that don't reflect real spatial relationships in the rent surface.

Weight Chart (click any tract)

Bar chart in the detail panel showing three weights per neighbor: gray (geometric, uniform), light gray (GAT no-MOE), teal (GAT+MOE). Where teal drops below gray, data reliability down-weighted that neighbor.

LISA Quadrants

HH/LL = spatial clusters. LH/HL = spatial outliers. NS = not significant at p < 0.05 (999 permutations).

2↔3 comparison is the methodological test. Tracts that change LISA classification between GAT no-MOE and GAT+MOE are where data reliability may have been shaping the spatial analysis.

Limitations

The no-MOE condition retains MOE% as a visible feature, so some implicit uncertainty awareness may persist even without perturbation. A stricter design would remove MOE% from that condition entirely. KNN neighborhoods are directional by construction, which is a known limitation of centroid-based neighbor definitions. This is exploratory work — the results suggest spatial weights should not be treated as neutral geometry, but they do not constitute a validated method.