GeoAI LISA — Atlanta Rent

GeoAI · Spatial Analysis

Does Your Weights Matrix Reflect Reality?

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

Every spatial analysis — LISA, spatial regression, spatial econometrics — assumes a weights matrix W defining which places are "neighbors." The standard approach is purely geometric: queen contiguity, rook, or k-nearest neighbors based on polygon borders.

But Census data carries uncertainty. A tract's rent estimate might have a margin of error of ±$800 — or ±$3,200. That uncertainty concentrates in small-population tracts, minority enclaves, and rapidly changing neighborhoods. When you weight those tracts equally, noisy data shapes your results invisibly.

This tool tests three conditions: geometric weights, a learned GAT without MOE awareness, and a learned GAT informed by data reliability.

Validation: MOE% vs Attention Weight

How to read this. Each dot is one tract-neighbor pair — an edge in the spatial graph. X-axis = the neighbor's MOE% (how uncertain its rent estimate is). Y-axis = the learned attention weight the GAT assigned to that edge.

GAT no-MOE (gray): inputs are fixed every epoch. The network sees rent and MOE% but has no training incentive to use uncertainty. Expect a flat slope — MOE% has no systematic effect on attention.

GAT + MOE ★ (teal): rent inputs are perturbed each epoch by noise scaled by MOE. High-MOE neighbors are unreliable teachers — their values change every epoch. The network learns to down-weight them. Expect a clearly negative slope — higher MOE% → lower attention weight.

The gap between slopes is the validation. If teal slopes down and gray is flat, the Monte Carlo perturbation is working as intended and the LISA differences between conditions are genuinely driven by data reliability — not random noise.

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 here 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 / KNN as-is. Uniform weights. Every neighbor counts equally. No learning. This is the standard approach used in virtually all published spatial analysis.

GAT, no MOE. Each tract has two input features: normalized rent and normalized MOE%. The GAT can see uncertainty but has no training incentive to act on it. Inputs are fixed every epoch. Attention weights reflect rent similarity — tracts with similar rents get high mutual attention regardless of how reliable those rents are.

GAT + MOE ★ (Monte Carlo perturbation). Same two-feature input. But each training epoch, the rent feature is perturbed by Gaussian noise scaled by the tract's standard error (MOE / 1.645). High-MOE tracts jump around dramatically every epoch. Low-MOE tracts are stable. The network learns to down-weight unstable neighbors — not because the loss function says to, but because noisy neighbors are unreliable teachers. Reliability awareness emerges from the training dynamics.

The key comparison is 2 vs 3. Any LISA difference between them is caused solely by MOE awareness — not by architecture, not by spatial structure.

Worked Example

Tract A — Example

Tract A has 4 neighbors. In Condition 1 (geometric), each gets weight 0.250. In Condition 3 (GAT+MOE), the network learned neighbor C has high MOE — its estimate isn't reliable. 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. High divergence = the GAT significantly re-weighted the geometric connections — either 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 sets of 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 classification between GAT no-MOE and GAT+MOE are where data reliability — not spatial structure — was shaping the spatial analysis.