Start with the global question: are nearby values alike? Global Moran’s I answers it with one number — essentially the correlation between each unit’s (mean-centred) value z_i and the average of its neighbours, Wz_i:
I = (n / S₀) · (Σ_i Σ_j w_ij z_i z_j) / (Σ_i z_i²), S₀ = Σ_i Σ_j w_ijWith a row-standardised W this simplifies to I = z'Wz / z'z. Positive I means like clusters with like (hot and cold zones); negative I means a checkerboard of highs beside lows.
Two things trip people up on the inference:
- Under the null of spatial randomness,
E[I] = −1/(n−1)— slightly negative, not zero (it tends to 0 only asngrows). Compare against that, not against 0. - Significance can come from a closed-form (normal / randomisation) variance, or from permutation: shuffle the values across locations many times and build an empirical null distribution. Permutation is safer when normality is doubtful.
But a global I has a blind spot: it is an average. One number can hide a map where clustering is fierce in one district and absent in another — or where two opposite patterns cancel out. That is what the local version is for.
LISA (Local Indicator of Spatial Association; Anselin, 1995) decomposes the global statistic to one value per unit:
I_i = z_i · Σ_j w_ij z_jand classifies each location by the sign of its own value against its neighbours’ average:
| type | unit | neighbours | reading |
|---|---|---|---|
| HH | high | high | hot spot |
| LL | low | low | cold spot |
| HL | high | low | a high outlier in a low area |
| LH | low | high | a low outlier in a high area |
HH/LL are clusters (positive local association); HL/LH are spatial outliers (negative). The local pieces sum back to the global I — it is an exact decomposition.
The trap that ruins LISA maps: one hypothesis test per location means thousands of simultaneous tests, and uncorrected pseudo p-values scatter false “significant” clusters everywhere. Apply a multiple-comparison correction (FDR / Benjamini–Hochberg, or the conditional adjustments Anselin discusses) before you believe a coloured map. And, as always, the whole picture depends on W — change the neighbour definition and the hot spots can move.
One last caution worth stating: if you compute Moran’s I on a variable that has already been spatially smoothed — for instance, coefficients estimated with overlapping kernels — part of the autocorrelation you measure is a mechanical artefact of the smoothing, not evidence of a real spatial process. Know what produced your variable before you celebrate its clustering.