Week 8: Local Moran’s I and LISA

Local Moran’s I, HH/HL/LH/LL quadrants, and LISA cluster maps as a model diagnostic

Published

October 23, 2026

What We Did

Last week, global Moran’s I on the residuals told us whether the model’s errors cluster. As the model improves, that number should fall. But a single number cannot tell us where the problems are. This week we broke it apart, one house at a time.

Reviewed global Moran’s I on residuals as a diagnostic to track across model versions.
Worked through a five-house example by hand: residual, deviation from the mean, neighbors’ average deviation, and local Moran’s I.
Classified each location into one of four quadrants (HH, HL, LH, LL) and interpreted them as prediction errors.
Discussed why a LISA value must be read together with its quadrant, its significance, the size of the errors, and a map.
Odds and ends: the RealPage case on algorithmic rent pricing.

Key Concepts

1. From Global to Local

Global Moran’s I asks: across the whole study area, do similar residuals sit next to each other?

Local Moran’s I, a LISA (Local Indicator of Spatial Association; Anselin, 1995), asks: does this particular location resemble its neighbors?

With row-standardized weights, the global statistic is the average of the local ones:

\[ I = \frac{1}{n}\sum_{i=1}^{n} I_i \]

LISA is therefore a decomposition of the global statistic, not a separate idea.

2. The Formula

Let $z_i = e_i - \bar{e}$ be the deviation of location $i$’s residual from the mean residual, and

\[ m_2 = \frac{1}{n}\sum_{i=1}^{n} z_i^2 \]

Then

\[ I_i = \frac{z_i}{m_2}\sum_{j} w_{ij} z_j \]

The sum $\sum_j w_{ij} z_j$ is the spatial lag of $z$. With row-standardized weights, it is the neighbors’ average deviation.

$I_i$ is large and positive when location $i$ and its neighbors deviate strongly in the same direction. It is negative when they deviate in opposite directions.

3. Worked Example: Five Houses in a Row

Houses A–E sit on a line, and each house’s neighbors are the houses directly next to it. All values are in thousands of dollars.

The mean residual is $(100 + 80 + 20 - 50 - 70)/5 = 16$, so $z_i = e_i - 16$.

House	Residual	Deviation $z_i$	Neighbors	Spatial lag (neighbors’ avg $z$)
A	+100	+84	B	+64
B	+80	+64	A, C	+44
C	+20	+4	B, D	−1
D	−50	−66	C, E	−41
E	−70	−86	D	−66

\[ m_2 = \frac{84^2 + 64^2 + 4^2 + 66^2 + 86^2}{5} = \frac{22{,}920}{5} = 4{,}584 \]

House	$z_i \times \text{lag}_i$	$I_i$	Quadrant
A	$84 \times 64 = 5{,}376$	1.17	High-High
B	$64 \times 44 = 2{,}816$	0.61	High-High
C	$4 \times (-1) = -4$	0.00	Nothing meaningful
D	$(-66) \times (-41) = 2{,}706$	0.59	Low-Low
E	$(-86) \times (-66) = 5{,}676$	1.24	Low-Low

Check: $(1.17 + 0.61 + 0.00 + 0.59 + 1.24)/5 \approx 0.72$, which equals the global Moran’s I for these five houses.

Two points from the example:

E has the largest $I_i$, even though A has the largest residual. $I_i$ rewards agreement with neighbors, not size alone.
C is technically High-Low ($z > 0$, lag $< 0$), but both values are close to zero, so $I_i \approx 0$. A quadrant label without a meaningful $I_i$ is noise.

4. The Four Quadrants (Moran Scatterplot)

The Moran scatterplot places $z_i$ on the x-axis and its spatial lag on the y-axis. Each quadrant describes a different pattern:

Quadrant	Location $i$	Neighbors	Sign of $I_i$	Pattern
High-High (HH)	Above mean	Above mean	+	Hot spot / cluster
Low-Low (LL)	Below mean	Below mean	+	Cold spot / cluster
High-Low (HL)	Above mean	Below mean	−	Spatial outlier
Low-High (LH)	Below mean	Above mean	−	Spatial outlier

Positive $I_i$: a cluster (HH or LL).
Negative $I_i$: an outlier (HL or LH).

The sign alone cannot distinguish HH from LL. The quadrant is needed.

5. Reading the Quadrants as Errors

When the variable is a residual, “high” and “low” have a specific meaning:

\[ e_i = y_i - \hat{y}_i \]

A high residual means the model underpredicted; a low residual means it overpredicted.

	Neighbors underpredicted	Neighbors overpredicted
Location $i$ underpredicted	High-High	High-Low
Location $i$ overpredicted	Low-High	Low-Low

HH: a pocket of underprediction. The model consistently guesses too low here. The area may be worth more than the features suggest because of an unmeasured amenity, a gentrifying block, or a strong school catchment.
LL: a pocket of overprediction. The model consistently guesses too high. Something the model does not see may be pulling prices down, such as vacancy, blight, a nearby disamenity, or a data issue.
HL / LH: isolated surprises. A single house does not match its neighbors, such as a renovated house on a distressed block, a teardown on an expensive block, or a bad record.

Strictly speaking, “high” and “low” are relative to the mean residual, not to zero. With OLS and an intercept, the mean residual is zero on the training data, so the two coincide. They can differ on test data or after filtering.

6. Reading LISA in Context

A LISA value on its own is not an answer. Read it together with:

The quadrant: a positive $I_i$ could be HH or LL, which represent opposite problems for the model.
Significance: a conditional permutation test keeps location $i$’s value fixed, randomly reassigns the remaining values to its neighbors, and recalculates $I_i$. Cluster maps usually show only significant locations.
The size of the residuals: a strong LL cluster of small overpredictions matters less than a weak cluster of $100,000 errors. LISA measures pattern; CV RMSE and MAE measure magnitude.
A map: the cluster map, with quadrant colors and non-significant locations in grey, shows where the patterns are.

Cautions:

Multiple testing. With thousands of houses, roughly 5% may appear significant at $p < 0.05$ by chance alone. Use a stricter cutoff or a false discovery rate (FDR) correction before drawing strong conclusions.
Neighbor definitions matter. Results change with the spatial weights, such as kNN with $k = 5$ versus $k = 10$, contiguity, or distance bands. Report the choice.
Clusters are not causes. An LL cluster shows where the model fails, not why. It indicates where to look for a missing feature.

7. Using LISA to Improve the Model

The cluster map can become a feature-engineering checklist:

What the map shows	Possible next step
HH cluster (underprediction)	Look for a missing amenity: parks, transit, school quality, or a neighborhood fixed effect
LL cluster (overprediction)	Look for a missing disamenity: vacancy, crime, industrial land, or flood risk
Clusters that follow neighborhood boundaries	Add neighborhood fixed effects
Scattered HL / LH outliers	Check for data errors, non-arm’s-length sales, or unusual properties

After adding features, rerun global Moran’s I and the LISA map, and compare CV RMSE using the same folds. A useful change should shrink the clusters without worsening out-of-sample error.

8. R Workflow

library(sf)
library(spdep)

# Residuals from the fitted model
sales$resid <- residuals(model)

# Neighbors: five nearest houses, row-standardized
coords <- st_coordinates(sales)
nb <- knn2nb(knearneigh(coords, k = 5))
lw <- nb2listw(nb, style = "W")

# Global Moran's I (last week)
moran.mc(sales$resid, lw, nsim = 499)

# Local Moran's I with a conditional permutation test
lisa <- localmoran_perm(sales$resid, lw, nsim = 499)

sales$Ii   <- lisa[, "Ii"]
sales$p    <- lisa[, "Pr(z != E(Ii)) Sim"]
sales$quad <- attr(lisa, "quadr")$mean   # High-High, Low-Low, High-Low, Low-High

# Keep only significant locations for the cluster map
sales$cluster <- ifelse(sales$p < 0.05, as.character(sales$quad), "Not significant")

Odds and Ends: The RealPage Case

RealPage sells revenue-management software that recommends daily rents to landlords. In August 2024, the DOJ and several states sued the company, alleging that the software pooled competing landlords’ nonpublic data and pushed them toward aligned, higher rents (“algorithmic collusion”).

In November 2025, the DOJ announced a proposed settlement with no fines and no admission of wrongdoing. Under its terms, RealPage must:

Stop using competitors’ nonpublic data for real-time recommendations; training data must be at least a year old.
Remove features that discouraged rent decreases or aligned prices among competitors.
Stop hyper-local pricing.

Several states that had joined the suit were not part of the settlement, and a number of cities, including Philadelphia, have passed their own ordinances restricting rent-setting algorithms.

Why this case belongs in a policy analytics course:

A pricing model becomes a public-policy actor once enough of the market relies on it.
The central issue is less the algorithm itself than the data it is allowed to see. This is the same question as target leakage, scaled up to an entire market.
Our house-price models ask, “What is this home worth?” RealPage’s model asks, “What can this landlord charge?” The tools are similar, but the consequences are very different.

--- title: "Week 8: Local Moran's I and LISA" description: "Local Moran's I, HH/HL/LH/LL quadrants, and LISA cluster maps as a model diagnostic" date: 2026-10-23 --- ## What We Did Last week, global Moran's I on the residuals told us whether the model's errors cluster. As the model improves, that number should fall. But a single number cannot tell us where the problems are. This week we broke it apart, one house at a time. - Reviewed global Moran's I on residuals as a diagnostic to track across model versions. - Worked through a five-house example by hand: residual, deviation from the mean, neighbors' average deviation, and local Moran's I. - Classified each location into one of four quadrants (HH, HL, LH, LL) and interpreted them as prediction errors. - Discussed why a LISA value must be read together with its quadrant, its significance, the size of the errors, and a map. - Odds and ends: the RealPage case on algorithmic rent pricing. ## Key Concepts ### 1. From Global to Local Global Moran's I asks: across the whole study area, do similar residuals sit next to each other? Local Moran's I, a **LISA** (Local Indicator of Spatial Association; Anselin, 1995), asks: does this particular location resemble its neighbors? With row-standardized weights, the global statistic is the average of the local ones: $$ I = \frac{1}{n}\sum_{i=1}^{n} I_i $$ LISA is therefore a decomposition of the global statistic, not a separate idea. ### 2. The Formula Let $z_i = e_i - \bar{e}$ be the deviation of location $i$'s residual from the mean residual, and $$ m_2 = \frac{1}{n}\sum_{i=1}^{n} z_i^2 $$ Then $$ I_i = \frac{z_i}{m_2}\sum_{j} w_{ij} z_j $$ The sum $\sum_j w_{ij} z_j$ is the **spatial lag** of $z$. With row-standardized weights, it is the neighbors' average deviation. $I_i$ is large and positive when location $i$ and its neighbors deviate strongly **in the same direction**. It is negative when they deviate in opposite directions. ### 3. Worked Example: Five Houses in a Row Houses A–E sit on a line, and each house's neighbors are the houses directly next to it. All values are in thousands of dollars. The mean residual is $(100 + 80 + 20 - 50 - 70)/5 = 16$, so $z_i = e_i - 16$. | House | Residual | Deviation $z_i$ | Neighbors | Spatial lag (neighbors' avg $z$) | |---|---|---|---|---| | A | +100 | +84 | B | +64 | | B | +80 | +64 | A, C | +44 | | C | +20 | +4 | B, D | −1 | | D | −50 | −66 | C, E | −41 | | E | −70 | −86 | D | −66 | $$ m_2 = \frac{84^2 + 64^2 + 4^2 + 66^2 + 86^2}{5} = \frac{22{,}920}{5} = 4{,}584 $$ | House | $z_i \times \text{lag}_i$ | $I_i$ | Quadrant | |---|---|---|---| | A | $84 \times 64 = 5{,}376$ | 1.17 | High-High | | B | $64 \times 44 = 2{,}816$ | 0.61 | High-High | | C | $4 \times (-1) = -4$ | 0.00 | Nothing meaningful | | D | $(-66) \times (-41) = 2{,}706$ | 0.59 | Low-Low | | E | $(-86) \times (-66) = 5{,}676$ | 1.24 | Low-Low | Check: $(1.17 + 0.61 + 0.00 + 0.59 + 1.24)/5 \approx 0.72$, which equals the global Moran's I for these five houses. Two points from the example: - **E has the largest $I_i$, even though A has the largest residual.** $I_i$ rewards agreement with neighbors, not size alone. - **C is technically High-Low** ($z > 0$, lag $< 0$), but both values are close to zero, so $I_i \approx 0$. A quadrant label without a meaningful $I_i$ is noise. ### 4. The Four Quadrants (Moran Scatterplot) The Moran scatterplot places $z_i$ on the x-axis and its spatial lag on the y-axis. Each quadrant describes a different pattern: | Quadrant | Location $i$ | Neighbors | Sign of $I_i$ | Pattern | |---|---|---|---|---| | High-High (HH) | Above mean | Above mean | + | Hot spot / cluster | | Low-Low (LL) | Below mean | Below mean | + | Cold spot / cluster | | High-Low (HL) | Above mean | Below mean | − | Spatial outlier | | Low-High (LH) | Below mean | Above mean | − | Spatial outlier | - **Positive $I_i$:** a cluster (HH or LL). - **Negative $I_i$:** an outlier (HL or LH). The sign alone cannot distinguish HH from LL. The quadrant is needed. ### 5. Reading the Quadrants as Errors When the variable is a residual, "high" and "low" have a specific meaning: $$ e_i = y_i - \hat{y}_i $$ A high residual means the model **underpredicted**; a low residual means it **overpredicted**. | | Neighbors underpredicted | Neighbors overpredicted | |---|---|---| | **Location $i$ underpredicted** | High-High | High-Low | | **Location $i$ overpredicted** | Low-High | Low-Low | - **HH: a pocket of underprediction.** The model consistently guesses too low here. The area may be worth more than the features suggest because of an unmeasured amenity, a gentrifying block, or a strong school catchment. - **LL: a pocket of overprediction.** The model consistently guesses too high. Something the model does not see may be pulling prices down, such as vacancy, blight, a nearby disamenity, or a data issue. - **HL / LH: isolated surprises.** A single house does not match its neighbors, such as a renovated house on a distressed block, a teardown on an expensive block, or a bad record. Strictly speaking, "high" and "low" are relative to the **mean residual**, not to zero. With OLS and an intercept, the mean residual is zero on the training data, so the two coincide. They can differ on test data or after filtering. ### 6. Reading LISA in Context A LISA value on its own is not an answer. Read it together with: 1. **The quadrant:** a positive $I_i$ could be HH or LL, which represent opposite problems for the model. 2. **Significance:** a conditional permutation test keeps location $i$'s value fixed, randomly reassigns the remaining values to its neighbors, and recalculates $I_i$. Cluster maps usually show only significant locations. 3. **The size of the residuals:** a strong LL cluster of small overpredictions matters less than a weak cluster of $100,000 errors. LISA measures **pattern**; CV RMSE and MAE measure **magnitude**. 4. **A map:** the cluster map, with quadrant colors and non-significant locations in grey, shows where the patterns are. Cautions: - **Multiple testing.** With thousands of houses, roughly 5% may appear significant at $p < 0.05$ by chance alone. Use a stricter cutoff or a false discovery rate (FDR) correction before drawing strong conclusions. - **Neighbor definitions matter.** Results change with the spatial weights, such as kNN with $k = 5$ versus $k = 10$, contiguity, or distance bands. Report the choice. - **Clusters are not causes.** An LL cluster shows where the model fails, not why. It indicates where to look for a missing feature. ### 7. Using LISA to Improve the Model The cluster map can become a feature-engineering checklist: | What the map shows | Possible next step | |---|---| | HH cluster (underprediction) | Look for a missing amenity: parks, transit, school quality, or a neighborhood fixed effect | | LL cluster (overprediction) | Look for a missing disamenity: vacancy, crime, industrial land, or flood risk | | Clusters that follow neighborhood boundaries | Add neighborhood fixed effects | | Scattered HL / LH outliers | Check for data errors, non-arm's-length sales, or unusual properties | After adding features, rerun global Moran's I and the LISA map, and compare CV RMSE using the same folds. A useful change should shrink the clusters **without** worsening out-of-sample error. ### 8. R Workflow ```r library(sf) library(spdep) # Residuals from the fitted model sales$resid <- residuals(model) # Neighbors: five nearest houses, row-standardized coords <- st_coordinates(sales) nb <- knn2nb(knearneigh(coords, k = 5)) lw <- nb2listw(nb, style = "W") # Global Moran's I (last week) moran.mc(sales$resid, lw, nsim = 499) # Local Moran's I with a conditional permutation test lisa <- localmoran_perm(sales$resid, lw, nsim = 499) sales$Ii <- lisa[, "Ii"] sales$p <- lisa[, "Pr(z != E(Ii)) Sim"] sales$quad <- attr(lisa, "quadr")$mean # High-High, Low-Low, High-Low, Low-High # Keep only significant locations for the cluster map sales$cluster <- ifelse(sales$p < 0.05, as.character(sales$quad), "Not significant") ``` ## Odds and Ends: The RealPage Case RealPage sells revenue-management software that recommends daily rents to landlords. In August 2024, the DOJ and several states sued the company, alleging that the software pooled competing landlords' **nonpublic** data and pushed them toward aligned, higher rents ("algorithmic collusion"). In November 2025, the DOJ announced a proposed settlement with no fines and no admission of wrongdoing. Under its terms, RealPage must: - Stop using competitors' nonpublic data for real-time recommendations; training data must be at least a year old. - Remove features that discouraged rent decreases or aligned prices among competitors. - Stop hyper-local pricing. Several states that had joined the suit were not part of the settlement, and a number of cities, including Philadelphia, have passed their own ordinances restricting rent-setting algorithms. Why this case belongs in a policy analytics course: - A pricing model becomes a public-policy actor once enough of the market relies on it. - The central issue is less the algorithm itself than **the data it is allowed to see**. This is the same question as target leakage, scaled up to an entire market. - Our house-price models ask, "What is this home worth?" RealPage's model asks, "What can this landlord charge?" The tools are similar, but the consequences are very different.