Week 7: Spatial Errors and House Price Prediction

Spatial features, residual maps, Moran’s I, permutation tests, and honest house-price prediction

Published

October 16, 2026

What We Did

This week focused on where a prediction model makes errors, rather than only how large its overall error is. We moved from census-tract analysis to individual house sales and used Philadelphia housing data to explore spatial features and residual patterns.

Prepared housing data: filtered invalid or unsuitable records and constructed predictors such as living area, bathrooms, and building age.
Created spatial features: counted crime incidents within a 660-foot buffer around each house and calculated the mean distance to the five nearest incidents.
Fitted a baseline regression: predicted sale prices using housing characteristics and a spatial crime feature.
Mapped residuals: examined where the model systematically underestimated or overestimated prices.
Measured spatial autocorrelation: defined neighboring houses, calculated their average residuals, and used Moran’s I to summarize residual clustering.
Performed a permutation test: shuffled residuals across fixed locations to assess whether the observed spatial association was unusually strong.
Introduced the Chicago house-price prediction task: use training-set cross-validation to compare models, construct spatial features, and submit predictions for a separate test set.

Key Concepts

1. Residuals

A residual is the actual value minus the predicted value:

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

Positive residual: the model underpredicts.
Negative residual: the model overpredicts.

For example, if a house sells for $240,000 and the model predicts $285,000, the residual is −$45,000. The model overpredicts by $45,000.

2. Buffer Counts and kNN Distances

Buffer counts and kNN distances describe nearby conditions in different ways.

Feature	What is fixed?	What is measured?
Buffer count	Search radius	Number of events inside the radius
Mean kNN distance	Number of nearby events	Average distance to those events

Two houses may both have zero crimes within a buffer but differ substantially in their distance to the nearest crimes. The kNN feature captures that difference.

Use an appropriate projected CRS and check its units. 660 feet equals 1/8 mile.

3. Spatial Lag and Spatial Weights

A spatial lag summarizes values at neighboring locations. With equally weighted, row-standardized neighbors, it is the average of their values.

For four neighbors with residuals of +40, +20, −10, and −30 thousand dollars:

\[ \text{Spatial lag} = \frac{40 + 20 - 10 - 30}{4} = 5 \]

The average neighboring residual is +$5,000.

Spatial lag refers to a spatial relationship, not a previous time period. Results depend on the neighbor definition and weighting method.

In this lesson, kNN serves two purposes:

Finding nearby crime incidents to construct a predictor.
Finding nearby houses to diagnose residual patterns.

4. Moran’s I

Moran’s I measures spatial association. Applied to residuals, it asks whether neighboring houses tend to have similar deviations from the mean residual.

Above the random baseline: similar residuals tend to cluster.
Near the random baseline: little overall spatial association under the chosen weights.
Below the random baseline: neighboring residuals tend to differ.

Residual clustering can suggest omitted local information, an unsuitable model form, or data problems. It does not establish a specific cause.

Moran’s I is neither prediction error in dollars nor the percentage of variation explained. Comparisons require consistent samples and spatial weights.

5. Permutation Testing

A permutation test evaluates whether the observed spatial association is unusual under random assignment of values to locations.

The procedure is:

Keep locations and spatial weights fixed.
Randomly reassign the observed residuals across locations.
Recalculate Moran’s I.
Repeat and compare the observed statistic with the simulated distribution.

With nsim = 499, the function performs 499 permutations. It does not refit the regression 499 times.

A small upper-tail p-value indicates that the observed Moran’s I is unusually large under the specified randomization null. It is not the probability that the null hypothesis is true.

6. Cross-Validation and RMSE

Cross-validation (CV) evaluates predictions on observations temporarily excluded from model fitting.

In ten-fold CV:

Divide the training data into ten folds.
Train on nine folds and predict the remaining fold.
Repeat until every fold has been held out once.
Summarize the held-out prediction errors.

CV RMSE measures prediction-error magnitude in the outcome’s units. Moran’s I measures the spatial structure of errors.

A model can have little spatial clustering while still making large prediction errors. Compare models using the same observations and CV folds.

7. Target Leakage

Target leakage occurs when held-out outcome information enters model training or feature construction.

For example, calculating neighboring mean sale prices from all observed prices before splitting CV folds can allow validation outcomes to enter features.

For this task, construct spatial predictors from available environmental information, such as crime exposure or transit access, rather than neighboring target prices.

Neighboring residual averages contain outcome information. They are useful for diagnosis but should not be reused as ordinary prediction features.

8. Improving and Evaluating the Model

Spatial error patterns can motivate additional features:

Suspected issue	Possible model change
Missing local amenities or disamenities	Add accessibility, distance, or buffer features
Overall differences between neighborhoods	Add neighborhood fixed effects
Relationships that vary across places	Add interaction terms
Nonlinear relationships	Add log transformations or polynomial terms

Evaluate changes using CV RMSE, residual maps, and Moran’s I. A lower training error or smaller Moran’s I alone does not establish better prediction.

Select the model using training-set CV, refit it on the available training data, and predict every required test observation. The separate test set provides the final independent evaluation.

--- title: "Week 7: Spatial Errors and House Price Prediction" description: "Spatial features, residual maps, Moran's I, permutation tests, and honest house-price prediction" date: 2026-10-16 --- ## What We Did This week focused on where a prediction model makes errors, rather than only how large its overall error is. We moved from census-tract analysis to individual house sales and used Philadelphia housing data to explore spatial features and residual patterns. - Prepared housing data: filtered invalid or unsuitable records and constructed predictors such as living area, bathrooms, and building age. - Created spatial features: counted crime incidents within a 660-foot buffer around each house and calculated the mean distance to the five nearest incidents. - Fitted a baseline regression: predicted sale prices using housing characteristics and a spatial crime feature. - Mapped residuals: examined where the model systematically underestimated or overestimated prices. - Measured spatial autocorrelation: defined neighboring houses, calculated their average residuals, and used Moran’s I to summarize residual clustering. - Performed a permutation test: shuffled residuals across fixed locations to assess whether the observed spatial association was unusually strong. - Introduced the Chicago house-price prediction task: use training-set cross-validation to compare models, construct spatial features, and submit predictions for a separate test set. ## Key Concepts ### 1. Residuals A residual is the actual value minus the predicted value: $$ e_i = y_i - \hat{y}_i $$ - **Positive residual:** the model underpredicts. - **Negative residual:** the model overpredicts. For example, if a house sells for $240,000 and the model predicts $285,000, the residual is **−$45,000**. The model overpredicts by $45,000. ### 2. Buffer Counts and kNN Distances Buffer counts and kNN distances describe nearby conditions in different ways. | Feature | What is fixed? | What is measured? | |---|---|---| | Buffer count | Search radius | Number of events inside the radius | | Mean kNN distance | Number of nearby events | Average distance to those events | Two houses may both have zero crimes within a buffer but differ substantially in their distance to the nearest crimes. The kNN feature captures that difference. Use an appropriate projected CRS and check its units. **660 feet equals 1/8 mile.** ### 3. Spatial Lag and Spatial Weights A spatial lag summarizes values at neighboring locations. With equally weighted, row-standardized neighbors, it is the average of their values. For four neighbors with residuals of +40, +20, −10, and −30 thousand dollars: $$ \text{Spatial lag} = \frac{40 + 20 - 10 - 30}{4} = 5 $$ The average neighboring residual is **+$5,000**. Spatial lag refers to a spatial relationship, not a previous time period. Results depend on the neighbor definition and weighting method. In this lesson, kNN serves two purposes: - Finding nearby crime incidents to construct a predictor. - Finding nearby houses to diagnose residual patterns. ### 4. Moran’s I Moran’s I measures spatial association. Applied to residuals, it asks whether neighboring houses tend to have similar deviations from the mean residual. - **Above the random baseline:** similar residuals tend to cluster. - **Near the random baseline:** little overall spatial association under the chosen weights. - **Below the random baseline:** neighboring residuals tend to differ. Residual clustering can suggest omitted local information, an unsuitable model form, or data problems. It does not establish a specific cause. Moran’s I is neither prediction error in dollars nor the percentage of variation explained. Comparisons require consistent samples and spatial weights. ### 5. Permutation Testing A permutation test evaluates whether the observed spatial association is unusual under random assignment of values to locations. The procedure is: 1. Keep locations and spatial weights fixed. 2. Randomly reassign the observed residuals across locations. 3. Recalculate Moran’s I. 4. Repeat and compare the observed statistic with the simulated distribution. With `nsim = 499`, the function performs 499 permutations. **It does not refit the regression 499 times.** A small upper-tail p-value indicates that the observed Moran’s I is unusually large under the specified randomization null. It is not the probability that the null hypothesis is true. ### 6. Cross-Validation and RMSE Cross-validation (CV) evaluates predictions on observations temporarily excluded from model fitting. In ten-fold CV: 1. Divide the training data into ten folds. 2. Train on nine folds and predict the remaining fold. 3. Repeat until every fold has been held out once. 4. Summarize the held-out prediction errors. **CV RMSE measures prediction-error magnitude in the outcome’s units.** Moran’s I measures the spatial structure of errors. A model can have little spatial clustering while still making large prediction errors. Compare models using the same observations and CV folds. ### 7. Target Leakage Target leakage occurs when held-out outcome information enters model training or feature construction. For example, calculating neighboring mean sale prices from all observed prices before splitting CV folds can allow validation outcomes to enter features. For this task, construct spatial predictors from available environmental information, such as crime exposure or transit access, rather than neighboring target prices. Neighboring residual averages contain outcome information. They are useful for diagnosis but should not be reused as ordinary prediction features. ### 8. Improving and Evaluating the Model Spatial error patterns can motivate additional features: | Suspected issue | Possible model change | |---|---| | Missing local amenities or disamenities | Add accessibility, distance, or buffer features | | Overall differences between neighborhoods | Add neighborhood fixed effects | | Relationships that vary across places | Add interaction terms | | Nonlinear relationships | Add log transformations or polynomial terms | Evaluate changes using CV RMSE, residual maps, and Moran’s I. A lower training error or smaller Moran’s I alone does not establish better prediction. Select the model using training-set CV, refit it on the available training data, and predict every required test observation. The separate test set provides the final independent evaluation.