Spatial Statistics in Action

Spatial Statistics in Action: Analyzing Geographic Data

Introduction

Spatial statistics encompass statistical techniques that account for location or heterogeneity

across space. By explicitly modeling spatial dependence and spatial variation, they allow

more powerful analysis of geographic datasets compared to aspatial methods. This report will

explore practical applications of spatial statistical methods through real-world examples.

We will first introduce key spatial concepts and some common spatial datasets used in

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

Foundational Concepts in Spatial Statistics

Two key properties characterize spatial datasets - location and neighbors. Location refers to

geometric coordinates like latitude/longitude defining where each observational unit is

positioned. Neighbors are the surrounding observational units that are assumed to share

proximity.

Proximity can be based on distance bands defining categorical neighbors or continuous

distance decay functions. The nature of neighbors impacts model specification - for example

nearest versus adjacent units. Spatial weights matrices encode neighbor connections.

Spatial dependence postulates that nearby things tend to be more alike than distant things due

to diffusion or common latent factors. This spatial autocorrelation violates independence

Moran's I and Local Indicators of Spatial Association (LISA) help detect dependence

patterns.

Example Spatial Datasets

Common spatial datasets analyzed with these methods include:

- Census data with indicators like income, poverty, education levels available at tract, county

or regional levels

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

or raster formats

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

- Network datasets representing roads, power lines, or hydrological features with linear

attribute values more than would be expected based on spatial randomness. Moran's I statistic

quantifies the amount of similarity between an observation and its neighbors' values relative

to the sample mean.

Values near +1 indicate positive autocorrelation or clustering, around 0 means spatial

randomness, and near -1 signals negative autocorrelation or dispersion. Under the null of

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Identifying naturally occurring spatial clusters enables characterizing common traits of

relationships worth exploring. Incorporating additional variables to "cluster on" enriches the

perspective on spatial patterns.

Spatial Regression

Regression models with spatial covariates help unpack sources of variation including

spatially lagged relationships. Key types are spatial error models addressing spatial

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

relationships with geographically weighted regression (GWR).

study.

Case Study: Analyzing Home Values in Metro County

Our example involves property transaction records for a mid-sized metro area within Metro

sale price (response), square footage, number of bedrooms/baths, lot size, year built, and

neighborhood identifiers (covariates) at the parcel level with x,y coordinates.

Visual inspection of sale prices mapped across the county reveals apparent spatial patterns

that may be worth exploring more rigorously with spatial statistics. Our goal is to build

statistical models to better understand drivers of housing values and their spatial dependence

structure.

We'll evaluate a sequence of Ordinary Least Squares (OLS), spatial error, spatial lag, and

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Model performance will be compared using information criteria and diagnostics.

Moran's I on the residuals (.25, p<.001) confirms significant positive autocorrelation. Non-

residuals pattern hints spatial effects are influencing sale prices beyond measured attributes.

Spatial Error Model

Augmenting OLS with a conditional autoregressive (CAR) error structure accounts for

spatially correlated disturbances. Model fit statistics including log likelihood and AIC/BIC all

significantly improve over the non-spatial OLS. Residual diagnostics now show randomness

as spatial clustering is addressed.

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

Alternatively, we could represent Tobler's law directly through a spatial lag of the dependent

prices. Yet residual diagnostics remain spatially correlated.

A lag model alone cannot fully disentangle spatially correlated errors from externalities. A

Spatial Durbin Model

Augmenting OLS with spatially lagged dependent and independent terms constitutes a spatial

Durbin model addressing diffusion effects. It further improves model fit substantially based

on all criteria. Residual testing now confirms independence.

Crucially, neighbor attributes like lot size and year built grow significant in the spatially

expanded framework. Spatial spillovers transmit comparative neighborhood quality

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

Understanding these spatially localized variations provides deeper strategic insights for urban

dynamics.

Conclusion

This case study showcased practical applications of techniques from the field of spatial

statistics. By modeling the inherent spatial properties in a real-world housing transaction

dataset, deeper insights were gleaned compared to aspatial methods. Key findings included:

- Significant evidence of spatial autocorrelation and clustering in prices beyond structural

attributes alone.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

accounting for the geographic dimension opens new analytical possibilities compared to

considerations are essential for understanding many real-world phenomena. The hope is this

practical example helps convey spatial statistics’ power to reveal hidden patterns from place-

based datasets.

Spatial statistics encompass statistical techniques that account for location or heterogeneity

across space. By explicitly modeling spatial dependence and spatial variation, they allow

more powerful analysis of geographic datasets compared to aspatial methods. This report will

explore practical applications of spatial statistical methods through real-world examples.

We will first introduce key spatial concepts and some common spatial datasets used in

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

geometric coordinates like latitude/longitude defining where each observational unit is

positioned. Neighbors are the surrounding observational units that are assumed to share

proximity.

Proximity can be based on distance bands defining categorical neighbors or continuous

distance decay functions. The nature of neighbors impacts model specification - for example

nearest versus adjacent units. Spatial weights matrices encode neighbor connections.

Spatial dependence postulates that nearby things tend to be more alike than distant things due

to diffusion or common latent factors. This spatial autocorrelation violates independence

assumptions in aspatial statistics.

Tobler's first law of geography states "everything is related to everything else, but near things

autocorrelation structures and allowing for spatially varying relationships. Techniques like

patterns.

Example Spatial Datasets

Common spatial datasets analyzed with these methods include:

- Census data with indicators like income, poverty, education levels available at tract, county

or regional levels

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

or raster formats

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

attribute values more than would be expected based on spatial randomness. Moran's I statistic

quantifies the amount of similarity between an observation and its neighbors' values relative

to the sample mean.

Values near +1 indicate positive autocorrelation or clustering, around 0 means spatial

randomness, and near -1 signals negative autocorrelation or dispersion. Under the null of

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

Cluster analysis aims to group spatially proximate observational units into non-overlapping

minimized. Popular techniques include hierarchical, k-means, and density-based spatial

members to better understand phenomena. Clusters may reveal previously unknown

relationships worth exploring. Incorporating additional variables to "cluster on" enriches the

perspective on spatial patterns.

Spatial Regression

Regression models with spatial covariates help unpack sources of variation including

spatially lagged relationships. Key types are spatial error models addressing spatial

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Our example involves property transaction records for a mid-sized metro area within Metro

sale price (response), square footage, number of bedrooms/baths, lot size, year built, and

neighborhood identifiers (covariates) at the parcel level with x,y coordinates.

Visual inspection of sale prices mapped across the county reveals apparent spatial patterns

that may be worth exploring more rigorously with spatial statistics. Our goal is to build

statistical models to better understand drivers of housing values and their spatial dependence

structure.

We'll evaluate a sequence of Ordinary Least Squares (OLS), spatial error, spatial lag, and

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Fitting a basic OLS model on the untransformed response and covariates establishes a

show clear spatial clustering remain - violating independence assumptions.

Moran's I on the residuals (.25, p<.001) confirms significant positive autocorrelation. Non-

residuals pattern hints spatial effects are influencing sale prices beyond measured attributes.

Spatial Error Model

Augmenting OLS with a conditional autoregressive (CAR) error structure accounts for

spatially correlated disturbances. Model fit statistics including log likelihood and AIC/BIC all

significantly improve over the non-spatial OLS. Residual diagnostics now show randomness

as spatial clustering is addressed.

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

"amenity premium" exists beyond structural attributes.

Augmenting OLS with spatially lagged dependent and independent terms constitutes a spatial

Durbin model addressing diffusion effects. It further improves model fit substantially based

on all criteria. Residual testing now confirms independence.

Crucially, neighbor attributes like lot size and year built grow significant in the spatially

expanded framework. Spatial spillovers transmit comparative neighborhood quality

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

stationarities - relationships are not consistently applicable everywhere.

urban neighborhoods. Meanwhile lot size plays a smaller role in denser areas amenable to

planning, real estate targeting, and more spatially tailored policies given location-dependent

Conclusion

This case study showcased practical applications of techniques from the field of spatial

statistics. By modeling the inherent spatial properties in a real-world housing transaction

dataset, deeper insights were gleaned compared to aspatial methods. Key findings included:

- Significant evidence of spatial autocorrelation and clustering in prices beyond structural

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

randomness, and near -1 signals negative autocorrelation or dispersion. Under the null of

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Spatial Clustering

Cluster analysis aims to group spatially proximate observational units into non-overlapping

minimized. Popular techniques include hierarchical, k-means, and density-based spatial

Identifying naturally occurring spatial clusters enables characterizing common traits of

relationships worth exploring. Incorporating additional variables to "cluster on" enriches the

Regression models with spatial covariates help unpack sources of variation including

spatially lagged relationships. Key types are spatial error models addressing spatial

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Allowing coefficient estimates to vary across space enables exploring spatially non-stationary

relationships with geographically weighted regression (GWR).

Equipped with these concepts and techniques, we are ready to apply them through a case

study.

Visual inspection of sale prices mapped across the county reveals apparent spatial patterns

statistical models to better understand drivers of housing values and their spatial dependence

structure.

We'll evaluate a sequence of Ordinary Least Squares (OLS), spatial error, spatial lag, and

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Model performance will be compared using information criteria and diagnostics.

OLS Regression as Baseline

Fitting a basic OLS model on the untransformed response and covariates establishes a

show clear spatial clustering remain - violating independence assumptions.

spatial OLS fails to capture key spatial dependence and its coefficients may be biased. The

spatially correlated disturbances. Model fit statistics including log likelihood and AIC/BIC all

significantly improve over the non-spatial OLS. Residual diagnostics now show randomness

as spatial clustering is addressed.

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

"amenity premium" exists beyond structural attributes.

Spatial Lag Model

Alternatively, we could represent Tobler's law directly through a spatial lag of the dependent

variable. Again all fit metrics increase notably, confirming the positive role of neighboring

Augmenting OLS with spatially lagged dependent and independent terms constitutes a spatial

on all criteria. Residual testing now confirms independence.

Crucially, neighbor attributes like lot size and year built grow significant in the spatially

expanded framework. Spatial spillovers transmit comparative neighborhood quality

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

Allowing coefficients to vary flexibly across the county reveals important spatial non-

stationarities - relationships are not consistently applicable everywhere.

urban neighborhoods. Meanwhile lot size plays a smaller role in denser areas amenable to

Understanding these spatially localized variations provides deeper strategic insights for urban

dynamics.

Conclusion

This case study showcased practical applications of techniques from the field of spatial

statistics. By modeling the inherent spatial properties in a real-world housing transaction

dataset, deeper insights were gleaned compared to aspatial methods. Key findings included:

- Significant evidence of spatial autocorrelation and clustering in prices beyond structural

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

- Relationships vary considerably depending on neighborhood location, violating stationarity.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

Spatial statistics encompass statistical techniques that account for location or heterogeneity

more powerful analysis of geographic datasets compared to aspatial methods. This report will

explore practical applications of spatial statistical methods through real-world examples.

We will first introduce key spatial concepts and some common spatial datasets used in

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

values in a metropolitan area. Different model specifications are compared to gain deeper

insights from the spatial dimension.

Overall, the aim is to illustrate how spatial statistics can be applied in action to better

understand relationships in geographic data and ultimately support improved decision making

key foundational concepts.

geometric coordinates like latitude/longitude defining where each observational unit is

proximity.

Proximity can be based on distance bands defining categorical neighbors or continuous

distance decay functions. The nature of neighbors impacts model specification - for example

nearest versus adjacent units. Spatial weights matrices encode neighbor connections.

Spatial dependence postulates that nearby things tend to be more alike than distant things due

to diffusion or common latent factors. This spatial autocorrelation violates independence

assumptions in aspatial statistics.

Tobler's first law of geography states "everything is related to everything else, but near things

are more related than distant things." Spatial heterogeneity means relationships can vary

across a study region due to spatially non-stationary covariates.

- Census data with indicators like income, poverty, education levels available at tract, county

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

or raster formats

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

- Network datasets representing roads, power lines, or hydrological features with linear

geometries

- Natural hazard monitoring involving spatially distributed measurements of earthquakes,

wildfires etc.

With location and neighbors defined, we can explore some key analytical techniques.

attribute values more than would be expected based on spatial randomness. Moran's I statistic

to the sample mean.

Values near +1 indicate positive autocorrelation or clustering, around 0 means spatial

randomness, and near -1 signals negative autocorrelation or dispersion. Under the null of

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Spatial Clustering

Cluster analysis aims to group spatially proximate observational units into non-overlapping

Spatial Regression

Regression models with spatial covariates help unpack sources of variation including

spatially lagged relationships. Key types are spatial error models addressing spatial

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Allowing coefficient estimates to vary across space enables exploring spatially non-stationary

relationships with geographically weighted regression (GWR).

Equipped with these concepts and techniques, we are ready to apply them through a case

study.

Case Study: Analyzing Home Values in Metro County

County over the past 5 years. The dataset contains over 25,000 home sales with attributes like

neighborhood identifiers (covariates) at the parcel level with x,y coordinates.

Visual inspection of sale prices mapped across the county reveals apparent spatial patterns

statistical models to better understand drivers of housing values and their spatial dependence

structure.

We'll evaluate a sequence of Ordinary Least Squares (OLS), spatial error, spatial lag, and

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Model performance will be compared using information criteria and diagnostics.

OLS Regression as Baseline

Fitting a basic OLS model on the untransformed response and covariates establishes a

baseline. While explanatory power is moderately high (adjusted R2 = 0.68), residual plots

Augmenting OLS with a conditional autoregressive (CAR) error structure accounts for

significantly improve over the non-spatial OLS. Residual diagnostics now show randomness

as spatial clustering is addressed.

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

"amenity premium" exists beyond structural attributes.

Spatial Lag Model

Alternatively, we could represent Tobler's law directly through a spatial lag of the dependent

variable. Again all fit metrics increase notably, confirming the positive role of neighboring

Durbin model addressing diffusion effects. It further improves model fit substantially based

Crucially, neighbor attributes like lot size and year built grow significant in the spatially

expanded framework. Spatial spillovers transmit comparative neighborhood quality

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

Allowing coefficients to vary flexibly across the county reveals important spatial non-

stationarities - relationships are not consistently applicable everywhere.

For example, square footage has a much stronger positive impact in outerlying versus central

urban neighborhoods. Meanwhile lot size plays a smaller role in denser areas amenable to

This case study showcased practical applications of techniques from the field of spatial

dataset, deeper insights were gleaned compared to aspatial methods. Key findings included:

- Significant evidence of spatial autocorrelation and clustering in prices beyond structural

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

- Relationships vary considerably depending on neighborhood location, violating stationarity.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

consistently improved model performance and diagnostics.

The techniques are easily extensible to myriad other spatial domains. Overall, explicitly

considerations are essential for understanding many real-world phenomena. The hope is this

based datasets.

Spatial statistics encompass statistical techniques that account for location or heterogeneity

more powerful analysis of geographic datasets compared to aspatial methods. This report will

explore practical applications of spatial statistical methods through real-world examples.

We will first introduce key spatial concepts and some common spatial datasets used in

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

values in a metropolitan area. Different model specifications are compared to gain deeper

insights from the spatial dimension.

Overall, the aim is to illustrate how spatial statistics can be applied in action to better

understand relationships in geographic data and ultimately support improved decision making

related to land use, public health, environmental monitoring and more. Let's begin with some

Proximity can be based on distance bands defining categorical neighbors or continuous

nearest versus adjacent units. Spatial weights matrices encode neighbor connections.

Spatial dependence postulates that nearby things tend to be more alike than distant things due

to diffusion or common latent factors. This spatial autocorrelation violates independence

assumptions in aspatial statistics.

Tobler's first law of geography states "everything is related to everything else, but near things

are more related than distant things." Spatial heterogeneity means relationships can vary

across a study region due to spatially non-stationary covariates.

Representing these concepts statistically requires specifying appropriate spatial

Moran's I and Local Indicators of Spatial Association (LISA) help detect dependence

Example Spatial Datasets

Common spatial datasets analyzed with these methods include:

- Census data with indicators like income, poverty, education levels available at tract, county

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

or raster formats

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

- Network datasets representing roads, power lines, or hydrological features with linear

geometries

- Natural hazard monitoring involving spatially distributed measurements of earthquakes,

wildfires etc.

Values near +1 indicate positive autocorrelation or clustering, around 0 means spatial

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Spatial Clustering

Cluster analysis aims to group spatially proximate observational units into non-overlapping

clusters so within-cluster similarities are maximized while between-cluster differences are

clustering of applications with noise (DBSCAN) algorithms.

members to better understand phenomena. Clusters may reveal previously unknown

perspective on spatial patterns.

Spatial Regression

Regression models with spatial covariates help unpack sources of variation including

spatially lagged relationships. Key types are spatial error models addressing spatial

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Allowing coefficient estimates to vary across space enables exploring spatially non-stationary

relationships with geographically weighted regression (GWR).

Equipped with these concepts and techniques, we are ready to apply them through a case

study.

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

- Relationships vary considerably depending on neighborhood location, violating stationarity.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

consistently improved model performance and diagnostics.

The techniques are easily extensible to myriad other spatial domains. Overall, explicitly

accounting for the geographic dimension opens new analytical possibilities compared to

considerations are essential for understanding many real-world phenomena. The hope is this

across space. By explicitly modeling spatial dependence and spatial variation, they allow

explore practical applications of spatial statistical methods through real-world examples.

We will first introduce key spatial concepts and some common spatial datasets used in

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

values in a metropolitan area. Different model specifications are compared to gain deeper

insights from the spatial dimension.

Overall, the aim is to illustrate how spatial statistics can be applied in action to better

understand relationships in geographic data and ultimately support improved decision making

related to land use, public health, environmental monitoring and more. Let's begin with some

key foundational concepts.

Foundational Concepts in Spatial Statistics

to diffusion or common latent factors. This spatial autocorrelation violates independence

assumptions in aspatial statistics.

Tobler's first law of geography states "everything is related to everything else, but near things

are more related than distant things." Spatial heterogeneity means relationships can vary

across a study region due to spatially non-stationary covariates.

Representing these concepts statistically requires specifying appropriate spatial

autocorrelation structures and allowing for spatially varying relationships. Techniques like

Moran's I and Local Indicators of Spatial Association (LISA) help detect dependence

patterns.

Example Spatial Datasets

or regional levels

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

- Network datasets representing roads, power lines, or hydrological features with linear

geometries

- Natural hazard monitoring involving spatially distributed measurements of earthquakes,

wildfires etc.

With location and neighbors defined, we can explore some key analytical techniques.

Spatial Autocorrelation

spatial randomness, Moran's I is asymptotically normally distributed, enabling hypothesis

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Spatial Clustering

Cluster analysis aims to group spatially proximate observational units into non-overlapping

clusters so within-cluster similarities are maximized while between-cluster differences are

minimized. Popular techniques include hierarchical, k-means, and density-based spatial

clustering of applications with noise (DBSCAN) algorithms.

Identifying naturally occurring spatial clusters enables characterizing common traits of

perspective on spatial patterns.

Spatial Regression

Regression models with spatial covariates help unpack sources of variation including

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Allowing coefficient estimates to vary across space enables exploring spatially non-stationary

relationships with geographically weighted regression (GWR).

Equipped with these concepts and techniques, we are ready to apply them through a case

study.

Case Study: Analyzing Home Values in Metro County

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Model performance will be compared using information criteria and diagnostics.

OLS Regression as Baseline

Fitting a basic OLS model on the untransformed response and covariates establishes a

baseline. While explanatory power is moderately high (adjusted R2 = 0.68), residual plots

show clear spatial clustering remain - violating independence assumptions.

Moran's I on the residuals (.25, p<.001) confirms significant positive autocorrelation. Non-

residuals pattern hints spatial effects are influencing sale prices beyond measured attributes.

spatially correlated disturbances. Model fit statistics including log likelihood and AIC/BIC all

as spatial clustering is addressed.

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

"amenity premium" exists beyond structural attributes.

Spatial Lag Model

Alternatively, we could represent Tobler's law directly through a spatial lag of the dependent

variable. Again all fit metrics increase notably, confirming the positive role of neighboring

prices. Yet residual diagnostics remain spatially correlated.

A lag model alone cannot fully disentangle spatially correlated errors from externalities. A

combination of both sources of spatial dependence may better depict the reality.

Crucially, neighbor attributes like lot size and year built grow significant in the spatially

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

Allowing coefficients to vary flexibly across the county reveals important spatial non-

stationarities - relationships are not consistently applicable everywhere.

For example, square footage has a much stronger positive impact in outerlying versus central

urban neighborhoods. Meanwhile lot size plays a smaller role in denser areas amenable to

smaller plots.

planning, real estate targeting, and more spatially tailored policies given location-dependent

Conclusion

This case study showcased practical applications of techniques from the field of spatial

dataset, deeper insights were gleaned compared to aspatial methods. Key findings included:

- Significant evidence of spatial autocorrelation and clustering in prices beyond structural

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

- Relationships vary considerably depending on neighborhood location, violating stationarity.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

consistently improved model performance and diagnostics.

The techniques are easily extensible to myriad other spatial domains. Overall, explicitly

accounting for the geographic dimension opens new analytical possibilities compared to

analysis. Techniques including spatial autocorrelation, clustering, and regression will then be

demonstrated. The bulk of the report focuses on analyzing a case study involving home

values in a metropolitan area. Different model specifications are compared to gain deeper

insights from the spatial dimension.

Overall, the aim is to illustrate how spatial statistics can be applied in action to better

understand relationships in geographic data and ultimately support improved decision making

related to land use, public health, environmental monitoring and more. Let's begin with some

key foundational concepts.

Foundational Concepts in Spatial Statistics

Two key properties characterize spatial datasets - location and neighbors. Location refers to

proximity.

Proximity can be based on distance bands defining categorical neighbors or continuous

nearest versus adjacent units. Spatial weights matrices encode neighbor connections.

Spatial dependence postulates that nearby things tend to be more alike than distant things due

to diffusion or common latent factors. This spatial autocorrelation violates independence

assumptions in aspatial statistics.

Tobler's first law of geography states "everything is related to everything else, but near things

are more related than distant things." Spatial heterogeneity means relationships can vary

across a study region due to spatially non-stationary covariates.

Representing these concepts statistically requires specifying appropriate spatial

autocorrelation structures and allowing for spatially varying relationships. Techniques like

Moran's I and Local Indicators of Spatial Association (LISA) help detect dependence

patterns.

- Remote sensing data on land use/land cover, vegetation indices, pollution levels with pixel

- Points of interest like locations of hospitals, schools, businesses geocoded with addresses

- Network datasets representing roads, power lines, or hydrological features with linear

geometries

- Natural hazard monitoring involving spatially distributed measurements of earthquakes,

wildfires etc.

With location and neighbors defined, we can explore some key analytical techniques.

quantifies the amount of similarity between an observation and its neighbors' values relative

randomness, and near -1 signals negative autocorrelation or dispersion. Under the null of

tests.

Visualizing I using LISA cluster maps reveals localized hot and cold spots to identify

significant spatial associations not apparent from a global indicator alone. Together these

help uncover hidden patterns to inform modeling or hypothesis generation.

Spatial Clustering

Cluster analysis aims to group spatially proximate observational units into non-overlapping

clusters so within-cluster similarities are maximized while between-cluster differences are

minimized. Popular techniques include hierarchical, k-means, and density-based spatial

clustering of applications with noise (DBSCAN) algorithms.

Regression models with spatial covariates help unpack sources of variation including

autocorrelation in the residuals and spatial lag models representing influence from

neighboring y-values per Tobler's law.

Combinations of these account for both spatial externalities and latent spatial effects.

Allowing coefficient estimates to vary across space enables exploring spatially non-stationary

relationships with geographically weighted regression (GWR).

Equipped with these concepts and techniques, we are ready to apply them through a case

study.

Case Study: Analyzing Home Values in Metro County

Our example involves property transaction records for a mid-sized metro area within Metro

neighborhood identifiers (covariates) at the parcel level with x,y coordinates.

Visual inspection of sale prices mapped across the county reveals apparent spatial patterns

statistical models to better understand drivers of housing values and their spatial dependence

We'll evaluate a sequence of Ordinary Least Squares (OLS), spatial error, spatial lag, and

geographically weighted regression specifications to see which provides the best fit and

insights into spatial relationships. All models control for structural attributes and year effects.

Model performance will be compared using information criteria and diagnostics.

OLS Regression as Baseline

Fitting a basic OLS model on the untransformed response and covariates establishes a

baseline. While explanatory power is moderately high (adjusted R2 = 0.68), residual plots

show clear spatial clustering remain - violating independence assumptions.

significantly improve over the non-spatial OLS. Residual diagnostics now show randomness

Error dependence within a given neighborhood boosts confidence nearby transactions share

common unobserved traits transmitted through proximity. This suggests a spatially varying

"amenity premium" exists beyond structural attributes.

Spatial Lag Model

Alternatively, we could represent Tobler's law directly through a spatial lag of the dependent

variable. Again all fit metrics increase notably, confirming the positive role of neighboring

prices. Yet residual diagnostics remain spatially correlated.

combination of both sources of spatial dependence may better depict the reality.

Durbin model addressing diffusion effects. It further improves model fit substantially based

expanded framework. Spatial spillovers transmit comparative neighborhood quality

throughout contiguous regions. A more complete understanding emerges.

Geographically Weighted Regression

Allowing coefficients to vary flexibly across the county reveals important spatial non-

stationarities - relationships are not consistently applicable everywhere.

For example, square footage has a much stronger positive impact in outerlying versus central

urban neighborhoods. Meanwhile lot size plays a smaller role in denser areas amenable to

smaller plots.

attributes alone.

- Spatially correlated errors and spillover effects both influence values through proximity.

- Relationships vary considerably depending on neighborhood location, violating stationarity.

- Sequentially expanding frameworks to represent spatial dependence and heterogeneity

consistently improved model performance and diagnostics.

The techniques are easily extensible to myriad other spatial domains. Overall, explicitly

accounting for the geographic dimension opens new analytical possibilities compared to

considerations are essential for understanding many real-world phenomena. The hope is this