Spatial Regression Flashcards

Labs and Course Overview

Labs are considered comprehensive sources of information. There's a mention of switching labs to Sweden and a potential bonus for the class. The lecturer acknowledges regular attendees and confirms a test.

Lecture Overview and Expectations

Today's lecture is described as the most challenging of the course, with direct relevance to upcoming labs. Students are encouraged to ask questions during and after the lecture to ensure understanding. The content will become less intense later in the semester. The lecture and lab materials are made available in advance, though slight adjustments may occur.

Spatial Regression Introduction

The lecture will cover spatial regression, briefly mentioned in a previous lecture. Topics include:

• Refresher on regression

• Basics of spatial regression: definition, function, types, and selection criteria

• Local Morans I: introduction and lab application

• ArcGIS capabilities comparable to GeoDa, specifically regression.

Regression Fundamentals

Regression involves explaining a phenomenon using explanatory variables. For example, understanding variations in asthma cases based on various factors. Variable selection can be based on additional information or experimentation.

An example equation is given:

$Crime = 68.61 - 159 <br>ewline Income - 0.27 <br>ewline House Value$

This suggests that income has a greater negative impact on crime rates compared to house value.

Limitations of Ordinary Least Squares (OLS) Regression

OLS regression assumes no spatial interaction or dependence in the data, which contradicts real-world observations where spatial correlation is common.

Examples of spatial correlation:

• Flu outbreaks

• Localized cancer clusters potentially linked to environmental factors like nuclear leaks or pollution

• Plant species distribution

• Chemical spills affecting neighboring areas

The Problem with Ignoring Spatial Dependencies

Ignoring spatial dependencies can lead to:

• Inaccurate models

• Overestimation of variable impacts

• Inflated R-squared values due to overcounted data from spatially correlated neighboring areas

• Weakened model performance and usability

Spatial Regression as an Alternative

Spatial regression addresses the limitations of OLS by:

• Checking the relationship between a variable of interest (e.g., crime) and its determinants.

• Accounting for local clustering and spatial dependency.

• Improving model accuracy by including spatial dependency in the model.

Assumptions of Regression

Key regression assumptions include:

• Linearity

• No outliers (or careful consideration of outliers)

• Stationarity (consistent relationships across different areas)

• No multicollinearity (independent variables should explain unique variance)

• Normally distributed errors (residuals forming a bell curve without skewness)

• No spatially correlated residuals

Residuals and Spatial Correlation

The residual or error in regression is the difference between predicted and observed values. For example, if a model predicts 10 crimes in an area but only 5 are observed, the model overestimates by 5.

It's problematic when these over- and under-predictions cluster together spatially. Residuals or errors should ideally be randomly distributed.

Consequences of Violated Assumptions

Violating OLS assumptions, particularly spatial correlation, can lead to:

• Incorrect coefficients

• Exaggerated R-squared values

• Invalid assumptions and inferences

• Potential issues in professional settings where regression models inform decisions (e.g., toll road construction, environmental policies).

Importance of Understanding Spatial Correlation

It's crucial to understand spatial correlation to avoid:

• Creating inaccurate models that can lead to flawed decisions.

• Misleading statistics, whether intentional or unintentional.

• Being unaware of how data manipulation or exclusion of participants can affect results.

Detecting Spatial Dependence in OLS

To determine if data is suitable for OLS, one must:

• Use software like GeoDa capable of diagnosing violations of OLS assumptions.

• Define neighboring areas using a weights table.

• Examine OLS diagnostics for spatial dependence.

Using GeoDa for OLS and Spatial Regression

In GeoDa, the process involves:

1. Inputting data and specifying the dependent and independent variables.

2. Defining the weights table (e.g., rook, queen contiguity) to inform the software about neighboring areas.

3. Running OLS regression and examining the Lagrange Multiplier test results.

4. Using the Lagrange Multiplier test to decide between a spatial error model or a spatial lag model.

Software Options: GeoDa and ArcGIS

• GeoDa: Can perform OLS, spatial lag, and spatial error regression.

• ArcGIS: Offers OLS and Geographically Weighted Regression (GWR) as an alternative to spatial regression.

Lagrange Multiplier Test and Model Selection

The Lagrange multiplier results guide the selection of the appropriate spatial regression model:

• Spatial Error Model: Used when there is spatial correlation in the residuals.

• Spatial Lag Model: Used when there is spatial correlation in the dependent variable.

Understanding Spatial Lag

Spatial lag refers to the value of the neighbors. For example, spatially lagged crime is the crime rate in the surrounding neighboring areas.

Spatial Error Regression Explained

Spatial error regression accounts for error terms that are correlated across neighboring areas. For instance, if areas $I$ and $J$ are neighbors, and a standard regression model over-predicts crime in both areas, spatial error regression adjusts for this spatial pattern in the errors.

Spatial Lag Regression Explained

Spatial lag regression incorporates the dependent variable values from neighboring areas as an additional predictor. In essence, it examines how including information from neighboring areas improves the model fit beyond what OLS can achieve with only independent variables.

Decision Process for Model Selection

The decision process involves assessing the significance of Lagrange Multiplier diagnostics:

1. Look at the Lagrange Multiplier (LM) test results.

2. If there is spatial correlation in the residuals, consider the Spatial Error Model.

3. If there is spatial correlation in the dependent variable, consider the Spatial Lag Model.

4. If both are significant, use the robust versions of the LM tests to decide between the Spatial Error or Spatial Lag Model.

   

Example: House Ownership in Manhattan

An example is presented using Manhattan census data to model house ownership. The goal is to identify factors that contribute to people buying a house.

Data and Variables

The dataset includes variables such as:

• Median income

• Percentage of Asian persons

• House ownership (dependent variable)

• Percentage of non-Hispanic whites

• Percentage of non-Hispanic blacks

• Education levels

• Unemployment rates

• Poverty rates

• Percentage of foreign-born individuals

Implementing OLS in GeoDa

The steps include:

1. Creating a weights table to define neighbors.

2. Running OLS regression in GeoDa.

3. Specifying house ownership as the dependent variable and the other listed variables as independent variables.

Interpreting OLS Results

Key outputs to examine include:

• R-squared and Adjusted R-squared: Adjusted R-squared is more trustworthy as it penalizes the use of more variables. It explains the proportion of variance in house ownership accounted for by the model. (higher number the better)

• Akaike Information Criterion (AIC): Used to compare models. Lower AIC indicates a better model. (lower number better)

• Sigma Square- the smaller the better

• log likelihood- the larger (or less negative) values the better.

Multicollinearity Check

The multicollinearity condition number should be below 20. If it's higher, it indicates that two or more explanatory variables may be explaining the same variance, and one should be removed.

Lagrange Multiplier Diagnostics

The Lagrange Multiplier (LM) diagnostics are used to detect spatial dependence. A significant Moran's I for error indicates spatial correlation in the residuals.

The LM diagnostics help determine whether to use a spatial lag or spatial error model:

• Spatial Lag: Models dependency in the dependent variable.

• Spatial Error: Models dependency in the error terms.

Comparing Models: OLS, Spatial Lag, and Spatial Error

In the example:

• OLS showed an adjusted R-squared of 0.48.

• Spatial Lag increased the R-squared to 0.526 and reduced the AIC.

• Spatial Error further increased the R-squared to 0.53 and reduced the AIC further.

However, the likelihood ratio test may indicate remaining spatial dependency even after running a spatial regression model.

Mathematical Representation: Spatial Lag and Error Models

Spatial Lag Model includes a weighted ownership variable:

$Y = \rho WY + X\beta + \epsilon$

where $<br>Y$ is the dependent variable, $<br>W$ is the spatial weights matrix, $<br>X$ represents the independent variables, and \[Beta] are their coefficients.

Spatial Error Model includes \
ewline Lambda, which accounts for spatially correlated errors:

$Y = X\beta + \lambda W\epsilon + \nu$

Flow Chart Decision-Making

The flow chart guides the selection of the appropriate model based on the significance of the LM diagnostics and their robust versions.

Issues and Limitations of the Analysis

• The R-squared might not be great due to the choice of variables or unobservable factors.

• Despite the limitations, there is an improvement of R-squared with spatial regression compared to OLS.

• Sigma square represents the variance of the error. Smaller values are better.

• Likelihood: Larger or less negative values are better.

Local Indicators of Spatial Association (LISA)

LISA is used to identify clusters of high and low values. It includes:

• High-High: Neighborhoods with high values surrounded by high values.

• Low-Low: Areas with low values surrounded by low values.

• High-Low: Areas with high values surrounded by low values.

  

Using LISA in GeoDa

LISA can be run in GeoDa to produce maps showing:

• Areas of clustering (high-high, low-low).

• P-values for these clusters, indicating their significance.

  

Benefits of LISA

LISA maps help visualize where spatial clusters occur and their statistical significance.

Transition to ArcGIS

Next section transitions to using ArcGIS for regression analysis.

Emergency Calls in a US Shot

An example uses emergency call data in a U.S. area. Each point represents a call. The data can be projected on a map to visualize call density by area.

Using ArcGIS Tools

ArcGIS offers tools like scatter plots to explore relationships between variables.

OLS Regression in ArcGIS

Running OLS in ArcGIS involves:

1. Selecting the dependent variable (number of emergency calls).

2. Choosing explanatory variables (population, number of jobs, low education, distance to urban center).

Interpreting ArcGIS Results

Key outputs include:

• Variables and Coefficients: Indicates the relationship between each variable and number of emergency calls.

• Probabilities: Indicates the significance of each variable.

• R-squared: Indicates the proportion of variance in emergency calls explained by the model.

Spatial Correlation Tool in ArcGIS

ArcGIS provides a warning to use the spatial correlation tool to check for spatial autocorrelation in the residuals. Unlike GeoDa, this step is not automated. The Moran's I tool is used on the residuals map.

Geographically Weighted Regression (GWR)

GWR creates a separate regression equation for each area, showing local model performance.

Benefits of GWR

GWR can improve model fit and provide localized insights, even when OLS does not show significant spatial autocorrelation.

The AIC will decrease and the R-squared becomes better.

Interpreting GWR Results

The software can:

• Print local R-squared values, showing model performance per area.

• Plot residuals and run Moran's I on them.

• Print individual coefficient changes over the area, i.e can identify a area where the job variable are actually high and important for the models.

Advantages of ArcGIS

ArcGIS allows the cooperation of independent variable coefficients into model builders.

Chicago Unemployment Lab

The lab focuses on identifying factors contributing to unemployment in Chicago.

Lab Activities

The lab involves:

• Selecting variables from census data to explain unemployment.

• Running regression in GeoDa.

• Interpreting and comparing results.

• Running LISA maps on the residuals.

• Creating a custom model with selected variables.

Running Error or Lag: How-to

The Error or Lag models are possible to create in the regression panel.

Overall Summary

The lecture provides a comprehensive overview of regression, spatial regression, and their implementation in GeoDa and ArcGIS.