This document discusses autoregressive processes in space, specifically conditional autoregressive (CAR) and simultaneous autoregressive (SAR) models. It introduces CAR and SAR models, noting they make assumptions about spatial homoscedasticity. Brook's lemma is discussed as it relates to the relationship between CAR and SAR models. Examples are provided and references listed, with the document appearing to be from a university department discussing a student's work on spatial autoregressive processes.

1. Department of Electrical Engineering
University of Arkansas
Autoregressive Processes in Space
Md Abul Hayat
mahayat@uark.edu
Feb 28, 2020

2. Contents
• Autoregressive Processes: Time
• Autoregressive Processes: Space
– Conditional Autoregressive (CAR) Models
– Simultaneous Autoregressive (SAR) Models
• Relationship Between CAR & SAR Models
• Estimating CAR Model Parameters
• Examples

3. Autoregressive Processes: Time
• AR(p) Model
– Formulation 1
– Formulation 2
– Compare
– YES

4. Autoregressive Processes: Space
• Are they the same?

5. Simultaneous Autoregressive (SAR) Models
Assumption: Homoscedasticity

6. Brook’s Lemma
• Statement

7. Brook’s Lemma

8. Conditional Autoregressive (CAR) Models
• Formulation
• Properties
• Besag (1974)

9. Simultaneous Autoregressive (SAR) Models
• Formulation
• Properties
• Whittle (1954)

10. Extension of Ideas

11. Relationship Between CAR & SAR Models

12. Relationship Between CAR & SAR Models

13. Example

14. Example

15. Example

16. Example

17. References
– Hierarchical Modeling and Analysis for Spatial Data - Sudipto Banerjee
– On the relationship between CAR and SAR models (2018)
– Spatio-Temporal Reconstruction of Remote Sensing Observations
– CARBayes: An R Package for Bayesian Spatial Modeling
– STATS 253: Analysis of Spatial and Temporal Data (Stanford)

18. Questions?
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