1. Why Geography is important.
Spatial Data Analysis
Why Geography is important.

2. What is spatial analysis?
From Data to Information
beyond mapping: added value
transformations, manipulations and application of analytical methods to spatial (geographic) data
Lack of locational invariance
analyses where the outcome changes when the locations of the objects under study changes
» median center, clusters, spatial autocorrelation
where matters
In an absolute sense (coordinates)
In a relative sense (spatial arrangement, distance)

3. Components of Spatial Analysis
Visualization
Showing interesting patterns
Exploratory Spatial Data Analysis (ESDA)
Finding interesting patterns
Spatial Modeling, Regression
Explaining interesting patterns

4. Implementation of Spatial Analysis
Beyond GIS
Analytical functionality not part of typical commercial GIS
» Analytical extensions
Exploration requires interactive approach
» Training requirements
» Software requirements
Spatial modeling requires specialized statistical methods
» Explicit treatment of spatial autocorrelation
» Space-time is not space + time
ESDA and Spatial Econometrics

5. What Is Special About Spatial Data?
Location, Location, Location
“where” matters
Dependence is the rule
spatial interaction, contagion, externalities, spill-overs, copycatting
First Law of Geography (Tobler)
everything depends on everything else, but closer things more so

6. Spatial heterogeneity
Lack of stationarity in first-order statistics
Pertains to the spatial or regional differentiation observed in the value of a variable
Spatial drift (e.g., a trend surface)
Spatial association

7. Nature of Spatial Data Spatially referenced data “georeferenced”
» “attribute” data associated with location
» where matters
Example: Spatial Objects
points: x, y coordinates
» cities, stores, crimes, accidents
lines: arcs, from node, to node
» road network, transmission lines
polygons: series of connected arcs
» provinces, cities, census tracts

8. GIS Data Model
Discretization of geographical reality necessitated by the nature of computing devices (Goodchild)
raster (grid) vs. vector (polygon)
field view (regions, segments) vs. object view (objects in a plane)
Data model implies spatial sampling and spatial errors

9. 3 Classes of Spatial Data
Geostatistical Data
points as sample locations (“field” data as opposed to “objects”)
Continuous variation over space
Lattice/Regional Data
polygons or points (centroids)
Discrete variation over space, observations associated with regular or irregular areal units

10. Point Patterns
points on a map (occurrences of events at locations in space)
Observations of a variable are made at location X
Assumption that the spatial arrangement is directly related to the interaction between units of observation

12. Visualization and ESDA
Objective
highlighting and detecting pattern
Visualization
mapping spatial distributions
outlier detection
smoothing rates
ESDA
dynamically linked windows
linking and brushing

13. Mapping patterns

14. ESDA

15. Spatial Process Spatial Random Field { Z(s): s ∈ D } Examples
» s ∈ Rd : generic data location (vector of coordinates)
» D ⊂ Rd : index set
(subset of potential locations)
» Z(s) random variable at s, with realization z(s)
Examples
s are x, y coordinates of house sales, Z sales price at s
s are counties, Z is crime rate in s

16. Point Pattern Analysis
Objective
assessing spatial randomness
Interest in location itself
complete spatial randomness
clustering, dispersion
Distance-based statistics
nearest neighbors
number of events within given radius

17. Point Patterns Spatial process Data Research question
index set D is point process, s is random
Data
mapped pattern
» examples: location of disease, gang shootings
Research question
interest focuses on detecting absence of spatial randomness (cluster statistics)
clustered points vs dispersed points

19. Geostatistical Data Spatial Process Data Research Question
index set D is fixed subset of Rd (continuous)
Data
sample points from underlying continuous surface
» examples: mining, air quality, house sales price
Research Question
interest focuses on modeling continuous spatial variation
spatial interpolation (kriging)

20. Variogram Modeling (Geostatistics)
Objective
modeling continuous variation across space
Variogram
estimating how spatial dependence varies with distance
modeling distance decay
Kriging
optimal spatial prediction

23. Lattice or Regional Data
Spatial process
index set D is fixed collection of countably many points in Rd
finite, discrete spatial units
Data
fixed points or discrete locations (regions)
» examples: county tax rates, state unemployment
Research question
interest focuses on statistical inference
estimation, specification tests

24. Spatial Autocorrelation
Objective
hypothesis test on spatial randomness of attributes = value and location
Global and local autocorrelation statistics: Moran’s I, Geary’s c, G(d), LISA
Visualization of spatial autocorrelation
Moran scatterplot
LISA maps

27. Spatial process models
How is the spatial association generated?
Spatial autoregressive process (SAR)
Y = ρWY + ε
Spatial moving average process (SMA)
Y = (I + ρW) ε
ε – vector of independent errors
W = distance weights matrix
In SAR, correlation is fairly persistent with increasing distance, whereas with SMA is decays to zero fairly quickly.

28. Spatial process—the rule governing the trajectory of the system as a chain of changes in state.
Spatial pattern—the map of a single realization of the underlying spatial process (the data available for analysis).
Say you conduct a regression analysis. If the residuals do not display spatial autocorrelation, then there is no need to add “space” to the model. Examine s.a. in the residuals using Moran’s I or Geary’s c or G(d).

29. Perspectives on spatial process models
Finding out how the variable Y relates to its value in surrounding locations (the spatial lag) while controlling for the influence of other explanatory variables.
When the interest is in the relation between the explanatory variables X and the dependent variable, after the spatial effect has been controlled for (this is referred to as spatial filtering or spatial screening).

30. The expected value of the dependent variable at each location is a function not only of explanatory variables at that location, but of the explanatory variables at all other locations as well.